url
string | fetch_time
int64 | content_mime_type
string | warc_filename
string | warc_record_offset
int32 | warc_record_length
int32 | text
string | token_count
int32 | char_count
int32 | metadata
string | score
float64 | int_score
int64 | crawl
string | snapshot_type
string | language
string | language_score
float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
https://www.physicsforums.com/threads/solving-one-tough-integral-elliptical-integrals-and-inverse-jacobian-functions.213058/
| 1,720,933,680,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-30/segments/1720763514548.45/warc/CC-MAIN-20240714032952-20240714062952-00205.warc.gz
| 833,230,373 | 17,859 |
# Solving One Tough Integral: Elliptical Integrals and Inverse Jacobian Functions
• pocaracas
In summary, the conversation revolves around a mathematical integral with conditions involving a variable 'n', and limitations on 'z' and 'r'. The integral has been solved analytically using hypergeometric functions and the result is zero. However, there is confusion about the conditions under which the integral is zero and if it can be solved using elementary functions. The conversation also includes a suggestion for further resources on the topic.
pocaracas
Hi,
I've been dealing with a really mind buster, at least, for me.
Here it is
$$I_n(r,z)=\int_z^r\frac{p^n}{\sqrt{r^2-p^2}\sqrt{p^2-z^2}}\,dp$$
where $$n$$ is an integer and $$0<z<r<1$$.
Mathematica tells me that the result is zero, but I'd like to know how to get there.
I've thought about elliptical integrals but Abramowitz doesn't help much beyond telling me that the Equivalent Inverse Jacobian Elliptic Function of
$$a\int_b^x\frac{dt}{\sqrt{(a^2-t^2)(t^2-b^2)}}$$
is
$$nd^{-1}\left(\frac{x}{b}|\frac{a^2-b^2}{a^2}\right)$$
and
$$a\int_x^a\frac{dt}{\sqrt{(a^2-t^2)(t^2-b^2)}}$$
is
$$dn^{-1}\left(\frac{x}{a}|\frac{a^2-b^2}{a^2}\right)$$
Whatever this means. I know nothing about these functions. nd and dn?
Any help?
It says the result is zero for any choice of n, r and z? That's certainly not true! Just pick some values and plot the integrand, e.g. n = 3, r = 2, z = 1. You'll see that in this example from 1 to 2 the integrand is only positive, and hence the integral can't be zero.
I'm assuming then that I misinterpreted the question. I doubt I can help you beyond what I pointed out there above, but reclarifying the question might help others see what your problem is.
Last edited:
Mute, look at the inequality condition on z and r.
I'm not sure if that's do able in terms of elementary functions, especially since it's the inverse of an elliptic function. Must it be solved analytically?
Gib Z said:
Mute, look at the inequality condition on z and r.
Doesn't really matter. r = 0.9 and z = 0.5 still gives a nonzero result when graphed, which is what I was getting at. We need to be given more information about what the problem is, because it's not clear under what conditions the integral is zero.
I'm not sure if that's do able in terms of elementary functions, especially since it's the inverse of an elliptic function. Must it be solved analytically?
I remember the book on Solitons by Drazin and Johnson had some discussion of these functions since some of them were solutions to the KdV equation, but I can't remember to what extent the functions were discussed (since it is primarily a book about the KdV equations and Solitons(, so the book might not be so usefull for this purposes. At any rate, the book at Amazon is
https://www.amazon.com/dp/0521336554/?tag=pfamazon01-20
Result of the integral
Hi,
Solving the integral as an indefinite integral results in a nice little function involving hypergeometric functions. Please find the result in the attached GIF file.
I guess it'll be easy now for everyone to evaluate it for all the values of r and z.
Regards
Chandranshu
#### Attachments
• Integral.gif
2.1 KB · Views: 421
Missing information
I forgot to add that the hypergeometric function in the result set is the Appell Hypergeometric function of two variables. See http://mathworld.wolfram.com/AppellHypergeometricFunction.html for more information. Also, I have used 'x' as the variable of integration in place of 'p'. This should be a small inconvenience.
Regards
Chandranshu
Last edited:
## 1. What are elliptical integrals?
Elliptical integrals are mathematical functions used to solve integrals involving elliptical curves, which are curves in the shape of an ellipse. They are commonly used in physics, engineering, and other fields to solve complex mathematical problems.
## 2. How are elliptical integrals different from other types of integrals?
Elliptical integrals are different from other types of integrals because they involve elliptical curves instead of the more common circular or hyperbolic curves. This makes them more complex and difficult to solve, but also allows them to model a wider range of real-world phenomena.
## 3. What are inverse Jacobian functions?
Inverse Jacobian functions are mathematical functions that are used to solve integrals involving inverse transformations of Jacobian matrices. They are closely related to elliptical integrals and are often used in conjunction with them to solve complex integrals.
## 4. Why are elliptical integrals and inverse Jacobian functions considered to be "tough" integrals?
Elliptical integrals and inverse Jacobian functions are considered to be "tough" integrals because they are more complex and difficult to solve compared to other types of integrals. They often require advanced mathematical techniques and extensive calculations to find solutions.
## 5. How are elliptical integrals and inverse Jacobian functions used in real-world applications?
Elliptical integrals and inverse Jacobian functions have a wide range of applications in various fields, such as physics, engineering, and statistics. They are used to model and solve problems related to motion, heat transfer, probability distributions, and more. They allow for more accurate and precise solutions to complex problems.
• Calculus
Replies
4
Views
837
• Calculus
Replies
3
Views
2K
• Calculus
Replies
6
Views
2K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
0
Views
408
• Calculus
Replies
3
Views
1K
• Calculus
Replies
16
Views
3K
• Calculus
Replies
20
Views
2K
• Calculus
Replies
19
Views
3K
• Calculus
Replies
3
Views
2K
| 1,436 | 5,692 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.703125 | 4 |
CC-MAIN-2024-30
|
latest
|
en
| 0.90982 |
http://oeis.org/A134968
| 1,580,122,961,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-05/segments/1579251696046.73/warc/CC-MAIN-20200127081933-20200127111933-00386.warc.gz
| 123,369,185 | 3,978 |
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A134968 Number of convex functions from {1,...,n} to itself. 0
1, 1, 4, 16, 54, 168, 462, 1212, 2937, 6832, 15135, 32430, 66898, 134710, 263466, 504308, 944208, 1736575, 3134832, 5574947, 9760954, 16868418, 28771587, 48513127, 80867486, 133455462, 218041708, 353039664, 566580113, 901958971, 1424480451, 2233367056 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,3 COMMENTS That is, the number of sequences of length n, taking values in {1,...,n} that have nondecreasing first differences (nonnegative second differences). LINKS FORMULA See Mathematica code. EXAMPLE a(3)=16: the 16 sequences are 111, 112, 113, 123, 211, 212, 213, 222, 223, 311, 312, 313, 321, 322, 323 and 333. MATHEMATICA (*P[n, k]=number of ways to partition n into exactly k parts*) P[n_Integer, n_Integer] = 1; P[n_Integer, k_Integer] := P[n, k] = Sum[P[n - k, r], {r, 1, Min[n - k, k]}] (*q[n, k]=number of ways to partition n into k-or-fewer parts*) q[0, 0] = 1; q[n_Integer, 0] = 0; q[n_Integer, k_Integer] := q[n, k] = q[n, k - 1] + P[n, k] a[n_] := Sum[(n - Max[f, r])*P[r, s]*q[f, n - 1 - s], {r, 0, n - 1}, {s, 0, n - 1}, {f, 0, n - 1}] CROSSREFS Sequence in context: A121159 A239032 A254823 * A238419 A267227 A223944 Adjacent sequences: A134965 A134966 A134967 * A134969 A134970 A134971 KEYWORD nonn,nice AUTHOR Jacob A. Siehler, Feb 04 2008, Feb 06 2008 STATUS approved
Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.
Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)
| 688 | 1,893 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.6875 | 4 |
CC-MAIN-2020-05
|
latest
|
en
| 0.578 |
http://slideplayer.com/slide/4645679/
| 1,642,934,681,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-05/segments/1642320304217.55/warc/CC-MAIN-20220123081226-20220123111226-00227.warc.gz
| 57,657,713 | 18,597 |
# 2.4.2 – Parallel, Perpendicular Lines
## Presentation on theme: "2.4.2 – Parallel, Perpendicular Lines"— Presentation transcript:
2.4.2 – Parallel, Perpendicular Lines
We know how to quickly graph certain linear equations
Y = mx + b Slope-intercept Y-intercept as a starting point; slope to find a second point Ax + by = c Standard Form Use the x and y intercepts as the two points
Now, we will compare two lines/equations and their graphs
Two lines may be: 1) Parallel 2) Perpendicular 3) Neither
Parallel Lines Given two lines, they are considered parallel if and only if they have the same slope; m1 = m2 The lines never intersect; run off in the same direction “Air” between them
Example. Tell whether the following lines are parallel or not.
1) y = 3x + 4, y – 3x = 10 2) -4x + y = -6, y = 4x 3) y – x = 1, y = -x
Example. Graph the equation y = 3x + 4
Example. Graph the equation y = 3x + 4. Then, graph a line that is parallel and passes through the point (1, 1).
Example. Graph the equation y = -x + 5
Example. Graph the equation y = -x + 5. Then, graph a line that is parallel and passes through the point (1, 1).
Perpendicular Lines Two lines are perpendicular, if and only if, their slopes are negative reciprocals OR their product is -1 OR m1(m2) = -1 Graphically, two perpendicular lines intersect at a 90 degree angle
Example. Tell whether the following lines are perpendicular, parallel, or neither.
1) y = 3x + 4, y = (-1/3)x – 5 2) y = 4x – 5, y = x 3) y = (x/2) – 9, y + (2/x) = -6 4) y = x – 1, y = -x
Example. Graph the equation y = 3x + 4
Example. Graph the equation y = 3x + 4. Then, graph a line that is perpendicular and passes through the point (1, 1).
Example. Graph the equation y = -x + 5
Example. Graph the equation y = -x + 5. Then, graph a line that is perpendicular and passes through the point (1, 1).
Horizontal and Vertical Lines
Horizontal Lines = lines of the form y = a; must have a “y-intercept” Vertical Lines = lines of the form x = a; must have a “x-intercept” General Rule; must be able to see the line, so cannot draw a line on top of the axis
Example. Graph the equation x = 3.
Example. Graph the equation y = -4.
Assignment Pg. 91 33-49 odd, 50-53, 60, 62, 69, 70
| 672 | 2,227 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.78125 | 5 |
CC-MAIN-2022-05
|
latest
|
en
| 0.872801 |
https://www.jiskha.com/search?query=What+is+a+regression+curve%3F
| 1,627,350,386,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-31/segments/1627046152168.38/warc/CC-MAIN-20210727010203-20210727040203-00104.warc.gz
| 861,329,834 | 14,867 |
# What is a regression curve?
4,467 results
1. ## Statistics
The following linear equation, y = b0 + b1x, is a regression line with y-intercept b0 and slope b1. Define These Terms 1) correlation coefficient 2) Linear regression equation
2. ## Calc
The slope of the tangent line to a curve is given by f'(x)=4x^2 + 7x -9. If the point (0,6) is on the curve, find an equation of the curve.
3. ## Calculus 1
The curve y = |x|/(sqrt(5- x^2)) is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (2, 2)
4. ## Statistics
Is is possible for the regression equation to have none of the actual (observed) data points located on the regression line?
5. ## STATISTICS
A) If a regression experiment has correlation coefficient r= 0.75, what precent of total variation is explained by the regression? B) if 85% of total variation is explained by the regression, and there is an inverse relationship between y and x, then what
Given the curve x^2-xy+y^2=9 A) write a general expression for the slope of the curve. B) find the coordinates of the points on the curve where the tangents are vertical C) at the point (0,3) find the rate of change in the slope of the curve with respect
7. ## Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y. What is the equation of the curve if (2, 1) is a point on the curve?
8. ## calculus
1. Given the curve a. Find an expression for the slope of the curve at any point (x, y) on the curve. b. Write an equation for the line tangent to the curve at the point (2, 1) c. Find the coordinates of all other points on this curve with slope equal to
9. ## pre-cal... compound interest and modeling data
Please help me with these 2 problems! 1. Complete the table for the time t necessary for P dollars to triple if interest is compounded continuously at rate r. r= 2%, t=? r= 4%, t=? r= 6%. t=? r= 8%, t=? r= 10%, t=? r= 12%, t=? 2. Draw a scatter plot of the
10. ## Math
A. If the correlation coefficient is 0.65, what is the sign of the slope of the regression line? B. As the correlation coefficient decreases from -0.97 to -0.99, do the points of the scatter plot move toward the regression line, or away from it?
11. ## Mathematics
The gradient of a curve is defined by dy/dx = 3x^(1/2) - 6 Given the point (9, 2) lies on the curve, find the equation of the curve
12. ## last calc question, i promise!
given the curve x + xy + 2y^2 = 6... a. find an expression for the slope of the curve. i got (-1-y)/(x + 4y) as my answer. b. write an equation for the line tangent to the curve at the point (2,1). i got y = (-1/3)x + (5/3). but i didn't any answer for c!
1. a.) Find an equation for the line perpendicular to the tangent curve y=x^3 - 9x + 5 at the point (3,5) [* for a. the answer that I obtained was y-5 = -1/18 (x-3) ] b.) What is the smallest slope on the curve? At what point on the curve does the curve
14. ## Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x/y. What is the equation of the curve if (4, 1) is a point on the curve? x2 − y2 = 15 x2 + y2 = 15 x + y = 15
15. ## Economics
The demand curve for a monopoly is: 1. the MR curve above the AVC curve. 2.above the MR curve. 3.the MR curve above the horizontal axis. 4. the entire MR curve.
16. ## Statistics
For the regression equation, Y=bX+a, which of the following X,Y points will be on the regression line? a, a,b b, b,a c, 0,a d, 0,b
17. ## calculus
Notice that the curve given by the parametric equations x=25−t^2 y=t^3−16t is symmetric about the x-axis. (If t gives us the point (x,y),then −t will give (x,−y)). At which x value is the tangent to this curve horizontal? x = ? At which t value is
18. ## math
The table shows the depth (d metres) of water in a harbour at certain times (t hours) after midnight on a particular day. time t (hours) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 and the in the next column corresponding to the figures above is depth d (m) 3.0 3.3
19. ## Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x/y. What is the equation of the curve if (3, 1) is a point on the curve? A. x^2 + y^2 =8
20. ## Psychological Statistics
TRUE OR FALSE. The Pearson correlation between X1 and Y is r = 0.50. When a second variable, X2, is added to the regression equation, we obtain R2 = 0.60. Adding the second variable increases the variability that is predicted by the regression equation for
21. ## calc
The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y. What is the equation of the curve if (3, 1) is a point on the curve?
22. ## Calculus 1
The curve y =|x|/(sqrt(5−x^2)) is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (2,2).
23. ## Calculus
There is a diagram with the curve y=(2x-5)^4.The point P has coordinates (4,81) and the tangent to the curve at P meets the x-axis at Q.Find the area of the region enclosed between curve ,PQ and the x-axis.
In each case, sketch the two specified normal curves on the same set of axes: a A normal curve with m 20 and s 3, and a normal curve with m 20 and s 6. b A normal curve with m 20 and s 3, and a normal curve with m 30 and s 3. c A normal curve with m 100
25. ## economic
colgate is one firm of many in the market for toothpaste, which is in long run equilibrium. a) draw diagram showing colgate's demand curve, MR curve, ATC curve and MC curve. label colgate profit maximizing output and price
26. ## Maths
The slope of a curve is equal to y divided by 4 more than x2 at any point (x, y) on the curve. A. Find a differential equation describing this curve. B. Solve the differential equation from part A. C. Suppose it’s known that as x goes to infinity on the
27. ## Calculus
Please help this is due tomorrow and I don't know how to Ive missed a lot of school sick Consider the curve given by the equation x^3+3xy^2+y^3=1 a.Find dy/dx b. Write an equation for the tangent line to the curve when x = 0. c. Write an equation for the
28. ## PHYSICS
A circular curve of highway with radius 200m is designed for traffic moving at 100km/h, and is banked so that cars can make the curve even without friction at that speed. What is the correct angle of banking for this curve? Show steps, please.
29. ## Stats
Observations are taken on sales of a certain mountain bike in 30 sporting goods stores. The regression model was Y = total sales (thousands of dollars), X1 = display floor space (square meters), X2 = competitors’ advertising expenditures (thousands of
30. ## statistics
If the correlation coefficient is -0.54, what is the sign of the slope of the regression line? - As the correlation coefficient decreases from 0.86 to 0.81, do the points of the scatter plot move toward the regression line, or away from it
31. ## maths
A curve y has gradient Dy/dx=3x^2-6x+2a)if the curve passes through the origin find it equation b)Find the area of the finite region included between the curve in a)and the x-axis.?
32. ## statistics
Job Sat. INTR. EXTR. Benefits 5.2 5.5 6.8 1.4 5.1 5.5 5.5 5.4 5.8 5.2 4.6 6.2 5.5 5.3 5.7 2.3 3.2 4.7 5.6 4.5 5.2 5.5 5.5 5.4 5.1 5.2 4.6 6.2 5.8 5.3 5.7 2.3 5.3 4.7 5.6 4.5 5.9 5.4 5.6 5.4 3.7 6.2 5.5 6.2 5.5 5.2 4.6 6.2 5.8 5.3 5.7 2.3 5.3 4.7 5.6 4.5
33. ## math
Data were recorded for the demand and revenue of a given product. Find the linear regression line which represents the revenue function. Find the quadratic regression curve which represents the revenue function. Data Demand x Revenue y 1 0 100 2 4 139.2 3
34. ## Statistics
A set of X and Y scores has SSx=10, SSY=20, and SP=8, what is the slope for the regression equation? a. 8/10 b. 8/20 c. 10/8 d. 20/8 For the regression equation, Y=bX+a, which of the following X,Y points will be on the regression line? a. a,b b. b,a c. 0,
35. ## Stats
x y 1 -10.0 2 -20.0 3 -30.0 4 -40.0 5 -50.0 - What would be the slope of this regression line? - Would the correlation between x and y be positive or negative? - How would you interpret these data in terms of linear regression?
36. ## Maths
y= ae^(bt)sin(kt) Use a sine regression to determine k, the period of oscillation of this system. State the regression equation and superimpose its graph on that of the data set. help?
37. ## college statistics
construct a scatterplot for x,y values x 1 2 3 4 5 y 0.5 0.0 -0.5-1.0-1.5 what would be the slope of this regression line would the correlation between xand y be postive oe negative how would you interpret these data in terms of regression
38. ## statistics
For the multiple regression equation ŷ = 100 + 20x©û - 3x©ü + 120x©ý a. Identify the y-intercept and partial regression coefficient. b. If x©û = 12, x©ü = 5, and x©ý c. If x©ý were to increase by 4, what change would be necessary in x©ü in
39. ## Math: Statistics
1. A regression analysis includes the effect of Age, which is categorised into five levels. How many dummy variables should be defined for including it in the analysis? 2. A regression analysis includes Sex as an explanatory variable. How many dummy
Data are collected on the average second-hand price of a particular model of car, from back copies of car trading magazines. The model of car was manufactured in 2000. Year 2001 2002 2003 2004 2005 Average price (£) 11015 5920 4123 3171 2587 Taking 2001
41. ## math
The path of a cliff diver can be modeled by a parabolic curve. The regression equation matching one particular dive is h(t)= -4.9^2+4.6x+24.8 h is height t is time in seconds at what point in meters did the diver reach maximum height?
42. ## Another graph
How do I make a graph of f'(-1) = f'(1) = 0, f'(x) > 0 on (-1,1), f'(x) < 0 for x < -1, f'(x) > 0 for x >1 Thanks. if the derivative is postive on -1 to 1, you have a curve that shope upward as x increases. If the derivative is zero at -1, and 1, thekn at
43. ## MATH
A data set has the following points: (0, 2), (4, 1), (6, 0) The slope of the linear regression line is -0.32. Find the y-intercept of the linear regression.
44. ## Algebra 2
A data set has the following point: (1,8)(2,7)(3,5) The slope of the linear regression line is -1.5 Find the y-intercept of the linear regression.
45. ## Statistics
I neep help on two questions! A condition that occurs in multiple regression analysis if the independent variables are themselves correlated is known as: 1. autocorrelation 2. stepwise regression 3. multicorrelation 4. multicollinearity (I think this is
46. ## statistics
For a given set of data, it is known that x = 10 and y = 4. The gradient of the regression line y on x is 0.6. Find the equation of the regression line and use it to estimate y when x = 8.
47. ## Math
A perfect positive or negative correlation means that A) the explanatory causes the y-variable. B) 100% of the variation is explained. C) we get the same regression equation if we switch the x and y variables. D) the slope of the regression equation is 1.0
48. ## Statistics
Hi Can anyone tell me in a regression model are the values of 0.13 0.10 A good indication of of the total y explained in a regression model ?? Thanks.
49. ## Math
Hello! Can you check my answers for these questions? Thank you in advance (= 1. The following ordered pairs give the scores on two consecutive 15-point quizzes for a class of 18 students. (7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (14, 11),
50. ## Math \ Algebra
Hello! Can you check my answers for these questions? Thank you in advance (= 1. The following ordered pairs give the scores on two consecutive 15-point quizzes for a class of 18 students. (7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (14, 11),
51. ## Calc.
sketch the curve using the parametric equation to plot the points. use an arrow to indicate the direction the curve is traced as t increases. Find the lenghth of the curve for o
52. ## Econ
For a typical negative externality market graph, with Demand curve (also labelled private value), and supply curve (private cost) and a social cost curve above the supply curve. What is the optimal quantity that maximizes total economic well being? Is it
53. ## Math
Hello! Can you check my answers for these 4 worksheet questions? Thank you in advance (: 1. The following ordered pairs give the scores on two consecutive 15-point quizzes for a class of 18 students. (7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7),
54. ## physics
A car is traveling at a constant speed along road ABCDE. (Curve BC where Bis beginning of curve and C is the end of the curve is in the shape of a "backward C." C is then the beginning of another curve and D is the end of the same curve in the shape of a
55. ## Pre-calculus
You are given a pair of equations, one representing a supply curve and the other representing a demand curve, where p is the unit price for x items. 80 p + x - 380 = 0 and 84 p - x - 40 = 0 Identify which is the supply curve and demand curve and the
56. ## maths
input the following data into your calculator and obtain the best fit power regression function for y in terms of x, for the data x 1 2 3 4 6 y 18.0 14.3 12.5 11.3 10.0 choose the one option which most closely describes the power relationship between x and
57. ## economics
Can you please exlain to me if I'm wrong not just correct me. ps. Ceteris paribusor the following markets, show whether change causes a shift in supply curve, a shift in demand curve, a movement along the supply curve, and/or a movement along the demand
58. ## Stats
Construct a scatterplot for the (x, y) values below, and answer the following questions. You do NOT need to submit your scatterplot with your answer; however, show all other work. x y 1 2.5 2 5.0 3 7.5 4 10.0 5 12.5 - What would be the slope of this
59. ## calculus
Compute the area of the region in the fi…rst quadrant bounded on the left by the curve y = sqrt(x), on the right by the curve y = 6 - x, and below by the curve y = 1.
60. ## calculus
Given the curve defined by the equation y=cos^2(x) + sqrt(2)* sin(x) with domain (0,pi) , find all points on the curve where the tangent line to the curve is horizontal
61. ## physics
Can I still use the equation that I generated from my curve when doing the experiment even if the number I want to calculate is bigger than the last point in my curve ( ps the curve is a best fit line) would my answer be accurate??
62. ## physics
The grath is a plot of work versus time. The power can determined from a)the xaxis, b)yaxis, c)the slope of curve,d)are under the curve, e)radius of curve
63. ## math help
Posted by grant on Saturday, May 12, 2007 at 9:20am. The table shows the depth (d metres) of water in a harbour at certain times (t hours) after midnight on a particular day. time t (hours) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 and the in the next column
64. ## statistics
Linear Regression In the babies.dta full dataset, generate a covariate called painind defined as 1 if the infant experienced severe pain upon receiving the shot (pain0 = 7) and as 0 otherwise. In Stata, you can use the commands: generate painind = 0
65. ## physics
The firetruck goes around a 180°, 162 m radius circular curve. It enters the curve with a speed of 12.6 m/s and leaves the curve with a speed of 38.8 m/s. Assuming the speed changes at a constant rate, what is the magnitude of the total acceleration of
66. ## Math
a curve is such that dy/dx=4x+7. the line y=2x meets the curve at point 'P'. Given that the gradient of the curve at P is 5. State the coordinates of P.
67. ## Math
a curve is such that dy/dx=4x+7. the line y=2x meets the curve at point 'P'. Given that the gradient of the curve at P is 5. State the coordinates of P.
68. ## Calculus
The length of a curve for 0
69. ## Statistics
A certain curve (that is NOT the normal curve) is shown below. The curve is symmetric around 0, and the total area under the curve is 100%. The area between -1 and 1 is 58%. What is the area to the right of 1? I do not know how to do this because it is not
70. ## PSYCH
Two common confounding variables that must be controlled in a nonequivalent control-group design are: A. selection and regression to the mean. B. selection and maturation. C. maturation and regression to the mean. D. sequence effects and maturation. MY
71. ## Economics
You are planning to estimate a short-run production function for your firm, and you have collected the following data on labor usage and output. Labor usage output 3 1 7 2 9 3 11 5 17 8 17 10 20 15 24 18 26 22 28 21 30 23 (a) Does a cubic equation appear
72. ## Math
Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. The table below shows the number of games sold, in thousands, from the years 2000–2010. Year 2000 2001 2002 2003 2004 2005 Number Sold
73. ## Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x/y. What is the equation of the curve if (4, 1) is a point on the curve? x2 − y2 = 15 x2 + y2 = 15 x + y = 15 xy = 15
74. ## Math
The least squares regression line is a unique line for each data set. True The least squares regression line minimizes the sum of the residuals. False
75. ## physics
A car going 30 m/s just barely makes it around a curve without skidding. This curve had a radius of 150 m. The next curve has a radius of 75 m. At what maximum velocity can it go around this curve without skidding?
76. ## statistics
An expert witness in a case of alleged racial discrimination in a state university school of nursing introduced a regression of the determinants of Salary of each professor for each year during an 8-year period (n = 423) with the following results, with
77. ## Economics
If an industry is highly profitable, new firms are likely to enter the market. This would be reflected with a shift of the A- demand curve to the right B-supply curve to the right C-supply curve to the left D- demand curve to the left
78. ## Statistics
We're doing a project. I have all my data, but I'm not sure about what this is asking for. Use your regression equation to determine the purchase price for the 1999 Jeep Grand Cherokee. How does the predicted value, ŷ, compare to the average selling
79. ## Calculus
There is a graph of y=1/sqrtx.Show that the area of the region between line and curve at x=1 and x=3 is 3-(5sqrt3/3). 👍 1 👎 1 👁 33 asked by Raj yesterday at 6:35am "Show that the area of the region between line and curve" . What line? Assuming you
80. ## college statistics
if the correlation co-efficient is o.32 what is the sign of the slope regression line as the correlation co efficient decreases from 0.78 to 0.71 do the oints of the scatter plot move toward the regression line or away from it
81. ## Physics
A banked circular highway curve is designed for traffic moving at 90 km/h. The radius of the curve is 70 m. If the coefficient of static friction is 0.30 (wet pavement), at what range of minimum and maximum speeds can a car safely make the curve?
82. ## College math
Highway curves If a circular curve without any banking has a radius of R feet, the speed limitL.in miles per hour for the curve is L=1.5 R. (a.)Find the speed limit for a curve having a radius of 400 feet. (b.) If the radius of a curve doubles, what
83. ## Math
. Given that x²cos y_sin y=0,(0,π). A. Verify that the given points on the curve. B.use implicit differention to find the slope of the above curve at the given point. C.find the equation of tangent and normal to the curve at that.
84. ## math
Given the curve of C defined by the equation x^2 - 2xy +y^4 =4. 1) Find the derivative dy/dx of the curve C by implicit differentiation at the point P= (1,-1). 2) Find a line through the origin that meets the curve perpendicularly.
85. ## physics!!!
The speed limit around a curve of radius 85 meters is 30 miles per hour. What angle should the curve be banked at if a car is to be able to go around the curve at that speed with no friction? Help!!!
86. ## AP Statistics
A certain density curve looks like an interverted letter V. The first segment goes fro the point (0,0.6) to the point (0.5,1.4). The segment goes from (0.5.1.44) to (1,0.6). (a) Sketch the curve. Verify that the area under the curve is 1, so that it is a
87. ## math
Use exponential regression to model the price P(t) as a function of time t since 1994. Include a sketch of the points and the regression curve. (Round the coefficients to 3 decimal places.)
88. ## Math
What is a regression curve?
89. ## Statistics
Determine the regression line of x+2y-5=0 and 3x+3y-8=0 (i) y on x and (ii) x on y, (iii) Find r using the regression coefficients
90. ## math-grd 11
what do they mean by sine regression? i am currently working on the grd 11 isu and i am just wondering how i am supposed to incorpporate sine regreession into it? my isu is, you have a picture you have to find a curve in it and figure out the equation of
91. ## statistics
A survey was taken in various Statistics classes and the following regression equation was obtained regarding the height and weight of students in the Statistics classes. Regression equation Height = 56.8 + 0.0712 Weight Given the regression equation
92. ## Calcus, Math
class of 18 students. (7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (14, 11), (14, 15), (8, 10), (9, 10), (15, 9), (10, 11), (11, 14), (7, 14), (11, 10), (14, 11), (10, 15), (9, 6) Create a scatter plot of the data. What kind of correlation does
93. ## statistics
In the following regression, X = total assets (\$ billions), Y = total revenue (\$ billions), and n = 64 large banks. (a) Write the fitted regression equation. (b) State the degrees of freedom for a twotailed test for zero slope, and use Appendix D to find
94. ## math
can someone please help me fast before midnight. a)I need the correlation between x and y b)slope of linear regression c)y intercept of linear regression for the points (1,8)(2,6)(3,4)(4,2)(5,5)
95. ## physics
highway curve in northern Minnesota has a radius of 230 m. The curve is banked so that a car traveling at 21 m/s and will not skid sideways, even if the curve is coated with a frictionless glaze of ice. At what angle to the horizontal is the curve banked?
96. ## Calculus
The slope of the tangent to a curve at any point (x, y) on the curve is -x/y . Find the equation of the curve if the point (3,-4) on the curve.
97. ## Calculus
The slope of the tangent line to a curve at any point (x, y) on the curve is x/y. What is the equation of the curve if (4, 1) is a point on the curve? a) x^2-y^2=15 b) x^2+y^2=15 c) x+y=15 d) xy=15
98. ## Statistics
A regression equation was calculated and is given below. The values for Miss America are values such as 0, 1, 2, 3, 4, etc. Regression Analysis: GPA versus Miss America The regression equation is GPA = 3.32 - 0.0890 Miss America b. What is the response
99. ## Physics
A motor scooter rounds a curve on the highway at a constant speed of 25.0 m/s. The original direction of the scooter was due east; after rounding the curve the scooter is heading 28 degrees north of east. The radius of curvature of the road at the location
100. ## math
consider the curve defined by the equation y=a(x^2)+bx+c. Take a point(h,k) on the curve. use Wallis's method of tangents to show that the slope of the line tangent to this curve at the point(h,k) will be m= 2ah+b. have to prove this for tow cases: a>0 and
| 6,741 | 23,505 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4 | 4 |
CC-MAIN-2021-31
|
latest
|
en
| 0.877724 |
https://www.physicsforums.com/threads/geometric-optics.410657/
| 1,521,376,889,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-13/segments/1521257645613.9/warc/CC-MAIN-20180318110736-20180318130736-00030.warc.gz
| 871,421,968 | 15,537 |
Geometric Optics
1. Jun 16, 2010
cyt91
1. The problem statement, all variables and given/known data
Given a convex lens of focal length of (x+5) cm and a concave lens of focal length x cm.
The 2 lenses are placed 30 cm apart coaxially i.e along the same axis with the convex lens on the left while the concave lens is on the right. A light bulb is placed to the left of the convex lens at a distance of 10 cm. It is observed that light rays that emerges from the concave lens are parallel to each other. Find the value of x.
2. Relevant equations
$$\frac{1}{f}$$=$$\frac{1}{u}$$+$$\frac{1}{v}$$
3. The attempt at a solution
Since the light rays that emerges from the concave lens are parallel,the light rays that emerges from the convex lens converges at the focal point of the concave lens.
Therefore,
$$\frac{1}{x+5}$$=$$\frac{1}{10}$$+$$\frac{1}{30-x}$$
$$x^{2}$$-45x+100=0
Solving for x, x=2.344 or 4.266
The answer given is x=2.655.
Part of the solution given involves the equation
$$\frac{1}{x+5}$$=$$\frac{1}{10}$$+$$\frac{1}{30+x}$$
instead of $$\frac{1}{x+5}$$=$$\frac{1}{10}$$+$$\frac{1}{30-x}$$
They've added a negative sign to the focal length of the concave lens i.e. -x instead of x.
Is the solution correct?
I feel the solution is wrong because we shouldn't add a negative sign to x (the focal length of the concave lens) since x is a variable.
2. Jun 16, 2010
Mindscrape
Yes, that is right, not only if you were to go through the geometric math of it, but you should also be able to simply think about it. A convex lens takes parallel light and converges it, while a concave lens takes parallel light and diverges it. They behave opposite of one another.
Basically it means you used the wrong focus point by making the focus on the the wrong side of the concave lens.
3. Jun 17, 2010
cyt91
But x is a variable. If it's a variable shouldn't the negative sign take care of itself?
E.g. x^2+6x+5=0
solving, (x+1)(x+5)=0
x=-1,-5
But,if I know x is negative (as in the case of the focal length of the concave lens), I couldn't simply add a negative sign in front of x :
(-x)^2+6(-x)+5=0
x^2-6x+5=0
(x-1)(x-5)=0
x=1,5 (a different set of solution)
So is the solution correct?
4. Jun 17, 2010
aim1732
Yes the solution is correct.
The variable x given is actually a modulus value - always positive. Think of it this way - if x were the variable you were saying it could be positive or negative but not both at the same time. Then the lenses would either be concave or convex but not both.
Actually x is distance and by definition a +ve quantity so when we say a concave lens has focal length x it is actually -x.
5. Jun 17, 2010
cyt91
Ok. I get your point. Thanks a lot. This is very helpful.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
| 836 | 2,831 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.15625 | 4 |
CC-MAIN-2018-13
|
longest
|
en
| 0.88724 |
https://testbook.com/question-answer/if-a-3b-2a-4b-3-5-then-a-b-a--5fed7810f2f757b42bb91183
| 1,627,530,274,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-31/segments/1627046153814.37/warc/CC-MAIN-20210729011903-20210729041903-00717.warc.gz
| 536,533,409 | 20,818 |
# If (a + 3b) : (2a + 4b) = 3 : 5, then (a - b) : (a + b) is equal to:
Free Practice With Testbook Mock Tests
## Options:
1. 2 : 1
2. 2 : 3
3. 3 : 2
4. 1 : 2
### Correct Answer: Option 4 (Solution Below)
This question was previously asked in
SSC CHSL Previous Paper 107 (Held On: 20 Oct 2020 Shift 3)
## Solution:
Given:
(a + 3b) : (2a + 4b) = 3 : 5
Formula used:
if $$\frac{a}{b} = \;\frac{c}{d}$$
then, $$\;\;\frac{{a - b}}{{a + b}} = \;\frac{{c - d}}{{c + d}}$$
Calculation:
$$\frac{{\left( {a + 3b} \right)}}{{\left( {2a + 4b} \right)}} = \;\frac{3}{5}$$
⇒5 × (a + 3b) = 3 × (2a + 4b)
⇒5a + 15b = 6a + 12b
⇒a = 3b
$$\frac{a}{b} = \;\frac{3}{1}$$
$$\frac{{a - b}}{{a + b}} = \;\frac{{3 - 1}}{{3 + 1}}$$
$$\frac{{a - b}}{{a + b}} = \;\frac{2}{4} = \;\frac{1}{2}$$
⇒(a - b) : (a + b) = 1 : 2
| 403 | 817 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.5 | 4 |
CC-MAIN-2021-31
|
latest
|
en
| 0.489258 |
https://junkyarddog.net/beverley-park/scalar-product-of-two-vectors-example.php
| 1,685,304,596,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-23/segments/1685224644506.21/warc/CC-MAIN-20230528182446-20230528212446-00657.warc.gz
| 378,825,030 | 8,665 |
# Scalar Product Of Two Vectors Example
Dot Product MatemГЎticas. 13/08/2011В В· Scalar Product - Example : Scalar Product - Perpendicular Vectors Vectors - equation of a line through two points :, Looking for examples of vector and scalar quantity in physics? Scalar and vector quantities are two of scalar's and vectors, along with examples and.
### Generate dot product of two vectors Simulink - MathWorks
Tensor Algebra People. Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are, Finding the Angle Between 2 3D Vectors 3D Vector Example Vector Product Definition Right Hand Rule Example Calculate the scalar product of the two vectors.
2. SCALARS, VECTORS, We define the scalar product of two vectors A and B as For example, consider the “double cross product” Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are
Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result
Triple Scalar Product: Definition, Formula & Example. When we take the cross product of two vectors вѓ—a and вѓ—b, Triple Scalar Product: Definition, Formula Note that the dot product is a scalar. Example: If a = [1,0,2] and b = [0,1,1] If the dot product of two vectors is zero, the vectors are perpendicular.
The vector product and the scalar product are the 2 methods for multiplying vectors which see the most application in astronomy and physics. ... we will take a look at the dot product of two vectors. multiplied by two methods: scalar product of vectors or Example of the dot product of vectors for
7/04/2015В В· If the scalar is zero, then the two vectors are at a right angle to each other. Example: Cross Product with Two Vectors: Dot Products and Projections. The Dot Product angle theta between the two vectors. Example The dot product of a=<1,3,-2 scalar projection of b onto a is the
Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are Dot product of two vectors. Example 2. Find the dot product of vectors a Dot product of two vectors Cross product of two vectors (vector product) Scalar
The dot product of two vectors is a scalar Example Compute v В·w knowing that v, w в€€R3, withv|= 2, w = h1,2,3i and the angle in between is Оё = ПЂ/4. 27/09/2012В В· Dot Product : Find Angle Between Two Vectors. Here I do another quick example of using the dot product to find the angle between two vectors. Category
Where to use scalar triple product further? Some solved examples on the scalar triple product. The scalar product of two vectors 27/09/2012В В· Dot Product : Find Angle Between Two Vectors. Here I do another quick example of using the dot product to find the angle between two vectors. Category
Note that the dot product is a scalar. Example: If a = [1,0,2] and b = [0,1,1] If the dot product of two vectors is zero, the vectors are perpendicular. This step-by-step online calculator will help you understand how to find cross product of two vectors. Scalar triple product Online calculator.
Where to use scalar triple product further? Some solved examples on the scalar triple product. The scalar product of two vectors The Cross Product a Г— b of two vectors is another vector that is at Example: The cross product of a But there is also the Dot Product which gives a scalar
The Scalar Product 9 internal.ncl.ac.uk. Dot product of two vectors. Example 2. Find the dot product of vectors a Dot product of two vectors Cross product of two vectors (vector product) Scalar, The dot product of two vectors gives you a scalar(a number). For example: v=i+j w=2i+2j Dot product of w*v = (2*1)+(2*1) =4.
### The Dot (or Scalar) Product UC Santa Cruz
Vectors Scalar or Dot Product ExamSolutions - YouTube. The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result, The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used.
### Dot Product Of Two Vectors Byju's Mathematics
Dot Product Of Two Vectors Byju's Mathematics. The scalar product or dot product of two vectors and is equal to: Example. It can also be expressed as: Example. Magnitude of a Vector. Examples Triple Scalar Product: Definition, Formula & Example. When we take the cross product of two vectors вѓ—a and вѓ—b, Triple Scalar Product: Definition, Formula.
• Vectors Scalar product or dot product or inner product
• The Scalar Product 9 internal.ncl.ac.uk
• Vectors Scalar or Dot Product ExamSolutions - YouTube
• Example Vectorsa andb Using the scalar product to п¬Ѓnd the angle between two vectors What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector
The Cross Product a × b of two vectors is another vector that is at Example: The cross product of a But there is also the Dot Product which gives a scalar •Do some examples . Cross Product When you take the cross product of two vectors a and b, If the triple scalar product is 0, then the vectors must lie in the
Looking for examples of vector and scalar quantity in physics? Scalar and vector quantities are two of scalar's and vectors, along with examples and Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get
The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical Tensors (especially 2nd Order Ones) Dyadic Product between two vectors This multiplication between two tensors results in a scalar. Another example:
7/04/2015В В· If the scalar is zero, then the two vectors are at a right angle to each other. Example: Cross Product with Two Vectors: Lorentz Invariance and the 4-vector Dot Product The 4-vector is a powerful tool because the dot product of two 4-vectors example, consider the case where I have
In this section we will define the dot product of two vectors. called the scalar product. The dot product is also an example of an inner product and so on The Dot Product block Generate dot product of two vectors. The Dot Product block generates the dot product of the input vectors. The scalar
2. SCALARS, VECTORS, We define the scalar product of two vectors A and B as For example, consider the “double cross product” This step-by-step online calculator will help you understand how to find cross product of two vectors. Scalar triple product Online calculator.
In this section we will define the dot product of two vectors. called the scalar product. The dot product is also an example of an inner product and so on Dot product, properties, geometric interpretation, angle between two vectors, orthogonal vectors, magnitude of vectors, vector projection, formulas, examples
The dot product, also called the scalar product, of two vectors is a number (scalar quantity) obtained by performing a specific operation on the vector components. The dot product of two vectors is a scalar Example Compute v В·w knowing that v, w в€€R3, withv|= 2, w = h1,2,3i and the angle in between is Оё = ПЂ/4.
Cross Product of two Vectors with Examples, definitions, theorems and proofs will be explained in detail in the pdf. You can also download the pdf file. The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical
Dot product, properties, geometric interpretation, angle between two vectors, orthogonal vectors, magnitude of vectors, vector projection, formulas, examples Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get
## The Scalar Product 9 internal.ncl.ac.uk
Tensor Algebra People. The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical, Calculating the Dot Product. A dot product is a scalar value that is For example if A and B were 3D vectors: the dot product of two vectors will be equal.
### Cross Product of two Vectors with Examples Hameroha.com
Cross Product of two Vectors with Examples Hameroha.com. This step-by-step online calculator will help you understand how to find cross product of two vectors. Scalar triple product Online calculator., Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are.
Where to use scalar triple product further? Some solved examples on the scalar triple product. The scalar product of two vectors Lorentz Invariance and the 4-vector Dot Product The 4-vector is a powerful tool because the dot product of two 4-vectors example, consider the case where I have
13/08/2011 · Scalar Product - Example : Scalar Product - Perpendicular Vectors Vectors - equation of a line through two points : 2. SCALARS, VECTORS, We define the scalar product of two vectors A and B as For example, consider the “double cross product”
13/08/2011 · Scalar Product - Example : Scalar Product - Perpendicular Vectors Vectors - equation of a line through two points : Finding the Angle Between 2 3D Vectors 3D Vector Example Vector Product Definition Right Hand Rule Example Calculate the scalar product of the two vectors
The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical Vectors. Products of Vectors. Vector Fields . Many physical quantities of interest are calculated by forming the scalar product of two vectors. Examples:
The Dot Product block Generate dot product of two vectors. The Dot Product block generates the dot product of the input vectors. The scalar Where to use scalar triple product further? Some solved examples on the scalar triple product. The scalar product of two vectors
Looking for examples of vector and scalar quantity in physics? Scalar and vector quantities are two of scalar's and vectors, along with examples and What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector
Calculating the Dot Product. A dot product is a scalar value that is For example if A and B were 3D vectors: the dot product of two vectors will be equal The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result
Where to use scalar triple product further? Some solved examples on the scalar triple product. The scalar product of two vectors Looking for examples of vector and scalar quantity in physics? Scalar and vector quantities are two of scalar's and vectors, along with examples and
Vectors - Cross Product Dot Product which is also called Scalar Product. Cross Product involving two vectors in component form The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used
Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get Triple Scalar Product: Definition, Formula & Example. When we take the cross product of two vectors вѓ—a and вѓ—b, Triple Scalar Product: Definition, Formula
The dot product, also called the scalar product, of two vectors is a number (scalar quantity) obtained by performing a specific operation on the vector components. Dot Product of Two Vectors Scalar Product of Two Vectors along with solved examples. Learn Dot Product of Vectors with Byju's
7/04/2015 · If the scalar is zero, then the two vectors are at a right angle to each other. Example: Cross Product with Two Vectors: What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector
Cross Product of two Vectors with Examples, definitions, theorems and proofs will be explained in detail in the pdf. You can also download the pdf file. The dot product of two vectors gives you a scalar(a number). For example: v=i+j w=2i+2j Dot product of w*v = (2*1)+(2*1) =4
Cross Product of two Vectors with Examples, definitions, theorems and proofs will be explained in detail in the pdf. You can also download the pdf file. The Scalar Product 9.3 Introduction There are two kinds of multiplication involving vectors. The п¬Ѓrst is known as the scalar product
This step-by-step online calculator will help you understand how to find cross product of two vectors. Scalar triple product Online calculator. Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get
Dot product of two vectors. Example 2. Find the dot product of vectors a Dot product of two vectors Cross product of two vectors (vector product) Scalar The vector product and the scalar product are the 2 methods for multiplying vectors which see the most application in astronomy and physics.
Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get The dot product, also called the scalar product, of two vectors is a number (scalar quantity) obtained by performing a specific operation on the vector components.
Dot Product of Two Vectors Scalar Product of Two Vectors along with solved examples. Learn Dot Product of Vectors with Byju's The dot product of two vectors gives you a scalar(a number). For example: v=i+j w=2i+2j Dot product of w*v = (2*1)+(2*1) =4
What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector Tensors (especially 2nd Order Ones) Dyadic Product between two vectors This multiplication between two tensors results in a scalar. Another example:
The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result The Cross Product a Г— b of two vectors is another vector that is at Example: The cross product of a But there is also the Dot Product which gives a scalar
### Scalar Product of Two Vectors Edu-Resource.com
Generate dot product of two vectors Simulink - MathWorks. Dot Product of Two Vectors Scalar Product of Two Vectors along with solved examples. Learn Dot Product of Vectors with Byju's, Vectors - Cross Product Dot Product which is also called Scalar Product. Cross Product involving two vectors in component form.
### The Scalar Product 9 internal.ncl.ac.uk
The Scalar Product 9 internal.ncl.ac.uk. The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used 13/08/2011 · Scalar Product - Example : Scalar Product - Perpendicular Vectors Vectors - equation of a line through two points :.
• The Dot (or Scalar) Product UC Santa Cruz
• Cross Product of two Vectors with Examples Hameroha.com
• Tensor Algebra People
• The Cross Product a Г— b of two vectors is another vector that is at Example: The cross product of a But there is also the Dot Product which gives a scalar The scalar product or dot product of two vectors and is equal to: Example. It can also be expressed as: Example. Magnitude of a Vector. Examples
The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result This MATLAB function returns the scalar dot product of A and B. the dot product of two complex vectors is also for example, C(1) = 54 is the dot product of A
The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical Lorentz Invariance and the 4-vector Dot Product The 4-vector is a powerful tool because the dot product of two 4-vectors example, consider the case where I have
Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are Finding the Angle Between 2 3D Vectors 3D Vector Example Vector Product Definition Right Hand Rule Example Calculate the scalar product of the two vectors
This step-by-step online calculator will help you understand how to find cross product of two vectors. Scalar triple product Online calculator. The vector product and the scalar product are the 2 methods for multiplying vectors which see the most application in astronomy and physics.
The vector product and the scalar product are the 2 methods for multiplying vectors which see the most application in astronomy and physics. The scalar product or dot product of two vectors and is equal to: Example. It can also be expressed as: Example. Magnitude of a Vector. Examples
Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors For example, they can help you get The Cross Product Of Two Vectors Example 2 Find two vectors orthogonal to This is known as the triple scalar product and it has physical and geometrical
The Dot Product block Generate dot product of two vectors. The Dot Product block generates the dot product of the input vectors. The scalar The scalar or the dot product of two vectors returns as the result scalar quantity as all three factors on the right side of the Scalar product of vectors examples
Dot product, properties, geometric interpretation, angle between two vectors, orthogonal vectors, magnitude of vectors, vector projection, formulas, examples Dot product, properties, geometric interpretation, angle between two vectors, orthogonal vectors, magnitude of vectors, vector projection, formulas, examples
What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector 27/09/2012 · Dot Product : Find Angle Between Two Vectors. Here I do another quick example of using the dot product to find the angle between two vectors. Category
Vector Algebra and Calculus 1. •If the scalar triple product of three vectors a· ♣Example 2.14 Question for civil engineers Two long straight pipes are The Cross Product a × b of two vectors is another vector that is at Example: The cross product of a But there is also the Dot Product which gives a scalar
Triple Scalar Product: Definition, Formula & Example. When we take the cross product of two vectors ⃗a and ⃗b, Triple Scalar Product: Definition, Formula The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used
Lorentz Invariance and the 4-vector Dot Product The 4-vector is a powerful tool because the dot product of two 4-vectors example, consider the case where I have Lorentz Invariance and the 4-vector Dot Product The 4-vector is a powerful tool because the dot product of two 4-vectors example, consider the case where I have
The scalar product One of the ways in which two vectors can be combined is known as the scalar product. When we calculate the scalar product of two vectors the result The vector product and the scalar product are the 2 methods for multiplying vectors which see the most application in astronomy and physics.
Vectors - Cross Product Dot Product which is also called Scalar Product. Cross Product involving two vectors in component form What is a Scalar Quantity? Before we learn about the Scalar product of two vectors, let’s refresh what we have already learned about the difference between a vector
2. SCALARS, VECTORS, We define the scalar product of two vectors A and B as For example, consider the “double cross product” Looking for examples of vector and scalar quantity in physics? Scalar and vector quantities are two of scalar's and vectors, along with examples and
The Dot Product block Generate dot product of two vectors. The Dot Product block generates the dot product of the input vectors. The scalar Calculating the Dot Product. A dot product is a scalar value that is For example if A and B were 3D vectors: the dot product of two vectors will be equal
Dot product, properties, geometric interpretation, angle between two vectors, orthogonal vectors, magnitude of vectors, vector projection, formulas, examples Calculating the Dot Product. A dot product is a scalar value that is For example if A and B were 3D vectors: the dot product of two vectors will be equal
Cross Product of two Vectors with Examples, definitions, theorems and proofs will be explained in detail in the pdf. You can also download the pdf file. Dot Product of Two Vectors Scalar Product of Two Vectors along with solved examples. Learn Dot Product of Vectors with Byju's
MATH1014 Quizzes You are here: This is a dot product of two vectors and the end quantity is a scalar. True This is a dot product of two vectors and the end Calculating the Dot Product. A dot product is a scalar value that is For example if A and B were 3D vectors: the dot product of two vectors will be equal
The scalar or the dot product of two vectors returns as the result scalar quantity as all three factors on the right side of the Scalar product of vectors examples 7/04/2015В В· If the scalar is zero, then the two vectors are at a right angle to each other. Example: Cross Product with Two Vectors:
| 4,932 | 22,355 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.9375 | 4 |
CC-MAIN-2023-23
|
latest
|
en
| 0.840109 |
https://proofwiki.org/wiki/Dimension_of_Proper_Subspace_is_Less_Than_its_Superspace
| 1,695,463,894,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-40/segments/1695233506480.7/warc/CC-MAIN-20230923094750-20230923124750-00664.warc.gz
| 528,049,060 | 12,315 |
# Dimension of Proper Subspace is Less Than its Superspace
## Theorem
Let $G$ be a vector space whose dimension is $n$.
Let $H$ be a subspace of $G$.
Then $H$ is finite dimensional, and $\map \dim H \le \map \dim G$.
If $H$ is a proper subspace of $G$, then $\map \dim H < \map \dim G$.
## Proof
Let $H$ be a subspace of $G$.
Every linearly independent subset of the vector space $H$ is a linearly independent subset of the vector space $G$.
Therefore, it has no more than $n$ elements by Size of Linearly Independent Subset is at Most Size of Finite Generator.
So the set of all natural numbers $k$ such that $H$ has a linearly independent subset of $k$ vectors has a largest element $m$, and $m \le n$.
Now, let $B$ be a linearly independent subset of $H$ having $m$ vectors.
If the subspace generated by $B$ were not $H$, then $H$ would contain a linearly independent subset of $m + 1$ vectors.
This would contradict the definition of $m$.
Hence $B$ is a generator for $H$ and is thus a basis for $H$.
Thus $H$ is finite dimensional and $\map \dim H \le \map \dim G$.
Now, if $\map \dim H = \map \dim G$, then a basis of $H$ is a basis of $G$ by Sufficient Conditions for Basis of Finite Dimensional Vector Space, and therefore $H = G$.
$\blacksquare$
| 371 | 1,271 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.859375 | 4 |
CC-MAIN-2023-40
|
latest
|
en
| 0.885037 |
https://courses.engr.illinois.edu/phys101/sp2018/practice/practice.pl?exam3/fa06
| 1,540,126,240,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-43/segments/1539583514005.65/warc/CC-MAIN-20181021115035-20181021140535-00426.warc.gz
| 625,867,313 | 4,797 |
### Fall 2006 Physics 101 Hour Exam 3(25 questions)
The grading button and a description of the scoring criteria are at the bottom of this page. Basic questions are marked by a single star *. More difficult questions are marked by two stars **. The most challenging questions are marked by three stars ***.
True-false questions are worth 2 points each, three-choice multiple choice questions are worth 3 points each, five-choice multiple choice questions are worth 6 points each. The maximum possible score is 102. The exam period was 90 minutes; the average score was 79.0; the median score was 83. Click here to see page1 page2 of the formula sheet that came with the exam.
Unless told otherwise, you should assume that the acceleration of gravity near the surface of the earth is 9.8 m/s2 downward and ignore any effects due to air resistance.
### QUESTION 1*
This and the following three questions relate to the same situation:
A 3 kg pendulum is placed on an elevator next to a 3 kg mass hung vertically by a spring with spring constant 200 N/m. When the elevator is at rest, the pendulum and the mass on the spring oscillate with the same period.
What is the length of the pendulum?
(a) 9 cm
(b) 15 cm
(c) 23 cm
(d) 27 cm
(e) 32 cm
### QUESTION 2*
The unstretched length of the spring without the mass is 0.4 meters. What is the equilibrium length of the spring with the 3 kg mass attached when the elevator is at rest?
(a) 0.55 m
(b) 0.69 m
(c) 0.83 m
### QUESTION 3**
The mass on the spring is observed to be moving 3 m/s when the spring is stretched 0.2 meters from its equilibrium position. What is the maximum speed of the mass as it oscillates up and down.
(a) 3.4 m/s
(b) 4.2 m/s
(c) 5.1 m/s
### QUESTION 4**
The elevator now accelerates upward with constant acceleration a = 2.5 m/s2.
Compare the period of the pendulum Tpendulum to the period of the mass on the spring Tmasss on spring when the elevator is accelerating up.
(a) Tpendulum < Tmass on spring
(b) Tpendulum = Tmass on spring
(c) Tpendulum > Tmass on spring
### QUESTION 5**
This and the following two questions relate to the same situation:
At an air show, a jet flies past you. The sound from the engine is 90 dB when it is 700 meters away.
How far away must the engine be, so that the intensity of the sound is reduced by a factor of 8?
(a) 8 × 700 m
(b) 4 × 700 m
(c) sqrt(8) × 700 m
### QUESTION 6**
How loud is the jet when the intensity of the sound has been reduced by a factor of 8?
(a) 89 dB
(b) 81 dB
(c) 11 dB
### QUESTION 7**
As the jet is flying toward you at 180 m/s, you hear the frequency of the sound from the jet as 420 Hz. What frequency will you hear as the jet flies away from you at 180 m/s? (Note that the speed of sound is v = 340 m/s).
(a) 129 Hz
(b) 240 Hz
(c) 310 Hz
(d) 380 Hz
(e) 420 Hz
### QUESTION 8*
This and the following two questions relate to the same situation:
A piano string is fixed at both ends 0.78 meters apart and has a mass of 0.0028 kg.
What is the fundamental wavelength of this string?
(a) 0.39 m
(b) 0.78 m
(c) 1.56 m
### QUESTION 9*
What is the tension in the string, if the fundamental frequency for the string is 450 Hz?
(a) 550 N
(b) 990 N
(c) 1220 N
(d) 1550 N
(e) 1770 N
### QUESTION 10*
What is the wavelength of the sound wave produced by this string vibrating at 450 Hz? (v = 340 m/s)
(a) 0.75 m
(b) 1.56 m
(c) 2.41 m
### QUESTION 11**
A balloon is filled with 0.8 liters of air, at 20°C and sealed. The balloon is then heated so the air inside is 60°C. What is the new volume of the balloon? (You may assume the pressure in the balloon remains constant as the balloon stretched).
(a) 0.91 liters
(b) 1.4 liters
(c) 2.4 liters
### QUESTION 12**
One liter of water at 20°C (β = 207 × 10-6 /K) is placed in a metal (α = 16 × 10-6 /K) cube, whose each side is length 0.1 m so that it can hold exactly one liter. The water and cube are heated to 80°C. How much water spills out of the pan?
(a) 5.03 × 10-4 liters
(b) 9.54 × 10-3 liters
(c) 14.1 × 10-3 liters
### QUESTION 13**
Fluids A and B are poured into a U-shaped pipe, shown below. Fluid A is water
(ρ = 1000 kg/m3), which rises 0.4 m above the bottom of the left pipe, and 0.2 m above the bottom of the right pipe. Fluid B in the right pipe rises 0.5 m above the level of water.
What is the density of Fluid B?
(a) 400 kg/m3
(b) 600 kg/m3
(c) 1000 kg/m3
(d) 1250 kg/m3
(e) 2500 kg/m3
### QUESTION 14**
A house has a 9 m wide and 11 m long flat roof. What is the lift on this roof when a 20 m/s wind blows over it? The density of air is 1.29 kg/m3.
(a) 19.8 × 103 N
(b) 21.5 × 103 N
(c) 23.0 × 103 N
(d) 25.5 × 103 N
(e) 27.0 × 103 N
### QUESTION 15*
This and the following two questions relate to the same situation:
A string holds an oak cube (ρO = 600 kg/m3, volume V = 0.8 m3) to the bottom of a pool. The pool is filled with water (ρW = 1000 kg/m3). The tension in the string is T = 2500 N.
What is the volume of the part of the cube under water, V1?
(a) V1 = 0.60
(b) V1 = 0.64
(c) V1 = 0.70
(d) V1 = 0.74
(e) V1 = 0.80
### QUESTION 16*
We cut the string. The volume of the part of the cube under water is now V2. What fraction of the volume of the cube is under water now?
(a) V2 / V = 0.4
(b) V2 / V = 0.6
(c) V2 / V = 0.8
### QUESTION 17*
We put a weight on the top of the oak cube so the cube gets immersed as deep as it was before the string was cut. What is the additional mass M?
(a) M = 155 kg
(b) M = 205 kg
(c) M = 255 kg
### QUESTION 18*
This and the following two questions relate to the same situation:
A hydraulic lift is filled with oil (ρ = 700 kg/m3). The cross section area of the large piston in the cylinder on the right is A2 = 180 cm2. The area of the small piston in the cylinder on the left is A1 = 0.4 cm2.
What force F1 do we need to apply to the small piston to lift a 20000 N weight so the bottom of both pistons is at the same height?
(a) F1 = 4.44 N
(b) F1 = 44.44 N
(c) F1 = 444.44 N
### QUESTION 19*
How much does the load go up when the small piston is pushed down 25 cm?
(a) 0.055 cm
(b) 0.075 cm
(c) 103.0 cm
### QUESTION 20***
Assume now that we apply force F1 again to the small piston, so the two pistons are at the same level. How much more force, ΔF, has to be applied to the small piston to hold the load 0.1 m above the level of the small piston?
(a) 0.027 N
(b) 0.225 N
(c) 314.0 N
### QUESTION 21*
This and the following two questions relate to the same situation:
Water flows through a pipe with the cross section area A1. The pipe is connected to a wider pipe with the cross section area A2 = 20 cm2. The velocity of water in the narrow pipe is v1 = 3.00 m/s, and in the wider pipe it is v2 = 0.6 m/s. The density of water is 1000 kg/m3.
Define P1 = the water pressure in the narrow pipe, and P2 = the water pressure in the wider pipe. Which one of these statements is correct?
(a) P1 < P2
(b) P1 = P2
(c) P1 > P2
### QUESTION 22*
What is the cross section of the narrow pipe, A1?
(a) A1 = 3.00 cm2
(b) A1 = 3.75 cm2
(c) A1 = 4.00 cm2
### QUESTION 23*
What is the absolute value of the water pressure difference, |ΔP| = |P1 -P2| between the wide and narrow parts of the pipe?
(a) |ΔP| = 0
(b) |ΔP| = 173 Pa
(c) |ΔP| = 208 Pa
(d) |ΔP| = 503 Pa
(e) |ΔP| = 4320 Pa
### QUESTION 24*
This and the following question relate to the same situation:
The submersible Alvin, which has been in service since 1964, has been used for biological research as well as for dives to the Titanic. It can dive down to the depth of about 4500 m.
On an average dive, Alvin dives to the depth of 2000 meters. What is the pressure at this depth, P2000? The density of the seawater, ρseawater = 1030 kg/m3.
(a) P2000 = 1.8 106 Pa
(b) P2000 = 1.0 107 Pa
(c) P2000 = 2.0 107 Pa
(d) P2000 = 3.5 107 Pa
(e) P2000 = 4.0 107 Pa
### QUESTION 25*
At another depth, the water pressure around Alvin is 4.9 × 107 Pa. Alvin's titanium pressure hull has a round flat window with a 10 cm radius. What is the total force F acting on the window at this depth?
(a) 0.5 × 105 N
(b) 1.0 × 105 N
(c) 7.5 × 105 N
(d) 1.5 × 106 N
(e) 2.0 × 107 N
Scoring of True-False (T,F) Questions:
• If you mark the correct answer, you get 2 points.
• If you mark the wrong answer or if you mark neither answer, you get 0 points.
Scoring of Multiple Choice I (a,b,c) Questions:
• If you mark the correct answer, you get 3 points.
• If you mark a wrong answer or if you mark none of the answers, you get 0 points.
Scoring of Multiple Choice II (a,b,c,d,e) Questions:
• If you mark one answer and it is the correct answer, you get 6 points.
• If you mark two answers and one of them is the correct answer, you get 3 points.
• If you mark three answers and one of them is the correct answer, you get 2 points.
• If you do not mark the correct answer, or if you mark more than three answers, or if you mark none of the answers, you get 0 points.
| 2,881 | 9,113 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.03125 | 4 |
CC-MAIN-2018-43
|
latest
|
en
| 0.877455 |
https://civilplanets.com/brickwork-calculation/
| 1,723,470,375,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-33/segments/1722641039579.74/warc/CC-MAIN-20240812124217-20240812154217-00785.warc.gz
| 126,859,218 | 21,902 |
Today we are going to discuss brickwork calculation. We are going to find out the below for 1m3 of brickwork
• Number of bricks
• Required cement quantity
• Required sand quantity
Contents of the Article
## Brickwork Basics
• Size of the Bricks – 190mm x 90mm x 90mm
For calculation, purposes convert the Size of bricks into Meter.
So now the measurement is 0.19m x 0.09m x 0.09m
Therefore Volume of 1 brick = 0.001539 m3
## Brickwork Calculation Formula
STEP 1 – Volume of single brick without mortar
Volume of 1 brick without mortar = 0.19 x 0.09 x 0.09 = 0.0015 m3
STEP 2 – Volume of single brick with mortar
While brickwork construction we will add cement mortar around 3 sides (left, right & top).
So adding 10mm thick on each side of brick as mortar thickness
The volume of brick with mortar = (0.19+0.010) x (0.09+0.010) x (0.09+0.010) = 0.20 x 0.1 x 0.1 = 0.002 m3
The volume of 1 brick with mortar = 0.002 m3
STEP 3 – Number of bricks for 1 m3 Brickwork
Number of bricks with mortar for 1 m3 = 1 cubic meter / volume of bricks with mortar = (1/0.002)= 500 Nos
So the required number of bricks for 1 m3 = 500 Nos
STEP 4 – Cement Mortar Volume for 1m3 of Brickwork
Cement Mortar Volume = Volume of brickwork with mortar – the volume of bricks without mortar
Volume of bricks with mortar in 1 m3 = 500 x 0.002 = 1 m3
Volume of bricks without mortar in 1 m3 = 500 x 0.001539 = 0. 7695 m3
Therefore required cement mortar = 1 m3 – 0.7695 m3 = 0.2305 m3
Step 5 – Calculate Cement & Sand Volume
As we know the required cement mortar = 0.2305 m3
Assuming cement mortar ratio as 1:5 (1 Part Cement: 5 Part Sand)
We have already discussed the cement mortar volume calculation in the plastering work post.
The above formula will give the dry volume of cement mortar, So to get wet concrete volume, we have to multiply the dry volume by 1.55
Total Parts = 1+5 =6
Required Volume of Cement = 1/6 x 1.55 (void ratio) x Mortar Volume = 1/6 x 1.55 x 0.2305= 0.0595 m3
To convert into cement bags – (0.0595 x 1440)/50 = 1.71 Bags or 85.68 Kg
Required Sand Volume = 5/6 x 1.55 (void ratio) x Mortar Volume = 5/6 x 1.55 x 0.2305 = 0.298 m3
## Result
Materials required for 1 m3 Brickwork is
• 500 Numbers of bricks
• 85.68 kg or 1.71 Bags of cement
• 0.298 m3 of sand
Make Use of this brickwork calculator
Hope you learned something from this post. Happy day 🙂
Bala is a Planning Engineer & he is the author and editor of Civil Planets.
#### 1 Comment
1. Shashank
For cement mortar the dry volume factor to be considered is 1.33 not 1.55
| 817 | 2,573 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.375 | 4 |
CC-MAIN-2024-33
|
latest
|
en
| 0.806934 |
https://www.physicsforums.com/threads/maximal-subgroups-of-solvable-groups-have-prime-power-index.594930/
| 1,511,062,091,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-47/segments/1510934805265.10/warc/CC-MAIN-20171119023719-20171119043719-00241.warc.gz
| 842,031,547 | 15,411 |
# Maximal subgroups of solvable groups have prime power index
1. Apr 9, 2012
### Barre
I would like to ask if somebody can verify the solution I wrote up to an exercise in my book. It's kind of long, but I have no one else to check it for me :)
1. The problem statement, all variables and given/known data
If $H$ is a maximal proper subgroup of a finite solvable group $G$, then $[G]$ is a prime power.
2. Relevant equations
Lemma 7.13 that I refer to is basicly this:
http://crazyproject.wordpress.com/2...-finite-solvable-group-is-elementary-abelian/
3. The attempt at a solution
Statement is true for abelian groups, so only nonabelian solvable groups are considered in the proof. All finite nonabelian solvable groups have at least one normal group (the commutator) and therefore contain a minimal normal subgroup. By Lemma 7.13 (iii), these subgroups have prime power order.
Assume there exists minimal normal subgroup $N$ such that $N \not\subseteq H$. Since $NH$ is a subgroup of $G$ properly containing $H$, $NH = G$. By Second Isomorphism Theorem, $G/N = HN/N \cong H/(N \cap H)$ and as all cardinalities are finite, $\frac{|G|}{|N|} = \frac{|H|}{|N \cap H|}$ which implies $\frac{|G|}{|H|} = [G] = \frac{|N|}{|N \cap H|}$, where the last one is a prime power.
Assume all minimal normal subgroups of $G$ are contained in $H$ and let $N$ be one such. We work by induction. If $|G| = 2$, then it is of prime order and the maximal proper subgroup $\langle e \rangle$ has index 2, certainly a prime power. In the quotient $G/N$, $H/N$ is maximal since if $H/N$ is properly contained in some subgroup $T/N$ then it follows that $H \subseteq T$, so $T = G$. Since order $|G/N|$ is strictly less than $|G|$, $[G/N/N] = \frac{[G/N]}{[H/N]} = \frac{[G:N]}{[H:N]} = p^n$ by induction. Also the identity $[G][H:N] = [G:N]$ implies $[G] = \frac{[G:N]}{[H:N]}$ so $[G] = p^n$.
2. Apr 9, 2012
### micromass
Staff Emeritus
Seems ok.
I also note that your proof also works in the abelian case, so there is no need to only look at nonabelian groups.
3. Apr 10, 2012
### Barre
Thanks for reading it and for the tip. An abelian group certainly has a normal minimal subgroup as long as it is not simple, and that is probably the case I could have handled separately.
| 679 | 2,269 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2017-47
|
longest
|
en
| 0.870411 |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-12-data-analysis-and-probability-12-7-thoretical-and-experimental-probability-practice-and-problem-solving-exercises-page-762/45
| 1,695,367,873,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-40/segments/1695233506339.10/warc/CC-MAIN-20230922070214-20230922100214-00397.warc.gz
| 897,090,312 | 13,910 |
## Algebra 1
Find the area of each region: White area=4 square units Red area=3 square units Blue area=6 square units Green area=3 square units Total area=16 square units P(events)=$\frac{favorableoutcomes}{possibleoutcomes}$ You have to find the ones which are not blue, so you subtract the total area and the blue area:16-6-10. The rest of the areas are 10 so it is the number of favorable outcomes. The area is 16 square units in total so this is the number of possible outcomes. P(blue)=$\frac{10}{16}$ -simplify- p(blue)=$\frac{5}{8}$
| 150 | 540 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.09375 | 4 |
CC-MAIN-2023-40
|
latest
|
en
| 0.763536 |
https://www.teacherspayteachers.com/Product/PDLs-Number-Building-Staircases-for-Cuisenaire-Rods-2737166?utm_source=addition%20guide%20blog%20post&utm_campaign=number%20building%20staircases
| 1,623,984,169,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-25/segments/1623487634616.65/warc/CC-MAIN-20210618013013-20210618043013-00211.warc.gz
| 925,850,765 | 32,248 |
DID YOU KNOW:
Seamlessly assign resources as digital activities
Learn how in 5 minutes with a tutorial resource. Try it Now
PDL's Number Building Staircases for Cuisenaire® Rods
PreK - 1st, Homeschool
Subjects
Resource Type
Formats Included
• Zip
Pages
124 pages
\$4.75
List Price:
\$12.50
You Save:
\$7.75
\$4.75
List Price:
\$12.50
You Save:
\$7.75
Description
Cuisenaire® Rods Staircases provide context for exploring common difference and describing quantities in relationship to other quantities. The following 10 activity mats and 100 worksheets are designed to emphasize these concepts.
Why common difference is Important?
Common difference is a strategy students may use to subtract quantities with greater ease. For example, which is easier to subtract 15 minus 9 or 16 -10. Both have the same common difference.
If students understanding that adding one to both numbers still creates the same answer, then students may avoid other strategies that are more complex.
This strategy continues for larger numbers. For example, take the problem 156 – 98. If the student adds 2 to both numbers, an easier problem is made available to the student. 158-100. Even for more complex problems like 1426 -679 can be solved using the same strategy. Adding 321 to both numbers creates the problem 1747 -1000.
Developing the student’s awareness for common difference prepares the student later to use this strategy for subtraction. You can see that understanding common difference carries the student a long way. It also provides for less error because regrouping is completely avoided.
Other topics explored
Many worksheets are provided to help create context to see the interconnectedness of multiplication, fractions and division. Students explore addition in the context of staircases as well to help them see the relationship between addition and subtraction.
The ability to read meaning into notation is a key factor of success in mathematics. Worksheets include both written word notation as well as symbol notation to help students read meaning into notation. For example, Five and four have a difference of one is the written word notation. 5 – 4 = 1.
Avoid Operating Language
Subtraction is what you do to a quantity. Difference is how you describe a relationship between two quantities. This entire packet avoids operation language like subtraction. Instead, the focus is on describing relationships of quantities to develop understanding of numeric behaviors.
We don’t go out into nature and manipulate (i.e. operate on) it without first learning to observe and describe the nature we see. Observing and describing quantities and their relationships develop student understanding on how quantities behave in relationship to each other.
When students observe basic behavior of quantities, they manipulate quantities with better understanding.
This product is included in the PDL's Interactive Math Notebook for Cuisenaire Rods
Check out these other activities:
(9 activities and 60 Worksheets)
(10 activities and 100 worksheets)
(28 activities and 10 worksheets)
(13 activities and 100 worksheets)
(20 activities and 80 student task cards)
(12 activities and 250 worksheets)
(10 activities and 50 worksheets)
Cuisenaire® Rods is a registered trademark of hand2mind, Inc. and is used with permission from hand2mind, Inc.
*Cuisenaire® Rods not included.
*IMPORTANT PRINTING NOTE* Some PDF software default the file to fit page. This changes the size of the activity so when you print, the cuisenaire rods will not fit within the desired area. Please change your settings so that they do not change the size of the page. This setting varies on your software. For Adobe, change the settings to NONE. For others, the setting should be changed to "use actual PDF dimensions for paper size."
Total Pages
124 pages
N/A
Teaching Duration
N/A
Report this Resource to TpT
Reported resources will be reviewed by our team. Report this resource to let us know if this resource violates TpT’s content guidelines.
| 844 | 4,031 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.03125 | 4 |
CC-MAIN-2021-25
|
latest
|
en
| 0.916179 |
http://perplexus.info/show.php?pid=6996&cid=45399
| 1,537,649,419,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-39/segments/1537267158691.56/warc/CC-MAIN-20180922201637-20180922222037-00507.warc.gz
| 189,675,934 | 5,230 |
All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
perplexus dot info
Find Fifty (Posted on 2010-11-06)
(I) N is a duodecimal (base 12) 50 digit positive integer of the form XX....Y, where X is repeated precisely 49 times followed by Y, such that N is divisible by the duodecimal number 137. Each of X and Y is a different duodecimal digit from 1 to B.
Determine the possible value(s) of N.
(II) P is a duodecimal (base 12) 50 digit positive integer of the form XX....Y, where X is repeated precisely 49 times followed by Y, such that P is divisible by the duodecimal number 147. Each of X and Y is a different duodecimal digit from 1 to B.
Determine the possible value(s) of P.
No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)
Comments: ( Back to comment list | You must be logged in to post comments.)
posible analytical solution Comment 3 of 3 |
Working in base 12 throughout, and for part I only:
1.137 = B*15 (both primes). 10^42 is 1 followed by 42 zeros; deducting 1 from that gives a string of 42 B's, which is naturally divisible by B, giving a string of 42 1's. Call that unwieldy number N1. N1 can now be multiplied by j = {1,2,...B} to provide the 'X' digits of N, with some number k between 0 and B being added or subtracted to generate Y (which is assumed to be different from X).
2. To satisfy the requirement of divisibility, jN1+/-k must be 0, mod B and also 0, mod 15. We know that 10^n = 1 mod B for every value of n. So N1 = 42 mod B which is congruent to 6 mod B; accordingly for (j=1,k=5), jN1+k = 0.
3.1 It follows from 2 by successive multiplication of j*k, and representation of their product, mod B, that the following numbers are congruent to 0, mod B:
{(j=1,k=5),(j=2, k=A),(j=3,k=4),(j=4,k=9),(j=5,k=3),(j=6,k=8),(j=7,k=2), (j=8,k=7), (j=9,k=1), (j=A,k=6),(j=B,k=0)}.
3.2 But where (j+k)>10, we must deduct rather than add, to avoid changing the 10's column:
(j=1,k=5),(j=2, k=A) write k=-1,(j=3,k=4),(j=4,k=9) write k=-2,(j=5,k=3),(j=6,k=8) write k=-3,(j=7,k=2), (j=8,k=7) write k=-4,(j=9,k=1), (j=A,k=6) write k=-5,(j=B,k=0).
3.3 This gives the last digit-pairs as{1..16,2..21,3..37,4..42,5..58,6..63,7..79,8..84,9..9A,A..A5,B..B0} so that these are the only numbers that require testing for divisibility by 15.
4. 10 and 15 are relatively prime,and 15 is prime, so that no power of 10 is divisible by 15, and checking the residues for small powers of 10 confirms that 10^n mod 15 will cycle through all values between 1 and 14 before repeating. Using the theorem a^(p-1) is congruent to 1(mod p) we can determine that 10^14 and hence also 10^40 leave a remainder of 1, mod 15. Since this is 3 complete cycles of values, pairing the residues Gauss-fashion informs us that a string of 40 1's is divisible by 15; that 41 1's leave a residue of 1, and hence that 42 1's leave a residue of (1+10) =11. We can now apply similar reasoning as in 3 to produce the pairs {(j=1,k=4),(j=2,k=8),(j=3,k=10)...etc} obtaining residues of {1,8,9,16,0,7,8,15,16,6,13)
5. It follows that only 5...58 is divisible by 137.
Edited on November 7, 2010, 8:22 am
Posted by broll on 2010-11-07 06:10:43
Search: Search body:
Forums (0)
| 1,085 | 3,211 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.59375 | 4 |
CC-MAIN-2018-39
|
longest
|
en
| 0.834052 |
https://www.aakash.ac.in/important-concepts/maths/geometric-tools
| 1,713,873,550,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-18/segments/1712296818474.95/warc/CC-MAIN-20240423095619-20240423125619-00100.warc.gz
| 562,475,889 | 31,031 |
• Call Now
1800-102-2727
•
# Geometric Tools
A ruler, often known as a straightedge, is used to draw straight lines and measure line segment lengths. A ruler, also known as a line gauge, is a mathematical instrument used in geometry and a variety of engineering applications. It's a straight edge that was originally solely used to create straight lines. It may also determine the object's distance. It is graded in centimeters and millimeters on one side and inches on the other for our convenience. It may be used to measure lengths in both metric and customary units. The “Hash Markings” are the intervals or marks specified in the ruler.
## Compass
It's a V-shaped gadget with a pointer on one side and a pencil on the other. It's possible to change the distance between the pencil and the pointer. Trace arcs, circles, and angles using it. It's also used to indicate equal lengths. A compass is a mathematical instrument that is used to design geometrical figures like circles. It's also used for the intersection of line segments and tools, which are used to intersect line segments and aid in the discovery of the forms' midpoints.
## Mathematical protractor
It's a semi-circular disc that's used to measure and draw angles. It has a graded range of 0 to 180 degrees and may be used to measure any angle that falls within that range. It contains two sets of marks, one from left to right and the other from right to left. The protractor's inner and outer readings complement each other. It indicates that the sum of the inner and outside readings equals 180 degrees.
## Divider
With its ‘V'-shaped structure, it resembles a compass. On both ends of the ‘V,' however, there are points. It's used to measure and compare lengths since the distance between them may be adjusted.
To mark the distance or designate divisions, the terms divider and compass are frequently employed. The main distinction between a compass and a divider is that a compass is a drafting device with one sharp point in the centre and a pen or pencil on the other point to delineate the circle or other functions. The divider, on the other hand, has two sharp points, one of which serves as a centre and the other as a marking tool.
## Set-Squares
Set squares are another mathematics tool that is often seen in a geometry box. These are the triangle plastic pieces that have had a part of the space between them eliminated. Set squares are offered in two varieties on the market. One has a 45-degree angle, while the other has a 30-60-degree angle. The 45-degree set square has a 90-degree angle, whereas the 30-degree and 60-degree set squares have a right angle. Vertical lines are drawn using the 45 degrees set square. Set squares can be used to create parallel lines, perpendicular lines, and standard angles, among other things
Talk to our expert
Resend OTP Timer =
By submitting up, I agree to receive all the Whatsapp communication on my registered number and Aakash terms and conditions and privacy policy
| 640 | 3,005 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.15625 | 4 |
CC-MAIN-2024-18
|
latest
|
en
| 0.952832 |
https://mathoverflow.net/questions/370748/multiplicities-of-the-laplacian-eigenvalues-of-a-graph
| 1,642,559,299,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-05/segments/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00623.warc.gz
| 449,461,984 | 25,929 |
# Multiplicities of the Laplacian eigenvalues of a graph
Let $$\lambda^G_1 > \lambda^G_2 > \dots$$ be the eigenvalues of the Laplacian matrix $$G$$ of a graph on $$n$$ vertices.
Let $$\mu(G)$$ be the composition $$a_1,\dots,a_k$$ of $$n$$ where $$a_i$$ is the multiplicity of $$\lambda^G_i$$.
Is $$\mu$$ surjective as a map from (finite) simple graphs to integer compositions?
• It barely works for $n=3$! Sep 3 '20 at 14:00
We can actually construct a graph with the desired composition of multiplicities by adding isolated vertices and taking complements:
As an initial remark, note that the smallest Laplacian eigenvalue of a graph $$G$$ is $$0$$ and that its multiplicity is $$1$$ if and only if $$G$$ is connected.
Let $$(a_1,\dots,a_k)$$ be the desired composition. If $$a_k > 1$$, we can choose a graph $$G$$ with $$\mu(G)=(a_1,\dots,a_{k-1},1)$$ and add $$a_k-1$$ isolated vertices.
If $$a_k = 1$$, choose a graph $$G$$ with $$\mu(G) = (a_{k-1},\dots,a_2, a_1+1)$$. Then the complement of $$G$$ has the desired multiplicities.
• (For people like me who are a bit slow, regarding the last paragraph: since $a_1+1 > 1$, the graph $G$ is disconnected; hence its complement is connected; hence $\lambda_1^{G} < n$; and so indeed the complement of $G$ has the desired $\mu$.) Sep 3 '20 at 16:00
• I believe it also follows from your argument that to get all the compositions we only need to consider threshold graphs: en.wikipedia.org/wiki/Threshold_graph Sep 3 '20 at 17:29
• Note, interestingly, that the number of isomorphism classes of threshold graphs on $n$ vertices is $2^{n-1}$, the same as the number of compositions of $n$. I guess your construction gives a bijection between these sets. Sep 3 '20 at 17:34
• This reminds me of "The critical group of a threshold graph" Hans Christianson and Victor Reiner. Connection? Sep 3 '20 at 18:42
• Theorem 5 of that paper (doi.org/10.1016/S0024-3795(02)00252-5) says "The eigenvalues of L(G) for G a threshold graph are the column lengths of the Ferrers diagram of the degree sequence of G," which is directly related to your eigenvalues multiplicity question. They attribute that result to Merris (doi.org/10.1016/0024-3795(94)90361-1). But I don't know if there's a direct connection to critical groups. Sep 3 '20 at 20:20
| 673 | 2,288 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.734375 | 4 |
CC-MAIN-2022-05
|
latest
|
en
| 0.852127 |
https://sciencing.com/measure-flagpole-6117722.html
| 1,719,062,900,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-26/segments/1718198862396.85/warc/CC-MAIN-20240622112740-20240622142740-00646.warc.gz
| 450,498,136 | 86,917 |
# How to Measure a Flagpole
••• Stars and stripes USA flag on a flagpole image by Steve Johnson from Fotolia.com
Print
You can easily measure the height of a flagpole without having to climb it by using the rule of similar triangles. The idea is that if two triangles have the same three angles, then the ratio between the sides’ lengths is the same between triangles as well. For example, if two triangles have angles 45, 45 and 90 degrees, then the two sides other than the hypotenuse are equal in each triangle.
Measure the length of the shadow cast by the flagpole on a sunny day. Use a yardstick or meter stick to do this. Denote the length with the letter S, for “shadow.”
Plant a stick vertically in the ground near the flagpole. Measure the length of its shadow and denote it with the lowercase letter s, to stand for smaller shadow.
Measure the stick’s vertical height. Denote it with the letter h.
Calculate the flagpole’s height, H, using the formula H/S = h/s. In other words, H = h(S/s).
For example, if S is 15 feet, h is 4 feet (perhaps you used a yardstick as the stick) and s is 3 feet. Then H is 4*(15/3) = 20 feet tall. This is the height of the flagpole.
#### Things You'll Need
• Measuring tape
• Stick
• Yardstick or meter stick
#### Tips
• The straighter, taller and more vertical the stick, the better will be the accuracy of your h and s measurement. You can use a level to make sure the stick is vertical.
| 354 | 1,442 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.703125 | 4 |
CC-MAIN-2024-26
|
latest
|
en
| 0.907306 |
https://www.physicsforums.com/threads/power-efficiency-type-question.197266/
| 1,508,563,464,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-43/segments/1508187824570.79/warc/CC-MAIN-20171021043111-20171021063111-00775.warc.gz
| 965,147,562 | 14,859 |
# Power Efficiency Type Question
1. Nov 10, 2007
### Kobayashi
A non-ideal voltage source supplies current of 250mA to load resistance of 3.3kOhms.
What is the magnitude of the source voltage if the power efficiency is 94%.
Also calculate the value of the additional resistance which must be added to the circuit to limit the current drawn from the supply to 200mA and calculate the new power efficiency (PL/PS).
So far I think I have worked out the supply voltage:
V=IR, (250mA)*(3300)Ohms, To get 825V
(825/94%)*100=877.6V
How to I carry on with this problem to find internal resistance and the new power effieciency. Please help. Thanks.
2. Nov 10, 2007
### dontharmanimals
internal resistance = (877.6 - 825 ) / 250 = 0.21 K =210 ohms
3. Nov 10, 2007
### dlgoff
"So far I think I have worked out the supply voltage:
V=IR, (250mA)*(3300)Ohms, To get 825V"
In your equation, what units should I and R be inorder to get V in volts?
Now what resistance would you need to add to get I=200mA?
4. Nov 11, 2007
### dontharmanimals
If current I is in milliamperes, resistance R must be in kilo-ohms to get Volts.
or
If current I is in amperes, resistance R must be in ohms to get Volts.
Total resistance R, required in the circuit to limit current to 200 mA is given by R = V / I.
R ( total ) = 877.6 volts / 200 mA = 4.388 kilo-ohms = 4388 ohms.
Additional resistance required = R ( total ) - internal resistance - resistance already
available in the circuit
= 4388 - 210 - 3300 = 878 ohms
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
| 468 | 1,606 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.03125 | 4 |
CC-MAIN-2017-43
|
longest
|
en
| 0.89499 |
http://research.omicsgroup.org/index.php/Central_limit_theorem
| 1,550,700,591,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-09/segments/1550247496694.82/warc/CC-MAIN-20190220210649-20190220232649-00456.warc.gz
| 227,810,692 | 35,840 |
# Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.[1][2] That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a "bell curve").
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.
In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.[3]
## Central limit theorems for independent sequences
File:Central limit thm.png
A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations; see Illustration of the central limit theorem for further details.
### Classical CLT
Let {X1, ..., Xn} be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ2. Suppose we are interested in the sample average
$S_n := \frac{X_1+\cdots+X_n}{n}$
of these random variables. By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n → ∞. The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More precisely, it states that as n gets larger, the distribution of the difference between the sample average Sn and its limit µ, when multiplied by the factor n (that is n(Sn − µ)), approximates the normal distribution with mean 0 and variance σ2. For large enough n, the distribution of Sn is close to the normal distribution with mean µ and variance σ2/n. The usefulness of the theorem is that the distribution of n(Sn − µ) approaches normality regardless of the shape of the distribution of the individual Xi’s. Formally, the theorem can be stated as follows:
Lindeberg–Lévy CLT. Suppose {X1, X2, ...} is a sequence of i.i.d. random variables with E[Xi] = µ and Var[Xi] = σ2 < ∞. Then as n approaches infinity, the random variables n(Sn − µ) converge in distribution to a normal N(0, σ2):[4]
$\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg) - \mu\bigg)\ \xrightarrow{d}\ N(0,\;\sigma^2).$
In the case σ > 0, convergence in distribution means that the cumulative distribution functions of n(Sn − µ) converge pointwise to the cdf of the N(0, σ2) distribution: for every real number z,
$\lim_{n\to\infty} \Pr[\sqrt{n}(S_n-\mu) \le z] = \Phi(z/\sigma),$
where Φ(x) is the standard normal cdf evaluated at x. Note that the convergence is uniform in z in the sense that
$\lim_{n\to\infty}\sup_{z\in{\mathbf R}}\bigl|\Pr[\sqrt{n}(S_n-\mu) \le z] - \Phi(z/\sigma)\bigr| = 0,$
where sup denotes the least upper bound (or supremum) of the set.[5]
### Lyapunov CLT
The theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the central limit theorem the random variables Xi have to be independent, but not necessarily identically distributed. The theorem also requires that random variables |Xi| have moments of some order (2 + δ), and that the rate of growth of these moments is limited by the Lyapunov condition given below.
Lyapunov CLT.[6] Suppose {X1, X2, ...} is a sequence of independent random variables, each with finite expected value μi and variance σ 2
i
. Define
$s_n^2 = \sum_{i=1}^n \sigma_i^2$
If for some δ > 0, the Lyapunov’s condition
$\lim_{n\to\infty} \frac{1}{s_{n}^{2+\delta}} \sum_{i=1}^{n} \operatorname{E}\big[\,|X_{i} - \mu_{i}|^{2+\delta}\,\big] = 0$
is satisfied, then a sum of (Xi − μi)/sn converges in distribution to a standard normal random variable, as n goes to infinity:
$\frac{1}{s_n} \sum_{i=1}^{n} (X_i - \mu_i) \ \xrightarrow{d}\ \mathcal{N}(0,\;1).$
In practice it is usually easiest to check the Lyapunov’s condition for δ = 1. If a sequence of random variables satisfies Lyapunov’s condition, then it also satisfies Lindeberg’s condition. The converse implication, however, does not hold.
### Lindeberg CLT
Main article: Lindeberg's condition
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920).
Suppose that for every ε > 0
$\lim_{n \to \infty} \frac{1}{s_n^2}\sum_{i = 1}^{n} \operatorname{E}\big[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\{ | X_i - \mu_i | > \varepsilon s_n \}} \big] = 0$
where 1{...} is the indicator function. Then the distribution of the standardized sums $\frac{1}{s_n}\sum_{i = 1}^n \left( X_i - \mu_i \right)$ converges towards the standard normal distribution N(0,1).
### Multidimensional CLT
Proofs that use characteristic functions can be extended to cases where each individual Xi is a random vector in Rk, with mean vector μ = E(Xi) and covariance matrix Σ (amongst the components of the vector), and these random vectors are independent and identically distributed. Summation of these vectors is being done componentwise. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution.[7]
Let
$\mathbf{X_i}=\begin{bmatrix} X_{i(1)} \\ \vdots \\ X_{i(k)} \end{bmatrix}$
be the k-vector. The bold in Xi means that it is a random vector, not a random (univariate) variable. Then the sum of the random vectors will be
$\begin{bmatrix} X_{1(1)} \\ \vdots \\ X_{1(k)} \end{bmatrix}+\begin{bmatrix} X_{2(1)} \\ \vdots \\ X_{2(k)} \end{bmatrix}+\cdots+\begin{bmatrix} X_{n(1)} \\ \vdots \\ X_{n(k)} \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^{n} \left [ X_{i(1)} \right ] \\ \vdots \\ \sum_{i=1}^{n} \left [ X_{i(k)} \right ] \end{bmatrix} = \sum_{i=1}^{n} \mathbf{X_i}$
and the average is
$\frac{1}{n} \sum_{i=1}^{n} \mathbf{X_i}= \frac{1}{n}\begin{bmatrix} \sum_{i=1}^{n} X_{i(1)} \\ \vdots \\ \sum_{i=1}^{n} X_{i(k)} \end{bmatrix} = \begin{bmatrix} \bar X_{i(1)} \\ \vdots \\ \bar X_{i(k)} \end{bmatrix}=\mathbf{\bar X_n}$
and therefore
$\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left [\mathbf{X_i} - E\left ( X_i\right ) \right ]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n} ( \mathbf{X_i} - \mu ) = \sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right)$.
The multivariate central limit theorem states that
$\sqrt{n}\left(\mathbf{\overline{X}}_n - \mu\right)\ \stackrel{D}{\rightarrow}\ \mathcal{N}_k(0,\Sigma)$
where the covariance matrix Σ is equal to
$\Sigma=\begin{bmatrix} {\operatorname{Var} \left (X_{1(1)} \right)} & \operatorname{Cov} \left (X_{1(1)},X_{1(2)} \right) & \operatorname{Cov} \left (X_{1(1)},X_{1(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1(1)},X_{1(k)} \right) \\ \operatorname{Cov} \left (X_{1(2)},X_{1(1)} \right) & \operatorname{Var} \left (X_{1(2)} \right) & \operatorname{Cov} \left(X_{1(2)},X_{1(3)} \right) & \cdots & \operatorname{Cov} \left(X_{1(2)},X_{1(k)} \right) \\ \operatorname{Cov}\left (X_{1(3)},X_{1(1)} \right) & \operatorname{Cov} \left (X_{1(3)},X_{1(2)} \right) & \operatorname{Var} \left (X_{1(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1(3)},X_{1(k)} \right) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \operatorname{Cov} \left (X_{1(k)},X_{1(1)} \right) & \operatorname{Cov} \left (X_{1(k)},X_{1(2)} \right) & \operatorname{Cov} \left (X_{1(k)},X_{1(3)} \right) & \cdots & \operatorname{Var} \left (X_{1(k)} \right) \\ \end{bmatrix}.$
## Central limit theorems for dependent processes
### CLT under weak dependence
A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called α-mixing) defined by α(n) → 0 where α(n) is so-called strong mixing coefficient.
A simplified formulation of the central limit theorem under strong mixing is:[8]
Theorem. Suppose that X1, X2, ... is stationary and α-mixing with αn = O(n−5) and that E(Xn) = 0 and E(Xn2) < ∞. Denote Sn = X1 + ... + Xn, then the limit
$\sigma^2 = \lim_n \frac{E(S_n^2)}{n}$
exists, and if σ ≠ 0 then $S_n / (\sigma \sqrt n)$ converges in distribution to N(0, 1).
In fact,
$\sigma^2 = E(X_1^2) + 2 \sum_{k=1}^{\infty} E(X_1 X_{1+k}),$
where the series converges absolutely.
The assumption σ ≠ 0 cannot be omitted, since the asymptotic normality fails for Xn = YnYn−1 where Yn are another stationary sequence.
There is a stronger version of the theorem:[9] the assumption E(Xn12) < ∞ is replaced with E(#REDIRECTmw:Help:Magic words#Other
and the assumption αn = O(n−5) is replaced with $\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.$ Existence of such δ > 0 ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (Bradley 2005).
### Martingale difference CLT
Theorem. Let a martingale Mn satisfy
• $\frac1n \sum_{k=1}^n \mathrm{E} ((M_k-M_{k-1})^2 | M_1,\dots,M_{k-1}) \to 1$ in probability as n tends to infinity,
• for every ε > 0, $\frac1n \sum_{k=1}^n \mathrm{E} \Big( (M_k-M_{k-1})^2; |M_k-M_{k-1}| > \varepsilon \sqrt n \Big) \to 0$ as n tends to infinity,
then $M_n / \sqrt n$ converges in distribution to N(0,1) as n → ∞.[10][11]
Caution: The restricted expectation[clarification needed] E(X; A) should not be confused with the conditional expectation E(X #REDIRECTmw:Help:Magic words#Other
## Remarks
### Proof of classical CLT
For a theorem of such fundamental importance to statistics and applied probability, the central limit theorem has a remarkably simple proof using characteristic functions. It is similar to the proof of a (weak) law of large numbers. For any random variable, Y, with zero mean and a unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,
$\varphi_Y(t) = 1 - {t^2 \over 2} + o(t^2), \quad t \rightarrow 0$
where o (t2) is "little o notation" for some function of t that goes to zero more rapidly than t2.
Letting Yi be (Xi − μ)/σ, the standardized value of Xi, it is easy to see that the standardized mean of the observations X1, X2, ..., Xn is
$Z_n = \frac{n\overline{X}_n-n\mu}{\sigma \sqrt{n}} =\sum_{i=1}^n {Y_i \over \sqrt{n}}$
By simple properties of characteristic functions, the characteristic function of the sum is:
\begin{align} \varphi_{Z_n} & =\varphi_{\sum_{i=1}^n {Y_i \over \sqrt{n}}}\left(t\right) = \varphi_{Y_1} \left(t / \sqrt{n} \right) \cdot \varphi_{Y_2} \left(t / \sqrt{n} \right)\cdots \varphi_{Y_n} \left(t / \sqrt{n} \right) \\[8pt] & = \left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n \end{align}
so that, by the limit of the exponential function ( ex= lim(1 + x/n)n) the characteristic function of Zn is
$\left[\varphi_Y\left({t \over \sqrt{n}}\right)\right]^n = \left[ 1 - {t^2 \over 2n} + o\left({t^2 \over n}\right) \right]^n \, \rightarrow \, e^{-t^2/2}, \quad n \rightarrow \infty.$
But this limit is just the characteristic function of a standard normal distribution N(0, 1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.
### Convergence to the limit
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
If the third central moment E((X1 − μ)3) exists and is finite, then the above convergence is uniform and the speed of convergence is at least on the order of 1/n1/2 (see Berry-Esseen theorem). Stein's method[12] can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.[13]
The convergence to the normal distribution is monotonic, in the sense that the entropy of Zn increases monotonically to that of the normal distribution.[14]
The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realisations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity, this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.
### Relation to the law of large numbers
The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behaviour of Sn as n approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions.
Suppose we have an asymptotic expansion of f(n):
$f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O(\varphi_{3}(n)) \qquad (n \rightarrow \infty).$
Dividing both parts by φ1(n) and taking the limit will produce a1, the coefficient of the highest-order term in the expansion, which represents the rate at which f(n) changes in its leading term.
$\lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_1.$
Informally, one can say: "f(n) grows approximately as a1 φ1(n)". Taking the difference between f(n) and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about f(n):
$\lim_{n\to\infty}\frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)}=a_2 .$
Here one can say that the difference between the function and its approximation grows approximately as a2 φ2(n). The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.
Informally, something along these lines is happening when the sum, Sn, of independent identically distributed random variables, X1, ..., Xn, is studied in classical probability theory.[citation needed] If each Xi has finite mean μ, then by the law of large numbers, Sn/n → μ.[15] If in addition each Xi has finite variance σ2, then by the central limit theorem,
$\frac{S_n-n\mu}{\sqrt{n}} \rightarrow \xi ,$
where ξ is distributed as N(0, σ2). This provides values of the first two constants in the informal expansion
$S_n \approx \mu n+\xi \sqrt{n}. \,$
In the case where the Xi's do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:
$\frac{S_n-a_n}{b_n} \rightarrow \Xi,$
or informally
$S_n \approx a_n+\Xi b_n. \,$
Distributions Ξ which can arise in this way are called stable.[16] Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor bn may be proportional to nc, for any c ≥ 1/2; it may also be multiplied by a slowly varying function of n.[17][18]
The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function $\sqrt{n\log\log n}$ intermediate in size between n of the law of large numbers and √n of the central limit theorem provides a non-trivial limiting behavior.
### Alternative statements of the theorem
#### Density functions
The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov[19] for a particular local limit theorem for sums of independent and identically distributed random variables.
#### Characteristic functions
Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. However, to state this more precisely, an appropriate scaling factor needs to be applied to the argument of the characteristic function.
An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.
## Extensions to the theorem
### Products of positive random variables
The logarithm of a product is simply the sum of the logarithms of the factors. Therefore when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution.
Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.[20]
## Beyond the classical framework
Asymptotic normality, that is, convergence to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.
### Convex body
Theorem. There exists a sequence εn ↓ 0 for which the following holds. Let n ≥ 1, and let random variables X1, ..., Xn have a log-concave joint density f such that f(x1, ..., xn) = f(#REDIRECTmw:Help:Magic words#Other
for all x1, ..., xn, and E(Xk2) = 1 for all k = 1, ..., n. Then the distribution of
$\frac{X_1+\cdots+X_n}{\sqrt n}$
is εn-close to N(0, 1) in the total variation distance.[21]
These two εn-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.
An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Another example: f(x1, …, xn) = const · exp( − (#REDIRECTmw:Help:Magic words#Other
where α > 1 and αβ > 1. If β = 1 then f(x1, …, xn) factorizes into const · exp ( − #REDIRECTmw:Help:Magic words#Other
which means independence of X1, …, Xn. In general, however, they are dependent.
The condition f(x1, …, xn) = f(#REDIRECTmw:Help:Magic words#Other
ensures that X1, …, Xn are of zero mean and uncorrelated;[citation needed] still, they need not be independent, nor even pairwise independent.[citation needed] By the way, pairwise independence cannot replace independence in the classical central limit theorem.[22]
Here is a Berry–Esseen type result.
Theorem. Let X1, …, Xn satisfy the assumptions of the previous theorem, then [23]
$\bigg| \mathbb{P} \Big( a \le \frac{ X_1+\cdots+X_n }{ \sqrt n } \le b \Big) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le \frac C n$
for all a < b; here C is a universal (absolute) constant. Moreover, for every c1, …, cn ∈ R such that c12 + … + cn2 = 1,
$\bigg| \mathbb{P} ( a \le c_1 X_1+\cdots+c_n X_n \le b ) - \frac1{\sqrt{2\pi}} \int_a^b \mathrm{e}^{-t^2/2} \, \mathrm{d} t \bigg| \le C ( c_1^4+\dots+c_n^4 ).$
The distribution of $(X_1+\cdots+X_n)/\sqrt n$ need not be approximately normal (in fact, it can be uniform).[24] However, the distribution of c1X1 + … + cnXn is close to N(0, 1) (in the total variation distance) for most of vectors (c1, …, cn) according to the uniform distribution on the sphere c12 + … + cn2 = 1.
### Lacunary trigonometric series
Theorem (SalemZygmund). Let U be a random variable distributed uniformly on (0, 2π), and Xk = rk cos(nkU + ak), where
• nk satisfy the lacunarity condition: there exists q > 1 such that nk+1qnk for all k,
• rk are such that
$r_1^2 + r_2^2 + \cdots = \infty \text{ and } \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0,$
• 0 ≤ ak < 2π.
Then[25][26]
$\frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} }$
converges in distribution to N(0, 1/2).
### Gaussian polytopes
Theorem Let A1, ..., An be independent random points on the plane R2 each having the two-dimensional standard normal distribution. Let Kn be the convex hull of these points, and Xn the area of Kn Then[27]
$\frac{ X_n - \mathrm{E} (X_n) }{ \sqrt{\operatorname{Var} (X_n)} }$
converges in distribution to N(0, 1) as n tends to infinity.
The same holds in all dimensions (2, 3, ...).
The polytope Kn is called Gaussian random polytope.
A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.[28]
### Linear functions of orthogonal matrices
A linear function of a matrix M is a linear combination of its elements (with given coefficients), M ↦ tr(AM) where A is the matrix of the coefficients; see Trace (linear algebra)#Inner product.
A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group O(n, R); see Rotation matrix#Uniform random rotation matrices.
Theorem. Let M be a random orthogonal n × n matrix distributed uniformly, and A a fixed n × n matrix such that tr(AA*) = n, and let X = tr(AM). Then[29] the distribution of X is close to N(0, 1) in the total variation metric up to[clarification needed] 23/(n − 1).
### Subsequences
Theorem. Let random variables X1, X2, … ∈ L2(Ω) be such that Xn → 0 weakly in L2(Ω) and Xn2 → 1 weakly in L1(Ω). Then there exist integers n1 < n2 < … such that $( X_{n_1}+\cdots+X_{n_k} ) / \sqrt k$ converges in distribution to N(0, 1) as k tends to infinity.[30]
### Tsallis statistics
A generalization of the classical central limit theorem to the context of Tsallis statistics has been described by Umarov, Tsallis and Steinberg[31] in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1. In analogy to the classical central limit theorem, such random variables with fixed mean and variance tend towards the q-Gaussian distribution, which maximizes the Tsallis entropy under these constraints. Umarov, Tsallis, Gell-Mann and Steinberg have defined similar generalizations of all symmetric alpha-stable distributions, and have formulated a number of conjectures regarding their relevance to an even more general central limit theorem.[32]
### Random walk on a crystal lattice
The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures. [33][34]
## Applications and examples
### Simple example
File:Dice sum central limit theorem.svg
Comparison of probability density functions, p(k) for the sum of n fair 6-sided dice to show their convergence to a normal distribution with increasing n, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
File:Central Limit Theorem.png
Another simulation with binomial distribution. 0 and 1 were generated and their means calculated for different sample sizes, from 1 to 512. It's possible to see that when the sample increases, the mean distribution tend to be more centered and with thinner tails.
A simple example of the central limit theorem is rolling a large number of identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.
File:Empirical CLT - Figure - 040711.jpg
This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the chi-squared values that quantify the goodness of the fit (the fit is good if the reduced chi-squared value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/n), which is called the standard deviation of the mean (since it refers to the spread of sample means).
### Real applications
File:UsaccHistogram.svg
A histogram plot of monthly accidental deaths in the US, between 1973 and 1978 exhibits normality, due to the central limit theorem
Published literature contains a number of useful and interesting examples and applications relating to the central limit theorem.[35] One source[36] states the following examples:
• The probability distribution for total distance covered in a random walk (biased or unbiased) will tend toward a normal distribution.
• Flipping a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).
From another viewpoint, the central limit theorem explains the common appearance of the "Bell Curve" in density estimates applied to real world data. In cases like electronic noise, examination grades, and so on, we can often regard a single measured value as the weighted average of a large number of small effects. Using generalisations of the central limit theorem, we can then see that this would often (though not always) produce a final distribution that is approximately normal.
In general, the more a measurement is like the sum of independent variables with equal influence on the result, the more normality it exhibits. This justifies the common use of this distribution to stand in for the effects of unobserved variables in models like the linear model.
## Regression
Regression analysis and in particular ordinary least squares specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of a large number of independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be assumed to be normally distributed.
### Other illustrations
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.[37]
## History
Tijms writes:[38]
The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie analytique des probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.
Sir Francis Galton described the Central Limit Theorem as:[39]
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The larger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper.[40][41] Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word central in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".[41] The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya[40] in 1920 translates as follows.
The occurrence of the Gaussian probability density 1 = ex2 in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by Liapounoff. [...]
A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald.[42] Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by von Mises, Pólya, Lindeberg, Lévy, and Cramér during the 1920s, are given by Hans Fischer.[43] Le Cam describes a period around 1935.[41] Bernstein[44] presents a historical discussion focusing on the work of Pafnuty Chebyshev and his students Andrey Markov and Aleksandr Lyapunov that led to the first proofs of the CLT in a general setting.
A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was never published.[45][46][47]
## Notes
1. ^ http://www.math.uah.edu/stat/sample/CLT.html
2. ^ Rice, John (1995), Mathematical Statistics and Data Analysis (Second ed.), Duxbury Press, ISBN 0-534-20934-3)[page needed]
3. ^ Voit, Johannes (2003), The Statistical Mechanics of Financial Markets, Springer-Verlag, p. 124, ISBN 3-540-00978-7
4. ^ Billingsley (1995, p. 357)
5. ^ Bauer (2001, Theorem 30.13, p.199)
6. ^ Billingsley (1995, p.362)
7. ^ Van der Vaart, A. W. (1998), Asymptotic statistics, New York: Cambridge University Press, ISBN 978-0-521-49603-2, LCCN 98015176
8. ^ Billingsley (1995, Theorem 27.4)
9. ^ Durrett (2004, Sect. 7.7(c), Theorem 7.8)
10. ^ Durrett (2004, Sect. 7.7, Theorem 7.4)
11. ^ Billingsley (1995, Theorem 35.12)
12. ^ Stein, C. (1972), "A bound for the error in the normal approximation to the distribution of a sum of dependent random variables", Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability: 583–602, MR 402873, Zbl 0278.60026
13. ^ Chen, L.H.Y., Goldstein, L., and Shao, Q.M (2011), Normal approximation by Stein's method, Springer, ISBN 978-3-642-15006-7
14. ^ Artstein, S.; Ball, K.; Barthe, F.; Naor, A. (2004), "Solution of Shannon's Problem on the Monotonicity of Entropy", Journal of the American Mathematical Society 17 (4): 975–982, doi:10.1090/S0894-0347-04-00459-X
15. ^ Rosenthal, Jeffrey Seth (2000) A first look at rigorous probability theory, World Scientific, ISBN 981-02-4322-7.(Theorem 5.3.4, p. 47)
16. ^ Johnson, Oliver Thomas (2004) Information theory and the central limit theorem, Imperial College Press, 2004, ISBN 1-86094-473-6. (p. 88)
17. ^ Vladimir V. Uchaikin and V. M. Zolotarev (1999) Chance and stability: stable distributions and their applications, VSP. ISBN 90-6764-301-7.(pp. 61–62)
18. ^ Borodin, A. N. ; Ibragimov, Il'dar Abdulovich; Sudakov, V. N. (1995) Limit theorems for functionals of random walks, AMS Bookstore, ISBN 0-8218-0438-3. (Theorem 1.1, p. 8 )
19. ^ Petrov, V.V. (1976), "7", Sums of Independent Random Variables, New York-Heidelberg: Springer-Verlag
20. ^ Rempala, G.; Wesolowski, J. (2002). "Asymptotics of products of sums and U-statistics" (PDF). Electronic Communications in Probability 7: 47–54. doi:10.1214/ecp.v7-1046.
21. ^ Klartag (2007, Theorem 1.2)
22. ^ Durrett (2004, Section 2.4, Example 4.5)
23. ^ Klartag (2008, Theorem 1)
24. ^ Klartag (2007, Theorem 1.1)
25. ^ Zygmund, Antoni (1959), Trigonometric series, Volume II, Cambridge. (2003 combined volume I,II: ISBN 0-521-89053-5) (Sect. XVI.5, Theorem 5-5)
26. ^ Gaposhkin (1966, Theorem 2.1.13)
27. ^ Bárány & Vu (2007, Theorem 1.1)
28. ^ Bárány & Vu (2007, Theorem 1.2)
29. ^ Meckes, Elizabeth (2008), "Linear functions on the classical matrix groups", Transactions of the American Mathematical Society 360 (10): 5355–5366, arXiv:math/0509441, doi:10.1090/S0002-9947-08-04444-9
30. ^ Gaposhkin (1966, Sect. 1.5)
31. ^ Umarov, Sabir; Tsallis, Constantino and Steinberg, Stanly (2008), "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF), Milan j. Math. (Birkhauser Verlag) 76: 307–328, doi:10.1007/s00032-008-0087-y, retrieved 2011-07-27.
32. ^ Umarov, Sabir; Tsallis, Constantino, Gell-Mann, Murray and Steinberg, Stanly (2010), "Generalization of symmetric α-stable Lévy distributions for q > 1", J Math Phys. (American Institute of Physics) 51 (3): 033502, PMC 2869267, PMID 20596232, doi:10.1063/1.3305292.
33. ^ Kotani, M.; Sunada, T (2003), Spectral geometry of crystal lattices, Contemporary Math., 338, 271–305.
34. ^ Sunada T. (2012), Topological Crystallography ---With a View Towards Discrete Geometric Analysis---", Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer
35. ^ Dinov, Christou & Sanchez (2008)
36. ^
37. ^ Marasinghe, M., Meeker, W., Cook, D. & Shin, T.S.(1994 August), "Using graphics and simulation to teach statistical concepts", Paper presented at the Annual meeting of the American Statistician Association, Toronto, Canada.
38. ^ Henk, Tijms (2004), Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press, p. 169, ISBN 0-521-54036-4
39. ^ Galton F. (1889) Natural Inheritance , p. 66
40. ^ a b
41. ^ a b c Le Cam, Lucien (1986), "The central limit theorem around 1935", Statistical Science 1 (1): 78–91, doi:10.2307/2245503
42. ^ Hald, Andreas A History of Mathematical Statistics from 1750 to 1930, Ch.17.[full citation needed]
43. ^ Fischer, Hans (2011), A History of the Central Limit Theorem: From Classical to Modern Probability Theory, Sources and Studies in the History of Mathematics and Physical Sciences, New York: Springer, ISBN 978-0-387-87856-0, MR 2743162, Zbl 1226.60004, doi:10.1007/978-0-387-87857-7 (Chapter 2: The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in Analytical Methods, Chapter 5.2: The Central Limit Theorem in the Twenties)
44. ^ Bernstein, S.N. (1945) On the work of P.L.Chebyshev in Probability Theory, Nauchnoe Nasledie P.L.Chebysheva. Vypusk Pervyi: Matematika. (Russian) [The Scientific Legacy of P. L. Chebyshev. First Part: Mathematics, Edited by S. N. Bernstein.] Academiya Nauk SSSR, Moscow-Leningrad, 174 pp.
45. ^ Hodges, Andrew (1983) Alan Turing: the enigma. London: Burnett Books., pp. 87-88.[full citation needed]
46. ^ Zabell, S.L. (2005) Symmetry and its discontents: essays on the history of inductive probability, Cambridge University Press. ISBN 0-521-44470-5. (pp. 199 ff.)
47. ^ Aldrich, John (2009) "England and Continental Probability in the Inter-War Years", Electronic Journ@l for History of Probability and Statistics, vol. 5/2, Decembre 2009. (Section 3)
48. ^ Jørgensen, Bent (1997). The theory of dispersion models. Chapman & Hall. ISBN 978-0412997112.
| 10,630 | 39,773 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2019-09
|
longest
|
en
| 0.915823 |
https://www.oreilly.com/library/view/fundamentals-of-technical/9780128020166/XHTML/B9780128019870000022/B9780128019870000022_a.xhtml
| 1,579,844,576,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-05/segments/1579250615407.46/warc/CC-MAIN-20200124040939-20200124065939-00273.warc.gz
| 1,015,223,663 | 21,215 |
## 2.3. Ratio, proportion, and variation
### 2.3.1. Ratio
A ratio (:) is the relation between two like numbers. For example, the ratio 5:7 can be written as a fraction, 5/7, or as division 5 ÷ 7. We read the ratio 5:7 as 5 to 7 or 5 per 7.
Definition of ratio
A ratio is a comparison of two numbers.
A ratio can be expressed by a fraction $(ab)$ or by a colon (:). So, the ratio of two numbers a and b can be written as a:b which is the same as the fraction $ab$, where b 0.
##### Example 1
Reduce the ratio 20:35 to the lowest term.
20:35 = 4/7.
##### Example 2
Reduce the ratio ...
Get Fundamentals of Technical Mathematics now with O’Reilly online learning.
O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.
| 225 | 777 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.765625 | 4 |
CC-MAIN-2020-05
|
longest
|
en
| 0.905989 |
http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/patrick2.html
| 1,675,866,623,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-06/segments/1674764500813.58/warc/CC-MAIN-20230208123621-20230208153621-00037.warc.gz
| 25,941,803 | 3,365 |
SEARCH HOME
Math Central Quandaries & Queries
Question from patrick: We have eight people, playing three rounds of golf. Would like to know the optimal pairings so that the most people can play with everyone. Thanks.
Well, everybody plays 3 times with 3 others [= 9 play-withs] and there are only 7 other players, so everybody will have to play two repeats.
An easy solution is
ABCD EFGH
ABGH EFCD
ABEF CDGH
This has the flaw that each person plays _three_ times with one other. But we can't avoid this, if everybody is to play with everybody else!
There cannot be three people who play together in both round 1 and round 2. Suppose there were:
ABCD EFGH
ABCH EFGD
Then none of ABC would have played with any of EFG; for round 3 to provide all the missing pairings for A, A would have to be in the same foursome as EFG in round 3. But the same is true of B, which is impossible.
No player can play with all different people in round 1 and round 2. If A plays with BCD in round 1 and with none of them in round
2, BCD play together in both rounds and we've shown this can't happen.
Let B be the player who plays with A in both rounds. We conclude that round 2 is (up to relabelling)
ABCD EFGH
ABGH EFCD
Now A and B must both play with E and F in round 3 and the solution is as given.
RD
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
| 361 | 1,410 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.609375 | 4 |
CC-MAIN-2023-06
|
latest
|
en
| 0.971624 |
http://mathhelpforum.com/number-theory/156292-binet-s-formula.html
| 1,480,893,565,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-50/segments/1480698541426.52/warc/CC-MAIN-20161202170901-00157-ip-10-31-129-80.ec2.internal.warc.gz
| 177,961,919 | 11,039 |
1. ## Binet's formula
I need to derive Binet's formula and I don't know how to start.
2. You should really start by telling us what tools you have to work with.
Either way, you should try Wikipedia first: Fibonacci number - Wikipedia, the free encyclopedia.
3. Thank you. Part of my problem is that the tools haven't really been defined. I am in a Combinatorics class and all I was told was to derive the formula. Actually, I have seen a version of the derivation that seeks a solution of the form f_n =q^n, q <> 0, but I am not sure what lead the writer to this approach. It seems to me that whatever prompted that approach is key to the whole thing. The rest of the steps are just algebra.
4. Hello, tarheelborn!
If you've never seen a derivation for a recursive functions,
. . these moves will seem like Black Magic.
I need to derive Binet's formula and I don't know how to start.
The Fibonacci Sequence $1,1,2,3,5,8,\:\hdots$ can be expressed like this:
. . $f(n\!+\!1) \:=\:f(n) + f(n\!-\!1),\;\;f(1) = 1,\;f(2)=1$
The sequence begins with 1 and 1; each subsequent term is the sum of the preceding two terms.
We assume that $f(n)$ is an exponential function: . $f(n) \,=\,X^n$
The equation $f(n\!+\!1) \:=\:f(n) + f(n\!-\!1)$ becomes:
. . $X^{n+1} \:=\:X^n + X^{n-1} \quad\Rightarrow\quad X^{n+1} - X^n - X^{n-1} \:=\:0$
Divide by $X^{n-1}\!:\;\;X^2 - X - 1 \:=\:0$
Quadratic Formula: . $X \;=\;\dfrac{1\pm\sqrt{5}}{2}$
Form a linear combination of the roots:
. . $f(n) \;=\;A\left(\dfrac{1+\sqrt{5}}{2}\right)^n + B\left(\dfrac{1-\sqrt{5}}{2}\right)^n$
Use the first two terms of the sequence:
. . $\begin{array}{cccccccc}f(1) = 1\!: & A\left(\frac{1+\sqrt{5}}{2}\right) + B\left(\frac{1-\sqrt{5}}{2}\right) & = & 1 \\ \\[-3mm]
f(2) = 1\!: & A\left(\frac{1+\sqrt{5}}{2}\right)^2 + B\left(\frac{1-\sqrt{5}}{2}\right)^2 &=& 1 \end{array}$
Solve the system of equations: . $A \,=\,\dfrac{1}{\sqrt{5}},\;\;B \,=\,-\dfrac{1}{\sqrt{5}}$
Hence: . $f(n) \;=\;\dfrac{1}{\sqrt{5}}\left(\dfrac{1+\sqrt{5}}{2 }\right)^n - \dfrac{1}{\sqrt{5}}\left(\dfrac{1-\sqrt{5}}{2}\right)^n$
. . . . . . $f(n) \;=\;\dfrac{(1+\sqrt{5})^n - (1 - \sqrt{5})^n}{2^n\sqrt{5}}$
| 808 | 2,173 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.09375 | 4 |
CC-MAIN-2016-50
|
longest
|
en
| 0.802609 |
https://socratic.org/questions/if-26-percent-of-a-number-is-65-find-the-number
| 1,638,386,691,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-49/segments/1637964360881.12/warc/CC-MAIN-20211201173718-20211201203718-00056.warc.gz
| 582,173,488 | 5,630 |
# If 26 percent of a number is 65, find the number ?
Mar 28, 2018
250
#### Explanation:
We know that 0.26 or 26% of a number (say x) is 65
$x \cdot 0.26 = 65$
So as you multiply 0.26 by the number to find out what x is divide both sides by 0.26
So $x = \frac{65}{0.26} = 250$
| 106 | 281 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.28125 | 4 |
CC-MAIN-2021-49
|
latest
|
en
| 0.87961 |
https://lifewithdata.com/2023/06/11/leetcode-minimum-depth-of-binary-tree-in-python/
| 1,695,938,212,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-40/segments/1695233510454.60/warc/CC-MAIN-20230928194838-20230928224838-00441.warc.gz
| 390,678,829 | 25,418 |
# Leetcode – Minimum Depth of Binary Tree in Python
## Introduction
Binary trees are one of the most fundamental data structures in computer science, and they form the basis for more advanced data structures such as binary search trees, heaps, and several others. The Minimum Depth of Binary Tree problem is a classic example of a problem that tests a candidate’s ability to work with binary trees. In this comprehensive article, we will explore various methods to solve the Minimum Depth of Binary Tree problem on LeetCode using Python.
## Understanding the Problem
Before diving into the solutions, let’s first understand the problem statement. The Minimum Depth of Binary Tree problem asks you to find the minimum depth of a binary tree. The minimum depth is the number of nodes along the shortest path from the root node down to the nearest leaf node. A leaf is a node with no children.
For example, consider the following binary tree:
3
/ \
9 20
/ \
15 7
The minimum depth of this tree is 2.
## Binary Tree Data Structure
In Python, a binary tree is usually implemented using nodes. Each node in the binary tree contains a value, and pointers to its left and right children, if they exist.
Here is a simple class definition for a binary tree node in Python:
class TreeNode:
def __init__(self, value=0, left=None, right=None):
self.value = value
self.left = left
self.right = right
## Approach 1: Recursive Depth-First Search (DFS)
One of the most intuitive ways to solve this problem is by using a depth-first search. We can recursively traverse the tree, and for each node, compute the minimum depth of its left and right subtrees.
Let’s look at the Python code for this approach:
class TreeNode:
def __init__(self, value=0, left=None, right=None):
self.value = value
self.left = left
self.right = right
def minDepth(root):
if not root:
return 0
left_depth = minDepth(root.left)
right_depth = minDepth(root.right)
# If a node has only one child, we must take the depth of the side with the child.
if not root.left or not root.right:
return 1 + left_depth + right_depth
return 1 + min(left_depth, right_depth)
This approach has a time complexity of O(N), where N is the number of nodes in the tree, since we visit each node once.
## Approach 2: Iterative Breadth-First Search (BFS)
Depth-first search can be inefficient if the tree is highly imbalanced. In such cases, breadth-first search can be more efficient. With BFS, we traverse the tree level by level, and the moment we encounter a leaf node, we know that we have found the minimum depth.
Here is Python code that uses BFS to solve the problem:
from collections import deque
class TreeNode:
def __init__(self, value=0, left=None, right=None):
self.value = value
self.left = left
self.right = right
def minDepth(root):
if not root:
return 0
queue = deque([(root, 1)])
while queue:
node, depth = queue.popleft()
# If this is a leaf node, return the depth.
if not node.left and not node.right:
return depth
if node.left:
queue.append((node.left, depth + 1))
if node.right:
queue.append((node.right, depth + 1))
This approach also has a time complexity of O(N), but in cases of highly imbalanced trees, it might find the minimum depth quicker than DFS.
## Analyzing Practical Considerations
When solving problems like this, especially in an interview setting, it’s important to consider the practical implications. For example, understanding the trade-offs between DFS and BFS in terms of memory consumption and speed for different shapes of input trees.
## Tips and Tricks
1. Understand the properties of binary trees.
2. Become familiar with different tree traversal algorithms.
3. Practice solving tree-based problems regularly.
## Conclusion
The Minimum Depth of Binary Tree problem is a classic example of problems that involve binary tree traversal. We discussed both the depth-first and breadth-first approaches to solving this problem, both of which have their own strengths and weaknesses depending on the nature of the input tree. It’s important to understand the fundamentals of binary tree traversals and to practice solving a variety of problems to become proficient in handling tree-based problems.
| 920 | 4,219 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.5 | 4 |
CC-MAIN-2023-40
|
latest
|
en
| 0.836165 |
https://getgocube.com/play/limit-to-world-record-speedcubing/
| 1,627,749,090,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-31/segments/1627046154089.68/warc/CC-MAIN-20210731141123-20210731171123-00713.warc.gz
| 299,559,552 | 190,309 |
5 Reasons why dinosaurs
GoCube
share:
share:
# Is There a Limit to the World Record for Speedcubing?
The world record for the fastest solve of the 3×3 Rubik’s Cube was set by Yusheng Du at 3.47 seconds. He beat Feliks Zemdegs’s previous record of 4.22 seconds in the process. Who do you think will be next to set a new world record? What if we told you that Du’s record is close to impossible to beat?
The Rubik’s Cube has thousands of combinations of moves or algorithms that can be used to solve it. However, the number of moves used to solve a cube as fast as possible is much smaller. The fewer the moves, the faster the solve time. Each solving method has an average number of moves to solve the cube:
The LBL (Layer By Layer) method is ideal for beginners and solves the cube with an average of 80 moves in half a minute.
The advanced CFOP (Fridrich) method solves the cube with an average of 60 moves in less than 10 seconds.
Another advanced method is the ROUX method, which solves the cube with an average of 48 moves in less than 10 seconds.
Finally, there is the ZZ advanced method, which can solve the cube with an average of 45 moves in less than 10 seconds as well.
Some solve methods are faster than others, but they all share something in common: a lowest possible solve time of two seconds. There are two limitations for speedcubers: physical and mental. It does not matter how good the speedcuber is, a human’s hands can only move a cube so fast. The fastest speedcubers in the world all maintain an average of ten turns per second. The other limitation is that speedcubers need to plan in advance which algorithms they will need to use and adapt as they solve.
The last element in this two-second-solve challenge is “God’s number”: any scrambled cube can be solved with a minimum of 20 moves. Even if we assume that a speedcuber knows the 20 moves in advance and executes them perfectly, the result would be between two and three seconds. In other words, luck plays a significant factor in reaching a solve in under three seconds.
Feliks Zemdegs stated in an interview in 2019 that he expects the next world record to eventually be 2.5 seconds. The reason he thinks this is possible is that there are always new techniques and methods being developed to shave off fractions of seconds. Additionally, new speedcube designs are always under development, which results in faster solve times.
For the time being the only way to reliably beat the two-second solve limit is by using a machine. Ben Katz and Jared Di Carlo designed a robot in 2018 that solved the cube in 0.38 seconds.
You can see this feat in action in this video:
Now that you know the true time limits of speedcubing solves, you can get a clearer picture for how fast you can eventually solve. We hope this article serves to inspire you to reach that limit rather than feeling restricted by it.
You might also like
| 655 | 2,910 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.53125 | 4 |
CC-MAIN-2021-31
|
longest
|
en
| 0.958933 |
https://www.jiskha.com/similar?question=Short+answer+find+the+quotient+6x%5E3-x%5E2-7x-9+------------+2x%2B3&page=803
| 1,566,696,807,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-35/segments/1566027322160.92/warc/CC-MAIN-20190825000550-20190825022550-00337.warc.gz
| 854,441,260 | 24,006 |
# Short answer find the quotient 6x^3-x^2-7x-9 ------------ 2x+3
143,010 questions, page 803
1. ## Mathematics
Find the size of angle C if angle A is 50 degrees. You have been give that triangle ABC AC = 8 cm and AB = 6 cm.
asked by Alizwa Nqakala on June 11, 2016
2. ## Math
If A and B are acute angle such that SinA=8/17 and CosB=3/5.Find 1, Cos(A+B) 2, Sin(A+B) 3, Sin(A-B)
asked by Ada Enugu on January 29, 2013
3. ## Calculus
An object is traveling along the x axis so that its speed is given by 2 v(t) t 7t 10 m/s. (a) Find the times when the object is at rest.
asked by Michael on May 14, 2013
4. ## Math
The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the two numbers.
asked by Scarlett on May 14, 2013
5. ## math
4.find to the nearest degree the celcius reading when the farenheit reading is 72 degrees
asked by chris on September 6, 2009
6. ## math
find whether the line 2x-y=0 is tangent, real chord on imaginary chord to the parabola y^2-2y+4y=0.
asked by sonu on August 8, 2011
7. ## Math
Find the slope of a line that is parallel to the line ontaining the points (3, 4) and 2, 6) A. m= 2 B. m= -2 C. m= -1/2 D. m= -1 I say it's C Please check
asked by Kel. Last post, please check on April 23, 2015
8. ## Math
Find the slope of a line that is parallel to the line ontaining the points (3, 4) and 2, 6) A. m= 2 B. m= -2 C. m= -1/2 D. m= -1 I say it's C or A Please check
asked by Kel on April 23, 2015
9. ## Geometric Sequence
Find the fourth term (a4) of geometric sequence if the first term (a1) is 2 and the common ratio, r is 4.
asked by Embraced on January 14, 2012
10. ## geometry
Find the value of x. x = 18x = 22.5x = 30x = 54 is like a like two line almost forming an X and on the left side it says (x+90) and on the right side it says 4x
asked by joey on June 24, 2016
11. ## Calculus
Find the derivative of y with respect to x. y= [(x^5)/5]lnx-[(x^5)/25] How would I go about doing this? Should I use the product rule to separate (x^5)/5]lnx ?
asked by Natalie on September 4, 2011
12. ## physics
from equqtion t equal to x+3 .find velocity and acceleration t equal 1second here x is distance
asked by add on September 23, 2012
13. ## PHI208: Ethics and Moral Reasoning
What other similarity can we find in Freeman’s stakeholder theory with Kant’s moral theory
asked by Erica on July 8, 2014
14. ## geometry
The endpoints of are A(9, 4) and B(5, –4). The endpoints of its image after a dilation are A'(6, 3) and B'(3, –3). can you please Explain how to find the scale factor.
asked by Cindy on December 19, 2011
15. ## Calculus and vectors-Help~David
Rectangular prism is defined by vectors (2,0,0)(0,9,5) and (0,0,3). Find the volume. I have done this so far, would anyone please verify this. (2,0,0)(0,9,6) (0-2, 9-0, 6-0) =(-2,9,6) (-2,9,6)(0,0,3) =(2,-9,-3) (2)(-9)(-3)=54 cm^3 Therefore volume is 54
asked by Jake on March 22, 2012
16. ## math-algebra 1
Find the 6th term of the geometric sequence whose common ratio is 1/3 and whose first term is 7.
asked by Emily on July 14, 2011
17. ## calculus
Find the points on the graph y = -x^3 + 6x^2 at which the tangent line is horizontal. I found -x^2 + 4 = 0 are the points [4,32]
asked by heather on February 8, 2017
18. ## Algebra
The base of a triangle in terms of x is: 2^2+4x+2 and the height is x^2+3x-4 Area=1/2(2x^2+4x+2) (x^2+3x-4)= 6x^2+2 (x^2+3x-4) Trying to find the area and not sure which direction to go
asked by Babydoll2007 on November 2, 2014
19. ## Math
A particle is moving along the x-axis so that its position at t>=0 is given by s(t)=(t)ln(5t). Find the acceleration of the particle when the velocity is first zero.
asked by Ke\$ha on December 29, 2016
20. ## physics
Find the ratio of the coulomb electrical force to the gravitation force between two electrons .
asked by Anonymous on November 6, 2017
21. ## probability
Let A and B two mutually exclusive events of a sample space S. If P(A)=0.6, P(B)=0.2 , find the probability of each of the following events a)P(A ∩ B)
asked by anonymous on July 25, 2010
22. ## Algebra II
Find the recursive formula for the following sequence and determine if the sequence is arithmetic or geometric. A) 3,7,11,15.... B) -5,-15,-75....
asked by Luisa on October 24, 2016
23. ## Math 7 A
Find the like terms in the expression 1.) y + 1.2y + 1.2z 2.) simplify 2 + 17x - 5x + 9 3.) simplify 3(5 =6) - 4 4.) Factor 81 – 27p
asked by Kooper on November 6, 2017
24. ## math
using the formula h(x) = 1.95 x + 72.85. Find the height of a female whose femur is 35 cm long x=length and H=height.
asked by Maryann on March 16, 2010
25. ## math
Find an equation of the line that satisfies the given conditions. Through (−7, −12); perpendicular to the line passing through (−4, 0) and (0, −2)
asked by alex on September 7, 2014
26. ## Trig FInd Translation Rule and Coordinates Thanks!
The vertices of a rectangle are R(–5, –5), S(–1, –5), T(–1, 1), and U(–5, 1). After translation, R' is the point (–11, –11). Find the translation rule and coordinates of U'. (x, y)--> (x – 6, y + 6); (–11, 7) (x, y)--> (x – 6, y –
asked by LeXi ;) on March 1, 2012
27. ## Pre calc
Evaluate using the Pythagorean indentities Find sinθ and cosθ if tanθ =1/6 and sinθ >0 Help please ! Thank you !
asked by Bob on January 18, 2018
28. ## Math
The measures of two supplementary angles are (10X-25) and (15X+50). Find the measure of the angles
asked by Anonymous on September 4, 2014
29. ## math
if the median to the hypotenuse of a right triangle measures 25 and a leg 14, find the measure of the other leg
asked by chelsea on March 15, 2010
30. ## math
Find a geometric sequence in which the 6th term is 28 and 10th term is 448
asked by shadrack on March 24, 2012
31. ## math
find an equation of the line passing through the pair of points. write the equation in Ax+By=C (-9,7), (-10,-4)
asked by ann on October 29, 2013
32. ## algebra
The value of a particular car dropped from 25,000 in may 21,000 in october.find the rate of change in value
asked by babygirl on September 29, 2011
33. ## Chemistry
Find the frequency if given energy 6.3x10^-19 Do I use the formula E=hf? 6.3x10^-19=6.626x10^-34(f) Then do I divide E and H?
asked by Anonymous on October 22, 2016
34. ## statitics
find the probility that when 4 managers are randomly selected from a group of 20, the 4 oldest are selected
asked by rose on February 11, 2010
35. ## Geometry
Help I need help to find the coordinates of the circumcenter of each triangle. Isosceles triangle CDE with vertices C(0, 6), D(0, –6), and E(12, 0)
asked by Zach on November 15, 2012
36. ## Pre-Calculus
g(x)= x^2+7 Find the inverse of g(x) and state the domain and range for the inverse of g(x) using interval notation
asked by Juniper on April 26, 2016
37. ## business
i am researching the prepackaged software industry and i need to find what the future holds for that industry.
asked by chris on January 25, 2012
38. ## statistics
find the sample variance s2 for the following sample data. Round to the nearest hundred x: 23 17 12 35 29
asked by Anonymous on July 22, 2010
39. ## statistics
A set of 50 data values has a mean of 15 and a variance of 36. Find the standard score of a data value = 30
asked by Kcr7 on August 28, 2012
40. ## trig
if x and y are positive acute angles, tan x=1/3and tan y =1/4, find the value of sin(x-y)
asked by Anonymous on May 25, 2011
41. ## math
4.find to the nearest degree the celcius reading when the farenheit reading is 72 degrees
asked by chris on September 6, 2009
42. ## social studies
in what state can you find an inland location that is lower than sea level? what is the name of the location?
asked by amy on August 30, 2007
43. ## Pre calc
Evaluate using the Pythagorean identities. Find sinθ and cosθ if tanθ = 1/4 and sinθ >0
asked by Alyssa on January 19, 2018
44. ## math
the lenght of a rectangle is twice its breadth. if its perimeter is 18cm, find its length and breadth
asked by Anonymous on August 27, 2016
45. ## Maths
the value of a machine depreciates every year by 5% .if the present value of the machine is 100000 ,find the value after 2 years
asked by Rahim on January 1, 2017
46. ## math
The fourth and ninth term of an a.p are -3 and 12 repectively find:the common different and the sum of the first seven term.
asked by Utibe on July 8, 2014
47. ## Maths
the value of a machine depreciates every year by 5% .if the present value of the machine is 100000 ,find the value after 2 years
asked by Anonymous on January 1, 2017
48. ## css
The height of a triangle is increased by 15% and the base is increased by 10%. Find the increase in its area.
asked by Anonymous on September 10, 2015
49. ## Geometry
Triangle STR is congruent to triangle XYZ, TR= 14c - 13, and YZ = 10c - 1, Find c and TR
asked by Maddy on July 23, 2018
50. ## math
Find the volume generated by rotating the area between y = cos( 3 x ) and the x axis from x = 0 to x = π/ 12 around the x axis
asked by sk on September 29, 2011
51. ## Calc
Let f(x)=sqrt(x) and g(x)=sin(x). Find f*g and determine where f*g is continuous on the interval (-4pi, 4pi)
asked by Maria on January 13, 2014
52. ## Physics
I have a forumla question. I have this formula x=1/2t^2. Is this the formula to find the acceleration that is caused by gravity?
asked by Soly on September 3, 2007
53. ## physic
A wheel 0.5m in radius is moving with a speed of 12m/s.find its angular speed ?
asked by Indrajeet raiger on December 26, 2016
54. ## gemoetry
how do I find the angles of 4 times one complementary angle decreased by 3 times the second is 3 degrees
asked by Anonymous on September 9, 2013
55. ## physics
from equation t equal to root x+3 find velocity and acceleration t equal to 1second
asked by ash on September 23, 2012
56. ## MATH
FIND THE 6TH AND 15TH TERM OF A AND B WHOSE 1ST TERM IS 6 AND COMMON DIFFERENCE IS 7
asked by ORJINTA on August 24, 2016
57. ## Vectors
Find the volume of a parallelepiped with 3 edges defined by Vectore a = (-2, 1, 4), Vector b = (5, 9, 0) and Vectore c = (0, 3, -7).
asked by Hera on July 20, 2019
58. ## calculus
Use the linear approximation (1+x)^k=1+kx to find an approximation for the function f(x)=1/square root of (4+x) for values of x near zero.
asked by Anonymous on March 6, 2008
59. ## MATH
FIND THE 6TH AND 15TH TERM OF THE A AND B WHOSE 1ST TERM IS 6 AND COMMON DIFFERENCE IS 7
asked by ORJINTA on August 24, 2016
60. ## Geometry
Line TV bisects
asked by Robert on August 19, 2015
61. ## Math
the perimeter and area of a rectangle are 48 cm and 140cm^2 respectively. Find the length and width of the rectangle
asked by Fawad on November 21, 2015
62. ## Math
Find the base of a parallelogram with a height of 9.7 meters and an area of 58.2 square meters.
asked by Mandy on April 25, 2012
63. ## math
The quadrilateral ABCD has area of 58 in2 and diagonal AC = 14.5 in. Find the length of diagonal BD if AC ⊥ BD.
asked by HELP!!! on April 19, 2017
64. ## math
find a commission on a \$1,250 sale with a commission rate of 5%. \$62.50 \$6.25 \$75.00 \$125.00 HELP PLSSS
asked by Ali on February 8, 2017
65. ## chemistry again
If you are given V1=410 ml T1= 27 C= 300 K P1= 740 mm Find V2 in ml at STP When given STP= 273K and 1 atm
asked by Bianca on October 22, 2011
66. ## Geometry
How do I find the area of a right triangle with a hypotenuse that measures 39 cm anad a leg that measures 15 cm?
asked by Cody on March 21, 2012
67. ## Algebra
Identify coefficients and the degrees of each term polynomial, and then find the degree of each polynomial. a^3b^8c^2 - c^22 + a^5c^10
asked by Miguel on March 10, 2011
68. ## Precalculus
Points A, B, and C are on a circle such that angle ABC=45 degrees, AB=6, and AC=8. Find the area of the circle.
asked by Katie on April 9, 2016
69. ## Algebra Help
Identify coefficients and the degrees of each term polynomial, and then find the degree of each polynomial. a^3b^8c^2 - c^22 + a^5c^10
asked by Miguel on March 10, 2011
70. ## physics
from equation t equal to root x+3 find velocity and acceleration t equal to 1second
asked by ash on September 23, 2012
71. ## Calculus - You wont get it
Find the local max and local min and saddle points of f(x, y) = x^4 + y^4 - 4xy +1
asked by Big Deal on May 23, 2011
72. ## math
Find the discriminant. Tell which conic section each graph will be. 4x^2 + 9y^2 +7x-2y-100=0 4x^2+14xy+7x-2y-100=0
asked by Rick on March 1, 2012
73. ## math
Find the % of Discount Regular Price \$125 Sale Price \$100
asked by Jessica on February 7, 2013
74. ## algebra
A man is 4 times as old as his son.After 16 years he will be only twice as old as his son. find their present ages.
asked by Raj on February 15, 2012
75. ## precalc
How would you find the inverse of these two functions: f(x)=4/(x-4)^2 and f(x)=x^2-8x+16 could you please show me step by step i have been trying to figure it out for hours.
asked by Diana on July 15, 2011
76. ## math
After 12 years manoj will be 3 times as old as he was 4 years ago. Find his present age.
asked by Anonymous on February 25, 2012
77. ## maths
Find the measurement of the first and third angles if the ratio of the three consecutive angles of a cyclic quadrilateral is 1:2:3
asked by Anamika Adhikari on January 2, 2016
78. ## Geometry
A cone with a radius of 3 inches has a total area of 24 pi sq in. Find the volume of the cone.
asked by April on April 30, 2012
79. ## Maths
8 people in the playground. How many ways can 6 people be selected to play? Find the rule.
asked by Anonymous on October 25, 2016
80. ## Math
Find the simple rate Principal \$2400 Rate 10 1/2% Time in years 1 1/2
asked by Muffin on February 21, 2013
81. ## calculus, math
Use Newton's method to approximate the value of (543)^(1/5) as follows: Let x1=2 be the initial approximation. find x2 and x3 =? approximation
asked by kayla on November 22, 2011
82. ## trig
If sinx = -3/5, tan x > 0, sec y = -13/5 and cot y < 0, find the following: a. csc(x + y) b. sec (x - y) How do I get the answers of a. -65/33 b. -65/16 Thank you
asked by Priya on May 1, 2008
83. ## chemistry
the density of ehter is 0.789 g/mL. Find the mas in kilograms for 475mL of ehter
asked by miichael on September 22, 2011
84. ## Math HELP
Find the like terms in the expression 1.) y + 1.2y + 1.2z 2.) simplify 2 + 17x - 5x + 9 3.) simplify 3(5 =6) - 4 4.) Factor 81 – 27p
asked by Tracy on November 6, 2017
85. ## math
Four times the sum of a number and 15 is at least 120. Let x represent the number. Find all possible values for x. A. x ≥ 26 B. x ≥ –15 C. x ≥ 15 D. x ≥ –26
asked by Anonymous on November 2, 2014
86. ## math
Hello. Please, help me solve this: Parabola y = ax^2 + bx + c passes through the point (2;-6). Find the values of a, b and c, if vertex of parabola is (4;-10).
asked by eve on May 19, 2012
87. ## Geometry
If line PQ bisects with Line ST at R. SR = 3x +3 and ST = 30 Find x I come up with x being 9 to equal 15 but the choice for answers are: x=8 x=6 x=4 x=2 What am I doing wrong?
asked by Jonathon on August 19, 2015
88. ## ALGEBRA
A line passes through the points whose coordinates are (c,3c) and (2c,5c).Assuming that c is not 0 find the slope of the line A-2 B-1/2 C1/2 D2
asked by RICARDO on July 9, 2012
89. ## Math
Maximizing Area: Among all rectangles that have a perimeter of 20 ft, find the dimensions of the one with the largest area.
asked by David on October 9, 2009
90. ## mathes
find an equation of the normal line to the curve y=radx-3 that is parallel to the line 6x+3y-4=0
asked by Anonymous on December 16, 2013
91. ## Calculus-derivatives
What is the slope of the tangent of each curve at given point y=√(16x^3), (4,32) If I find the derivative of y=√(16x^3) Will it be 16/-x^3
asked by Jake on February 15, 2012
92. ## no idea...but plz help
Find the sum of all real numbers x such that (absolute value(x – 2009)) + (absolute value(x – 2010)) = 3.
asked by Ryan on November 29, 2009
93. ## Chemistry
I had to find the root mean square velocity of a gas using the only formula KE=1/2MV^2 . Is this even possible using only this formula?
asked by Shadow on March 11, 2013
94. ## Math
how to find out length of a rectangle area given 18900cm'2 width 1.8 what is length in metres
asked by Alex on April 27, 2016
95. ## math
find the straight line that passes through point (2,4) and is perpendicular to line 2x + 12y - 6 = 0
asked by lily on July 5, 2014
96. ## Chemistry
I had to find the root mean square velocity of a gas using the only formula KE=1/2MV^2 . Is this even possible using only this formula?
asked by Shadow on March 11, 2013
97. ## calculus check
Find the limit as x approaches 3 of [(2x^3-5x^2+7)/(3sin(xpi/6)-cos(xpi))] When I plugged in 3, I got 4.
asked by Jeff on April 26, 2015
98. ## Geometry
A triangle has vertices (3, 0) and (6, 4). One of its angles is bisected by the x axis. Find the perimeter of the triangle.
asked by Momo on May 18, 2016
99. ## algebra 1
thr ratio of two integers is 6:13 the smaller interger is 54. find the larger interger
asked by john on February 21, 2013
100. ## Math
Find the simple interest on a loan of \$100 at an interest rate of 2% for 7 years.
asked by Julie on March 12, 2010
| 5,749 | 17,308 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.765625 | 4 |
CC-MAIN-2019-35
|
latest
|
en
| 0.935822 |
https://decimalworksheets.co/write-each-fraction-as-a-decimal-worksheet/
| 1,632,376,789,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780057417.10/warc/CC-MAIN-20210923044248-20210923074248-00652.warc.gz
| 252,738,481 | 13,861 |
# Write Each Fraction As A Decimal Worksheet
Write Each Fraction As A Decimal Worksheet – Fantastic, exciting and free of charge math worksheets should be able to provide a statistical difficulty in a different way. Math is after all nothing more than a numeric expression of some of life’s most basic concerns: How much money do I have remaining if I purchase a soft drinks? At the end of every week, the amount of my daily allowance will I have the ability to save if I don’t?
When a child understands to relate math to each day queries, he will likely be great at it from the simplest addition all the way to trigonometry. To transform rates, decimals and fractions is hence one particular essential talent. The amount of an apple company cake has become eaten? The reply to this question may be expressed in rates, 50%; or perhaps in decimals, .5; or in fraction, ½. Quite simply, [%half of|50 % of|one half of%] mom’s scrumptious apple company pie is gone. How many kids in school did their groundwork? Once again this could be answered in several methods: in rates, 70%; or even in ratio, 7:10; These two imply out of ten youngsters in school you can find 7 good types who performed and a few not-so-excellent ones who didn’t. The end result is that children learn math far better when it makes sense.
Hence, the Write Each Fraction As A Decimal Worksheet which you get to your youngsters ought to include intriguing phrase conditions that enable them to with the sensible use of the lessons they find out. It should also provide the same difficulty in a number of techniques to make sure that a child’s understanding of a topic is further and extensive.
There are many normal workouts which teach pupils to transform rates, decimals and fractions. Converting percentage to decimals as an example is actually as basic as shifting the decimal point two places left and losing the [%percent|%|percentage%] signal “%.” Therefore 89% is equal to .89. Indicated in fraction, that would be 89/100. Once you drill kids to do this frequently sufficient, they learn how to do conversion process practically naturally.
Ratios and proportions are likewise fantastic math classes with plenty fascinating useful applications. If three pans of pizza, a single kilo of spaghetti, two buckets of poultry can properly feed 20 starving friends, then how much pizzas, pasta and chicken does mommy need to plan for birthday party with 30 youngsters?
## Start Teaching your kids with Write Each Fraction As A Decimal Worksheet
Wouldn’t it be fantastic if your little one figured out how you can mathematically physique this out? To help train him, start by providing him lots of totally free Write Each Fraction As A Decimal Worksheet!
| 568 | 2,727 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.125 | 4 |
CC-MAIN-2021-39
|
latest
|
en
| 0.95219 |
https://math.answers.com/Q/How_many_km_are_in_4.5_cm
| 1,718,645,344,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-26/segments/1718198861733.59/warc/CC-MAIN-20240617153913-20240617183913-00154.warc.gz
| 349,833,907 | 47,672 |
0
How many km are in 4.5 cm?
Updated: 10/26/2022
Wiki User
13y ago
A kilometer is 1,000 meters, and there are 100 centimeters in a meter.
So there are 100,000 centimeters in a kilometer.
That means one centimeter is 0.00001 kilometers.
So there are 0.000045 kilometers in 4.5 centimeters.
If you meant to ask how many centimeters in 4.5 kilometers, that's 100,000 x 4.5 = 450, 000
Wiki User
13y ago
| 136 | 409 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.59375 | 4 |
CC-MAIN-2024-26
|
latest
|
en
| 0.879084 |
https://betterlesson.com/lesson/resource/2988994/monster-subtraction
| 1,628,109,330,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-31/segments/1627046154897.82/warc/CC-MAIN-20210804174229-20210804204229-00049.warc.gz
| 150,000,411 | 27,904 |
# Subtraction Number Stories- Day 3
2 teachers like this lesson
Print Lesson
## Objective
Students will be able to create their own number stories and represent them using one of the methods we have learned in this unit.
#### Big Idea
After reading If You Were a Minus Sign, students create their own number stories, by representing subtraction using pictures and possibly equations.
## Problem of the Day
5 minutes
I start each math lesson with a Problem of the Day. I use the procedures outlined here on Problem of the Day Procedures.
Today's Problem of the Day:
Jake has 9 skateboards. He lost 5 of them. How many skateboards does Jake have left?
I set this problem up with some structures to help the students organize their thinking. I gave a blank number sentence frames to help the students to write their answer as an equation. On the Notebook file, there is also a picture of a skateboard set to Infinite Cloner. This way the student can use it represent the problem. If you do not have a SMART Board, you can use the PDF and manipulatives, pictures, or students' drawings.
Since we do this whole group, I have one students come up and work on this problem. I remind the student to check their work when they are finished and have the class tell if they agree or disagree by showing a thumbs up or thumbs down. Students also have the opportunity to share why they agree or disagree.
## Presentation of Lesson
25 minutes
I start this lesson reminding students that we have been working on our "If You Were A Minus Sign" book. I remind students that a minus sign is the symbol that we put between two numbers in an equation to show that we are subtracting. By this point, we have been working with subtraction for a few weeks, so most students are comfortable with the name and purpose of a minus sign.
We have been working with writing subtraction number sentences which are also called equations. Today we are going to continue to work on our books. Let's start with a group number story.
I have students give me an idea for a number story. I illustrate it on the same paper that they will be using and project it on the SMART board. I have the students help me write the equation to go with it. I also model how to solve the same number story with manipulatives and on a ten frame. I make sure to model how I want the students to complete their number story. I include a picture that clearly represents the subtraction. I also include the equation at the bottom of the page. While writing equations is not required in kindergarten, it is encouraged, so I set students up with the opportunity to write an equation for their number story.
You are going to get your own "If You Were A Minus Sign" Book that you have been working on. Your book has four pages. Today we are going to work on the third page, but if you get finished, you may do the last page as well. You can begin as soon as you have your book.
I walk around and make sure that students have come up with a number story and are drawing it and writing their equations. I have ten frames and manipulatives available for who want to use those to represent the number story. If students use the manipulatives, I take a picture of their work and add it to their book after I print the picture. Subtraction Number Story Student Example
I got the idea for this activity from Mrs. Ricca's Kindergarten. The "If You Were A Minus Sign" student book that I am using in this lesson is available as a free download from her blog.
## Practice
20 minutes
The centers for this week are:
I am not sure if or for how long I will pull groups today. I anticipate that some students will need help with their number stories and that some may just take longer on this task than our usual independent work. I circulate to help students with their number stories and to make sure students are engaged and do not have any questions about how to complete the centers if they have finished their number story. If time allows, I pull groups during centers and work with them depending on the time they need (5 - 10 minutes).
Today I am focusing on subtraction with all of the groups. While my students are doing well on our subtraction lessons and centers, as we near the end of unit assessment, I would like to observe them more closely as their work through word problems. I verbally give the group a word problem. I have them solve it using manipulatives and write the equation. With students who are able to do this easily, I also have them try with drawing pictures instead of manipulatives.
Prior to clean up, I check in with each table to see how the centers are going. My students continue to struggle with getting cleaned up quickly and quietly after centers. I have been using counting down from 20 slowly instead of a clean up song. Counting backwards is as critical as counting up. Students need to be able to know the number that comes before, as well as after, any given number (w/i 10, w/i 20, etc.). Counting back is a critical strategy for subtraction.
The students like to count backwards with me as they clean up and I can lengthen or reduce the clean up time based on how students are doing and how much time we have.
## Closing
10 minutes
To close, I put a student's book on the document camera and project it on the SMARTBoard and have the students explain their work. I choose a student who clearly represented subtraction in their picture and wrote a correct equation. I mention positive things noticed during centers as well as something that needs to be better next time.
I review what we did during our whole group lesson. "Today we learned about subtraction number stories and working on creating our own. Tomorrow you will show me what you have learned about subtraction on a short assessment."
| 1,263 | 5,840 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.71875 | 5 |
CC-MAIN-2021-31
|
latest
|
en
| 0.944728 |
https://studysoup.com/tsg/calculus/141/calculus-early-transcendental-functions/chapter/16935/6
| 1,603,665,916,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-45/segments/1603107890028.58/warc/CC-MAIN-20201025212948-20201026002948-00266.warc.gz
| 550,832,238 | 10,791 |
×
×
# Solutions for Chapter 6: Differential Equations
## Full solutions for Calculus: Early Transcendental Functions | 6th Edition
ISBN: 9781285774770
Solutions for Chapter 6: Differential Equations
Solutions for Chapter 6
4 5 0 372 Reviews
12
0
##### ISBN: 9781285774770
Since 76 problems in chapter 6: Differential Equations have been answered, more than 45289 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendental Functions, edition: 6. Chapter 6: Differential Equations includes 76 full step-by-step solutions. Calculus: Early Transcendental Functions was written by and is associated to the ISBN: 9781285774770.
Key Calculus Terms and definitions covered in this textbook
• Arccosecant function
See Inverse cosecant function.
• Combinations of n objects taken r at a time
There are nCr = n! r!1n - r2! such combinations,
• Explanatory variable
A variable that affects a response variable.
• Finite series
Sum of a finite number of terms.
• Fitting a line or curve to data
Finding a line or curve that comes close to passing through all the points in a scatter plot.
• Five-number summary
The minimum, first quartile, median, third quartile, and maximum of a data set.
• General form (of a line)
Ax + By + C = 0, where A and B are not both zero.
• Graph of an equation in x and y
The set of all points in the coordinate plane corresponding to the pairs x, y that are solutions of the equation.
• Graph of an inequality in x and y
The set of all points in the coordinate plane corresponding to the solutions x, y of the inequality.
• Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x:- q ƒ(x) = or lim x: q ƒ(x) = b
• Inverse composition rule
The composition of a one-toone function with its inverse results in the identity function.
• Logarithm
An expression of the form logb x (see Logarithmic function)
• Multiplicity
The multiplicity of a zero c of a polynomial ƒ(x) of degree n > 0 is the number of times the factor (x - c) (x - z 2) Á (x - z n)
• Negative linear correlation
See Linear correlation.
• Pie chart
See Circle graph.
• Projection of u onto v
The vector projv u = au # vƒvƒb2v
• Stem
The initial digit or digits of a number in a stemplot.
• Sum of a finite arithmetic series
Sn = na a1 + a2 2 b = n 2 32a1 + 1n - 12d4,
• Summation notation
The series a nk=1ak, where n is a natural number ( or ?) is in summation notation and is read "the sum of ak from k = 1 to n(or infinity).” k is the index of summation, and ak is the kth term of the series
• Variation
See Power function.
×
| 724 | 2,784 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.90625 | 4 |
CC-MAIN-2020-45
|
latest
|
en
| 0.86791 |
https://vdocument.in/number-rhymes-rhymes-2020pdf-number-rhymes-counting-down-from-and-up-to-10.html
| 1,695,349,363,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-40/segments/1695233506320.28/warc/CC-MAIN-20230922002008-20230922032008-00485.warc.gz
| 670,965,891 | 22,500 |
# number rhymes rhymes 2020.pdf · number rhymes counting down from, and up to, 10 saying how many...
of 2 /2
Teachers: Early Years Number Rhymes Counting down from, and up to, 10 Saying how many there are altogether Children often enjoy singing and saying rhymes and telling familiar stories. Adults could share the song ‘Ten Green Bottles’ in order to involve the children in singing and counting. The Activity Provide a collection of ten green bottles, partly filled with sand. Stand them in a row for all the children to see. Sing the song and act it out. Encouraging mathematical thinking and reasoning: Describing What is happening to the number of bottles each time one falls? Reasoning Two bottles have fallen off the wall. How many are there left? How do you know that? What if you count the bottles on the wall and those that have fallen off? Can you see a pattern? Opening Out What if two fell off at once? Imagine how many bottles there will be on the wall if three have fallen off. What if we add five more bottles, how many would there be then? Recording Can you show on your fingers how many there are/how many will be left? Can you find the numeral, dotty card or Numicon to match the number left? Can you draw a picture/make a mark, to show me how many bottles there are on the wall now? NRICH
Post on 09-Aug-2020
3 views
Category:
## Documents
Embed Size (px)
TRANSCRIPT
Teachers: Early Years
Number Rhymes
Counting down from, and up to, 10Saying how many there are altogether
Children often enjoy singing and saying rhymes and telling familiar stories.
Adults could share the song ‘Ten Green Bottles’ in order to involve the children in singing and counting.
The ActivityProvide a collection of ten green bottles, partly filled with sand. Stand them in a row for all thechildren to see. Sing the song and act it out.
Encouraging mathematical thinking and reasoning:
DescribingWhat is happening to the number of bottles each time one falls?
ReasoningTwo bottles have fallen off the wall. How many are there left? How do you know that? What if youcount the bottles on the wall and those that have fallen off? Can you see a pattern?
Opening OutWhat if two fell off at once?Imagine how many bottles there will be on the wall if three have fallen off.What if we add five more bottles, how many would there be then?
RecordingCan you show on your fingers how many there are/how many will be left?Can you find the numeral, dotty card or Numicon to match the number left?Can you draw a picture/make a mark, to show me how many bottles there are on the wall now?
NRICH
• counting them all to find out how many are left• using the language of subtraction: saying how many are left• knowing that one less is the next number counting backwards e.g. predicting the next number before the next bottle falls
Describing position• using positional language e.g. on, off, next to, before, after, left, right
Development and VariationIf two fall off at once, children may realise they can count back to subtract: you could support this with a number line.Children could show with fingers how many there will be.Counting up and down from a given number in the context of the number of children in the group e.g. ten children and two are away today so there are eight here.Counting sets and collections of objects and adding or removing some by hiding objects under a cloth or in a bag.
Story, rhyme and song linksFive Little Ducks Went Swimming One Day, Ten Fat Sausages
ResourcesGreen plastic bottles partly filled with sand or water to weigh them down Numerals, dotty cards, NumiconWhiteboards and pensCamera or video camera for recording
• using number words and language about counting e.g. none, zero, next door number/number neighbour
• showing on fingers how many there are
• finding numerals to match the number left
Subtracting
| 857 | 3,861 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.515625 | 4 |
CC-MAIN-2023-40
|
latest
|
en
| 0.931266 |
www.nationalfishpharm.com
| 1,723,468,401,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-33/segments/1722641039579.74/warc/CC-MAIN-20240812124217-20240812154217-00236.warc.gz
| 699,075,311 | 4,372 |
# Water Volume Calculation & Conversion Charts
Calculating Water Volume In U.S. Gallons
1. Mathematical Formulas
a. If computed in inches, divide the total by 231.
b. If computed in feet, multiply the total by 7.5
Rectangular shapes: length X width X depth.
Circular shapes: 3.14 X radius² X depth
2. Using a domestic water meter: Use a wrench to turn the locking nut on the meter to remove the lid. Record the reading before filling the pond. Subtract this figure from the figure recorded after filling is completed. No water at all should be used in the house during filling.
3. Using hose output: Fill a large bucket for exactly 60 seconds. Measure the water in pints and divide by 8 to compute U.S. gallons. Record start and stop times of filling the pond. Multiply the total minutes required for filling by the number of gallons the hose discharges in one minute.
Calculating Water Volume In Liters
1. Multiply length X width X depth in meters X 28.41 for rectangular volume.
2. Multiply depth in meters X metric diameter² X 20.75 for circular volume.
Frequently Used Equivalents ppm (parts per million) is equivalent to one milligram per liter of water. 5ml = 1 teaspoon 20 drops = 1 ml. 60 drops = 1 teaspoon 15ml = 1 tablespoon 2 tsp. = 1 dessert spoon 2 dessert spoons = 1 tablespoon 2 tbsp. = 1 fl. oz. 8 fl. oz. = 1 cup 1 cup = 48 teaspoons 1 cup = 16 tablespoons 1 cup = 237ml. *These figures are commonly supplied by most chart sources. However, using a standard, pharmacist-supplied, two-milliliter eye dropper and a standard kitchen measuring teaspoon, the equivalency was found to be 3ml/tsp. and 9ml/tbsp. *Culinary, or food industry measuring devices and weights are different from pharmaceutical and laboratory measuring devices. Table of Liquid Equivalents 1 milliliter (ml) = 1 cm³ = 1cc = 20 drops = 0.20 tsp. = 0.061 in.³ = 0.001 L = 1gm. of water = 0.002 lb. of water = 0.0003 U.S. gallons 1 U.S. Gallon (gal.) = 3.785 L = 0.1339 ft.³ = 231 in.³ = 8.345 lb. of water = 3785.4 gm. of water = 4qt. = 8 pt. = 135.52 oz. = 128 fluid oz. = 3785.4 ml 1 fluid ounce = 6 tsp. = 2 tbsp. = 0.0078 U.S. gal. = 0.031 qt. = 29.57 gm. = 0.062 pt. = 0.065 lb. = 1.04 oz. 1 Imperial gallon = 4.5459L = 0.1605 ft.³ = 277.42 in. ³ 4.845 qt.
Weight Equivalents Parts Per Million 1 grain (gr.) = 64.8 mg. = 0.065 gm.= 0.35 oz.1 gram (gm.) = 15.432 gr.= 0.0353 fl. oz. = 0.0022 lb.= 0.002 pt.= 0.001 L = 1000 mg.= 0.001 kg.1 ounce (oz.) = 480 gr.= 28.35 gm.= 0.0075 gal.= 0.03 qt.= 0.06 pt. = 0.0625 lb.=0.96 fl. oz. 1 pound (lb.) = 5760 gr. = 373.24 gm. 1 pound, avoirdupois = 7000 gr. = 453.6 grams = 16 oz. = 0.12 gal = 0.016 ft.³ of water = 0.48 qt. = 0.96 pt. = 15.35 fl. oz = 458.59 cc or cm³ = 453.59 ml. = 453.59 gm. = 0.454 L Leave this text in this area for sizing purposes 1 ppm = 1 mg/L= 3.8 mg/gal. = 2.7 lb./acre foot= 0.0038 gm/gal. = 0.0283 gm/ft.³ 1 oz./1000 ft.³ = 0.0000623 lb./ft.³ = 0.0586 gr./gal. = 8.34 lb./million gallons water = 0.134 oz./1000 gal. = 1 oz./1000 ft.³ water = 1 gm./264 gal. water = 1 gm./m³ water 1 gr./gal. = 19.12 ppm.
©1995-2006 www.nationalfishpharm.com - Site Map - Site Design by Brian Aukes & AFAM LLC.
.
| 1,127 | 3,185 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.84375 | 4 |
CC-MAIN-2024-33
|
latest
|
en
| 0.78506 |
https://www.netexplanations.com/ml-aggarwal-cbse-solutions-class-6-math-ninth-chapter-ratio-and-proportion-exercise-9-1/
| 1,600,774,070,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-40/segments/1600400205950.35/warc/CC-MAIN-20200922094539-20200922124539-00759.warc.gz
| 988,959,567 | 11,267 |
# ML Aggarwal CBSE Solutions Class 6 Math Ninth Chapter Ratio and Proportion Exercise 9.1
### ML Aggarwal CBSE Solutions Class 6 Math 9th Chapter Ratio and Proportion Exercise 9.1
Ex. 9.1
(1) Express the following ratios in simplest form:
(i) 20 : 40
Solution: = 1 : 2
(ii)40 : 20
Solution: 2 : 1
(iii) 81 : 108
Solution: 9 : 12
= 3 : 4
(iv) 98 : 63
Solution: 14 : 9
(2) Fill in the missing numbers in the following equivalent ratios:
(3) Find the ratio of each of the following in simplest form:
(i) 2.1 m to 1.2 m
Solution: 2.1: 1.2
= 21/10 : 12/20
= 21 : 12
= 7 : 4
(ii) 91 cm to 1.04 m
Solution: 1.04 m = 104 cm
Now, 91 cm : 104 cm
= 7 : 8
(iii) 3.5 kg to 250 g
Solution: 3.5 kg = 3500 g
Now, 3500 g : 250 g
= 700 : 50
= 140 : 10
= 28 : 2
= 14 : 1
(iv) 60 paise to 4 rupees
Solution: 4 rupees = 400 paise
Now, 60 : 400
= 15 : 100
= 3 : 20
(v) 1 minute to 15 seconds
Solution: 1 minute = 60 seconds
Now, 60 : 15
= 4 : 1
(vi) 15 mm to 2 cm
Solution: 2 cm = 20 mm
Now, 15 mm : 20 mm
= 3 : 4
(4) The length and the breadth of a rectangular park are 125 m and 60 m respectively. What is the ratio of the length to the breadth of the park?
Solution: The ration of the length to the breadth of the park = 125 m : 60 m
= 25 : 12
(5) The population of village is 4800. If the number of females is 2160, find the ratio of the males to that of females.
Solution: Total population of the village = 4800
Number of females = 2160
∴ Number of males = 4800 – 2160 = 2640
The ratio of the males to that of females = 2640 : 2160
= 528 : 432
= 264 : 216
= 132 : 108
= 66 : 54
= 33 : 27
= 11 : 9
(6) In a class, there are 30 boys and 25 girls. Find the ratio of the number of
(i) boys to that of girls
(ii) girls to that of total number of students.
(iii) boys to that of total number of students.
Solution: (i) boys to that of girls =
30 : 25
= 6 : 5
(ii) Total number of students = 30 + 25 = 55
The ratio of girls to that of total number of students = 25 : 55
= 5 : 11
(iii) Total number of students = 55
The ratio of boys to that of total number of students = 30 : 55
= 6 : 11
(7) In a year, Reena earns Rs. 1,50,000 and saves Rs. 50,000. Find the ratio of
(i) money she earns to the money she saves
(ii) money that she saves to the money she spends.
Solution: Reena earns = 1,50,000
She saves = 50,000
That’s mean, she spends 1,50,000 – 50,000 = 1,00,000
(i) The ratio of money she earns to the money she saves = 1,50,000 : 50,000
= 30000 : 10000
= 6000 : 2000
= 3 : 1
(ii) The ratio of money that she saves to the money that she spends =
50,000 : 1,00,000
= 1 : 2
(8) The monthly expenses of a student have increased from Rs. 350 to Rs. 500. Find the ratio of
(i) increase in expenses to original expenses
(ii) original expenses to increased expenses
(iii) increased expenses to increase in expenses
Solution: Increase in expenses = 500 – 350 = 150
(i) The ratio of increase in expenses to original expenses = 150 : 350
= 30 : 70
= 6 : 14
= 3 : 7
(ii) original expenses to increased expenses = 350 : 500
= 70 : 100
= 14 : 20
= 7 : 10
(iii) The ratio of increased expenses to increase in expenses = 500 : 150
= 100 : 30
= 50 : 15
= 10 : 3
(9) Mr Mahajan and his wife are both school teachers and earn Rs. 20900 and Rs. 18700 per month respectively. Find the ratio of:
(i) Mr Mahajan’s income to his wife’s income
(ii) Mrs Mahajan’s income to the total income of both
Solution: Mr Mahajan’s income = Rs. 20900
Mrs. Mahajan income = Rs. 18700
Their total income = 20900 + 18700
= 39600
(i) The ratio of Mr Mahajan’s income to his wife income = 20900 : 18700
= 4180 : 3740
= 836 : 748
= 418 : 374
= 209 : 187
(ii) The ratio of Mrs Mahajan income to the total income of both =
18700 : 39600
= 3720 : 7920
= 744 : 1584
= 372 : 792
= 186 : 396
= 93 : 198
(10) Out of 30 students in a class, 6 like football, 12 like cricket and remainining likes tennis. Find the ratio of
(i) number of students liking football to number of students liking tennis.
(ii) number of students liking cricket to total number of students.
Solution: Total students = 30
Number of students like football = 6
Number of students like cricket = 12
Number of students like tennis = 30 – (6 + 12)
= 30 – 18
= 12
(11) Divide Rs. 560 between Ramu and Munni in the ratio 3 : 2
Solution: Total Rs. 560
Ramu get = 560 x 3/5
= 112 x 3
= 336
Munni get = 560 x 2/5
= 112 x 2
= 224
(12) Two people invested Rs. 15000 and Rs. 25000 respectively to start a business. They decided to share the profits in the ratio of their investments. If their profit is Rs. 12000, how much does each get.
Solution: Total investment = 15000 + 25000 = 40000
1st people get in profit = 12000 x 15000/40000
= 3 x 1500
= 4500
2nd people get in profit = 12000 x 25000/40000
= 7500
(13) The ratio of Ankur’s money to Roma’s is 9 : 11. If Ankur has Rs. 540, how much money does Roma have?
Solution: Let, the total money be x
Ankur has Rs. 540
Therefore,
X x 9/20 = 540
9x/20 = 540
=> 9x = 10800
=> x = 10800 / 9
=> x = 1200
∴ Roma get = 1200 – 540
= 1160
(14) The ratio of weights of tin and zinc in an alloy is 2:5. How much zinc is there in 31.5 of alloy?
Solution: The ratio of weights of tin and zinc in an alloy is 2:5
Tin have = 2/7
Zinc have = 5/7
∴ zinc have in an alloy = 31.5 x 5/7
= 315/10 x 5/7
= 22.5 g
Updated: December 11, 2019 — 2:03 pm
| 1,942 | 5,437 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.65625 | 5 |
CC-MAIN-2020-40
|
latest
|
en
| 0.822329 |
https://mersenneforum.org/printthread.php?s=eafc936076703210eb76f274be3d193f&t=13093
| 1,620,568,689,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-21/segments/1620243988986.98/warc/CC-MAIN-20210509122756-20210509152756-00377.warc.gz
| 416,694,188 | 2,930 |
mersenneforum.org (https://www.mersenneforum.org/index.php)
- Aliquot Sequences (https://www.mersenneforum.org/forumdisplay.php?f=90)
- - A new driver? (or type of driver?) (https://www.mersenneforum.org/showthread.php?t=13093)
10metreh 2010-02-15 08:03
A new driver? (or type of driver?)
2^8 * 7 * 73 is a guide. According to the [url=http://www.lafn.org/~ax810/analysis.htm]rules[/url], it is class 4, so it isn't too hard to escape.
However, if you add in a 5, a 19 and a 37 to produce 2^8 * 5 * 7 * 19 * 37 * 73, then the 2^8 keeps the 7 and the 73 there, the 19 keeps the 5 there, the 37 keeps the 19 there and the 73 keeps the 37 there. The main difference is that the new primes raise the power of 2 of the sigma to 8, and other prime factors will raise it to 9 or more, meaning that it will keep the 2^8.
As far as I can see, you can only escape this structure when one of the factors in it is squared (like a driver that isn't 2^3 * 3 or the downdriver), but according to Clifford Stern's page (linked above), the guide is just the 2^8 * 7 * 73. The original definition of drivers and guides only allows for drivers formed from factors of the sigma of the power of 2, but clearly other primes can have a huge effect.
I apologise if this is incorrect.
Mini-Geek 2010-02-15 12:00
A few experiments on your new "driver": (2^8*5*7*19*37*73 = 459818240)
[URL]http://factordb.com/search.php?se=1&aq=459818240*2639365095828977842437628607597655323897574150129623567*124032856897&action=range&fr=0&to=2[/URL] (37^2 at index 1; no 19, and so lost the driver, at index 2)
[URL]http://factordb.com/search.php?se=1&aq=459818240*101*107&action=range&fr=0&to=10[/URL] (7^2 at index 5; 2^7, and so lost the driver, at index 6)
[URL]http://factordb.com/search.php?se=1&aq=459818240*3&action=range&fr=0&to=2[/URL] (index 1=driver*3^2; 2^9, and so lost the driver, at index 2; it was lost without any of the driver's factors being squared, just that the non-driver cofactor was a square)
Doesn't seem to have much staying power.
10metreh 2010-02-15 14:06
Of course it's not like ordinary drivers in that it can keep the 2^8 while losing one of its factors, and it doesn't seem to stay for long, it's just the fact that it needs a squared factor to disappear that interests me.
Greebley 2010-02-15 15:57
The main difference is that a square will have a cascade effect in that all those dependent on the square term can be lost. So in MiniGeek's example the 37^2 lost the 19 which will eventually lose the 5 term and the 3 term (which wasn't listed but is based on the 5). Once the 19 is lost it is unlikely to get it back (1/19 every time we have a 37^2 term?)
Since the 37 would be squared every 37 iterations on average (I think), and keeping the 19 isn't very high if we do get a square, the long term stability is much lower than for a true driver. In the shorter term though, it is likely to last
All times are UTC. The time now is 13:58.
| 925 | 2,956 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.515625 | 4 |
CC-MAIN-2021-21
|
latest
|
en
| 0.852442 |
https://m.hausarbeiten.de/document/88496
| 1,624,336,434,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-25/segments/1623488507640.82/warc/CC-MAIN-20210622033023-20210622063023-00141.warc.gz
| 341,673,267 | 19,571 |
# Beitrag zum Jahr der Mathematik 2008
Neue bisher unbekannte, elementare Formeln für Dreiecke
Wissenschaftliche Studie 2008 174 Seiten
## CONTENT
PBI
Preface
Content
Part I: Triangles
The Theorems of W E H R L E
Area of right-angled triangles A = w + ¼ w*
w = A - r
Product of catheti ab = 2r(2R+r)
Diameter of incircle 2 r = a+b – c
Sum of catheti a+b = 2 (R+r)
Catheti only by radi a= R+r ±√ (R– r – 2rR) = R+r ± √ (R– A)
Theorems of W E H R L E od differences w = (2r) = 4r
The smalles rational triangles with integer sides x= l - m y = l (l - m) and z =m (l +1)
Every rational triangle consists of two rational rectangular ones
The distance of the centers of in- and circum circle d = R(R-2r) or d = √ (R + r - A)
Proving the Theorem of WEHRLE
Trigonomical Wehrles:
Sinus-Wehrle
The Wehrle-number of the differences
of the sinus-values is (r/R)²
Other Theories of tigomometrical Wehlre
The sum of squares of the sides
( a²+b²+c² ) =2( ab+ac+cb ) –( w* +8w)
Proof for the area-theorem: A = w + r
The triangle build by the touching-points
a´= 2 tA r / √( tA+r)=(b+c-a)r / √(¼(b+c-a)+r) etc.
It seems, that something still remains irrational
Proof for w = R² - d² (d is the distance of radii)
The bent of incircle is the sum of bents
of circles with the radii R+d and R-d .
A similar closure problem from J. Steiner
(R - d)-² + (R + d)-²= r-²
Cicle octagon
Exercises
The theorem of Wehrle for quadrangles
Wehrle for circle kites 2rR = abc / (a+b)
Wehrle for circle trapeziums 2rR = (a+c)√(a²+c²+6ac)
Exercises:
PART III: Pyramides
Still a dimension more
Just the volume is very simple
Thus the double product of radiu is
The Theorem of Fehringer-Wehrle
Conclusion
Exercises:
Part IV: n and infinite dimensions
From four to infinite dimensional space
Theorem of Pythagoras for n dimensions
Volume and surface of n-dim. spheres
Vanishing spheres
Appendix: The Theorems of F E H R I N G E R 113
Um- und Inkugelradien am allgemeinen Tetraeder
Erweiterung der Euklidischen Flächensätze auf das allgemeine Dreieck und die Fehringerschen Gleichungen nebst Anwendung zur Volumenbestimmung des allgemeinen Tetraeders.
References 144
## DESCRIPTION:
illustration not visible in this excerpt
Portrait from Erich Meyer
www.wehrle-formeln.net
The author, Hugo Wehrle, studied 1975-1982 mathematics and physics at the Albert-Ludwigs University Freiburg with the university state examination. Than he learned computer languages (programming: first COBOL, Assembler, Pascal, Unix, C) and later a course as systemadministrator for Windows- and Unix netwoks: the Microsoft certified system engineer (MCSC). Finally, after some other steps, I worked 1989-1996 in this IT-section by the AEG (general electric company) in Konstanz, producing postal sorting machines for the whole world; 1989 - 2001 by INTERNOLIX, producing software for online-jobs.
The author of the appendix is Arno Fehringer, who studies mathematics and physics at the Albert-Ludwigs University in Freiburg too, at about the same time as I did. He is still teacher for mathematics at the higher school (called „Gymnasiallehrer“) in Gailingen.
The subject of my book is elementary geometry in two, three, four and finally infinte dimensions. Brief examples of the content:
The area of a rectangular triangle is the sum of w and a quarter of w*
A = w + 1/4 w*
And there are some bradnew diagrams of Thoery of trigonomical Wehrle
The analogy of a right-angled triangle yields the three-dimensional theorem of Pythagoras ! If all branches at one edge of a polygonal pyramid are equal and the double of the height, than the radius of the circum sphere is just the length of these branches!
The circum radius R of any tetrahedron is formulated in the Theorem of Fehringer-Wehrle only by its branches:
illustration not visible in this excerpt
Than we leave the tree dimensional space, discuss four dimensional bodies and elevate finally to the simplices of infinite dimensional space, to show, that all the hypershperes have disappeared completely?
My book proves at the very end, that eucledean geometry can not be the last truth, or in other words, eucledean geometry is not the correct model for the really to describe the real space exactly, we live in[1] and physicists deal with! But nobody would ever notice this, if I would´nt write explicitely in the conclusion chapter. Without my logical conclusions, the book will be a work of elementary geometry for triangles, quadrangles, tetrahedrons and simplices, with some very new formulae (for example the one called Fehringer-Wehrle for general tetrahedrons), which never have been published before in any other book; last not least with a chapter of infinite dimensional eucledean space, where something strange happens: The spheres are diasappearing.
Purpose of the book is to present new ideas and to make people to be amazed! They shall be able, to come to the logical conclusion, that mathematics is reallity, but euclidean geometry is only a simplifying model. The book is a monograph and should be comprehensive for everybody, who really are interested in mathematics!
For certain, the reader must not have studied mathematics to understand, what I´m talkig about!
## MARKETING:
The book is a research monograph with many exercises at the end of the first three chapters. Research aereas are mew formulae for triangles, quadrangles and pyramides.
For example, that for any triangle the square of the relation of the incircle to circum circle radius is
illustration not visible in this excerpt
The radius of circum circle of a circle-trapezium is always c2/(2ab) times the length of the diagonals!
r = ac/(a+c), and of circum circle R = ½√(a²+c²) and distance of centers
|MuMi | = ½ {(r -x i) /(r+xi)} √(r²+xi²)
The distance of centers of the spheres of right-angled tetrahedrons is
|MiMu| = √[R2 + 3r2 - (a+b+c)r ] with the product of radii
4rR =[ab+ac+bc-√(a²b²+a²c²+b²c²)]√(a²+b² +c²) / (a+b+c)
For general tetrahedrons the radii product is
Why is this subject area important ?
Mathematics isn´t important for most people, only “things that nobody needs”! But we live in a contradictionary world. People use the technics, based in constructions, using the knowledge of mathematics, for example to look TV, or hear DVDs, or even to fly to the moon (IMPOSSIBLE WITHOUT COPMUTERS; thus impossible without mathematics of gravitation etc). But almost nobody cares and it isn`t honored, doing mahematics and searching for the very basic and ultimate reasons of the things, that keep the univers together, rather than playing football, baseball, basketball, rugby, tennis, beachball, soccer or other sports, or playing songs, where you can earn millions of dollars, standing on the stage or producing movies like R. Reagan (or Arnold Schwarzenegger), getting well-known by this way and therefore president.(or governor).
But it is very importand for western world, to be at the top in knowledge and technical know-how, to use newest machinery for the production of export articles, because otherwise the third world will overrun us! And mathematics is the basic language of all science and technology.
But I needn`t explain you the necessity of mathematics or geometry, or should I? My book is very good illustrated and comprehensible, it has colours and delights, a storehouse of new ideas, a productive ground for a lot of rosy geometical trees!
The best available book for elementary geometry is Donald COXETER`s >>Introduction to Geometry<<. For more philosophical questions like the question of Euklid reality for example or for metamathematics, it is, - as far as I know -, W. N. Molodschi`s >> Studien zu philosophischen Problemen der Mathematik << VEB Deutscher Verlag der Wissenschaften, Berlin 1977.
## Preface
Can you imagine, that there are elementary and still unrevealed theorems for the triangle, although mathematicians handle with these most simple objects of planimetry since over 2 500 years?
Remember all the greek geometrical genius like the first philosopher Thales of Milet (≈625-547 BC), the vegetarian school of the brotherhood of Pythagoras (born at about 600 BC), the thirteen volumes of the elements of Euclid (living in Alexandria about 300 BC), Zenon from Elea (≈490-430) or Archimedes from Syracus (≈285-212)! The world-outlookings Aristarchos from Samos and Plolemäus from Alexandria. And finally Hypatia from Alexandria: her murderers have killed the mathematics too - for about thousand years, the mathematics disappears with her completely: The roman empire could not live without soldiers, but without mathematicians and this longer as any other on the world!
All these knew triangles, quadrangles and tetrahedrons! Could it be, that there is something still unfound, unsolved or not to find in any book of mathematics or formulary? You know, that the product of the sides of any triangle divided through their sum (called cicumference) is just the double product of their radius of the circumscribed and inscribed circles? That the sum of the sides of catheti is just the sum of the diameters: a+b = 2(r+R)? Their half product (the area) is the sum of the Wehrle-number w and a quarter of the Wehrle-number of the differnces w*:
A = w + 1/4 w*
Where w* is the square of the diameter of the incircle (2r) power 2.
The diameter of the incircle is just the sum of the two minor sides subtracted by the largest one. Can you express the right-angled sides only by their in- and circumscribed radius? You know the smallest of the rational (not rectangled) triangles, having natural numbers for the sides[2] ; or the smallest rational isosceles triangles with integer lengths of sides? Both can be constructed by using “pythagorean[3] ” triangles only
During the matter in chapter I are triangles, it`s quadrangles in chapter II. Witch of the quadrangles have an in circle or circum circle and what´s the product of their radii 2rR? Witch kind of quadrangles exists, having integer sides and areas, altitudes and in- or circum circles all rational?
You dont know the theorems of the Sinus-Wehrle or the Sinus-Wehrle of the differences, indicating the relation of the radii of the incircle to the circum circle of any triangle?
Than in chapter III for pyramides, we leave the plane and therefore we need one more dimension. The analogy of a right-angled triangles yields the right angled tetrahedron and the three-dimensional Theorem of Pythagoras ! You surely dont know the radius product of the in- and circum sphere of any tetrahedron deduced only by its branches?
If all branches at top vertice of any polygon-pyramide are equal, than its radius of circum sphere is just the square of this length divided through the double altitude of pyramide: Specially if the altitude is just half of length l of equal branches at the top, its circum sphere radius is just R=l!
You know, that the content of a sphere in any dimension relates to its boundary always like the dimension n to its radius r.
In the last chapter we leave the tree dimensional space and finally elevate to infinite dimensional space. Would you last not least understand, why the hypershperes there have vanished all completely? Everybody will think intuitively, that this is just impossible! That there exists no points of a constant distance to a fixed point, - the middle -, nobody can imagine! So, only one conclusion is possible: Something must be wrong with the euclidean geometry! That’s the reason why Albert Einstein, who tried decades of his last years to get the G and U nification T heory (GUT or TOE: Theory Of almost Everything), excused his failing with this statement:
“A new mathematics would be necessary”!
In this book, the basic problem about the number of dimensions of euclidean spaces up to the infinity of dimensions is approached. You must know, that there exist proofs in mathematics, that are only nominal. For example, almost all mathematicians will tell you, that there is much more than the infinity. They prove you, that there are much, much more real numbers (א 1) as integers (א 0).
But indeed, it´s infinite, and how can something be more as this potential and never reachable infinity, only existing in mind, but not real?
Yes, they will make you believe, that there is very, very much, much more and even infinite infinitely more as the infinity of the integer numbers: that there is no pope among the cardinals of infinities, this never ending infinity of infinities. They call this transfinite mathematics, and they handle with this infinities א n as like as with integers and, - bringing their own mindgames with them -, asking for the infinity א ½ according to the fraction ½, the midst between the first integers zero and unit! A cardinal error, because they will never ever find or prove it!
It might be, if there is a space with more than “only” one infinity of dimensions, - they could explain to you -, there could be two or more limiting values to infinity (for example for the volumes of n-dimensional unit-spheres).
Finally they make you sure to find statements, you never will be able to say, if it`s true or not (like Kurt Gödel does), anyway in wich sytem of axioms you will move. They tried to breaking down the whole mathematical building of causality! And I‘m sure, you will never, nerver really understand.
I will call a spade a spade! You will find no ambiguity at all, but only brilliant clarity. Perhaps you will suddenly understand mathematical statements, you never would dream of doing!
illustration not visible in this excerpt
Arthur Schopenhauer
## Part I The Theorems of >>W E H R L E<<
illustration not visible in this excerpt
What have all these triangles common
The Wehrle-number is simple the product divided through the sum:
illustration not visible in this excerpt
For example, let’s take the first three natural numbers 1, 2 and 3.
Its Wehrle-number is w (1, 2, 3) = 1 x 2 x 3 : (1+2+3) = 1
Add now respectively two of these first natural numbers 1, 2 and 3, than you get as sums the three sides of a triangle with a =1+2 = 3, b =1+3 = 4 and c =2+3= 5. In this case it is even right angled, that means it has an angle of 90° degrees, because of the Theorem of Pythagoras: 3+4=5.
Its incircle-radius r is just this number of Wehrle 1.
(In general r is the square root of the Wehrle-number of these tangents!)
The Wehrle-number of the sides 3, 4 and 5 of this pythagorean triangle is
w (3, 4, 5) = 3 x 4 x 5 : ((4-1) + 4 + (4+1)) = 3 4 5 : (3 4) = 5
and all triangles of the figure above have the same Wehrle-number five !
In other words, all those triangles have the same product of the radius of incircle r and circum circle R.
You know, the Wehrle-number of the sides of a triangle, (shorter: of the triangle) is just the double product of the two radius, that is of the incircle, touching all sides, and of the circum circle, passing through all three edges. (Two points define a straight line uniquely, and three a circle, which center is the crossing point of the perpendicular bisctors of the sides defined by these three edges!)
illustration not visible in this excerpt
Example: 3 x 4 x 5 : (3 + 4 + 5) = 2 x 1 x 2,5 where r = 1 and R = 2,5
illustration not visible in this excerpt
Tangents are 1, 2 and 3
illustration not visible in this excerpt
The area of w and w* (a quater of w* is r 2 )
Cause the Wehrle-number has the unit of an area, it could be visualized by an area too. Now, the Wehrle-number of an rectangular triangle is just the area of the triangle, subtracting the quarter of another Wehrle-number, the Differences-Wehrle w*:
A = w + ¼ w* .
Because this Wehrle-number of differences w* is exactly the square of the diameter of the incircle, as we will see later, it follows:
w = A - r 2
The area of rectangular triangles is the sum of the Wehrle-number and the square of the incicrle-radius!
Example: A = ½ x 3 x 4 = 6 and because r = 1, it is w = 6 - 1 = 5
The double area 2A is just the area of a rectangle with the sides, joining the right angle. Therefore
illustration not visible in this excerpt
and the product of the two catheti is with w=2rR logical
illustration not visible in this excerpt
Example: For a=3 and b=4 follows r=1 and R=2,5: 3x4 = 2x(5+1)
The largest side of a rectangular triangle is called hypotenuse. Because the longest side opposites the largest angel, the smallest side opposites the smallest angel, the hypotenuse opposites always the greatest right angle (The sum of angles is 180°, if the postulate of parallels is valid)
Let the hypotenuse be c. This is the diameter of the circum circle too, and we will understand soon, that the diameter of the incircle is exactly
2 r = a+b – c ( = differences of sides dc)
The roundabout way over the two catheti a and b is exactly 2r more (the diameter of the incircle) than the direct way by hypotenuse c.
Cause the hypotenuse is the diameter of the circum circle- (circle of Thales), we get
The sum of catheti is just the sum of diameters!
a+b = 2 (R+r)
Example: a=3 and b=4 a+b = 3+4 = 5+2 = 2R+2r
Both equations have the two variables a and b containing only terms with the radius r and R and yield together a theorem for the catheti a1 and a2 only depending from the radii:
illustration not visible in this excerpt
where a = a1 and b = a2 , or vis versa
Example: R=2,5 and r=1 yields for the sum of radii R+r = 2,5+1 = 3,5
The term under then radical sign is 2,5 - 1 - 5 = 0,25 a square here, their root 0,5. Hence the catheti are a= 3,5 +0,5 =4 and b = 3,5 - 0,5 = 3 (hypotenuse is 2R=5).
Back to general triangle. Let’s call these expressions of three sides, where from the sum of two the third is to be subtracted
illustration not visible in this excerpt
as differences.
These are precisely the double tangents, in which the touching point of incircle divides the side. The very small one is the radius of the incircle, cause the hypotenuse c is the largest side, opposite the right angle (at the opposite edge C the tangents form a square).
These differences express the so-called „un-equalities of the triangle“, that means, that the sum of two sides has always to be greater than the third side:
illustration not visible in this excerpt
therefore dc = a+b-c has to be positiv etc.
Construct now the Wehrle-number of the differences w * („w-star“),
illustration not visible in this excerpt
where the denominator is
illustration not visible in this excerpt
We result the square, surrounding the incircle, for any triangle
illustration not visible in this excerpt
The largest side opposing edge and both touching points of the together with the incircle-center Mi form a square, and the Wehrle-number of the differences w * is the circumscribed square of the incircle 4r2 .
### The smalles rational triangles with integer sides
If the Wehrle-number of tangents is a square, we can construct a rational or discreet triangle (no square roots to count), that means with rational area, altitudes, sinus values of the angles and radii too. If the sides of the triangle shall be natural (integer), than the smallest I proved, which is not rectangular, has the tangents 1, 3 and 12 with the Wehrle-number w(1,3,12) = (1 3 12) : (1+3+12)= (3:2)2.
Its radius for the incircle is therefore 1,5.
The sides 4, 13 and 15 are the three two-sums of its tangents 1, 3 and 12:
illustration not visible in this excerpt
r=1 x 3 x 12/(1+3+12)=9/4, thus r= 1,5 Its radius of circum circle is R=w/2r= 4x13x15 /(4+13+15) :3 = 65 : 8 = 8,125
and its area is A = ½ru =½1,5x32 = 24
The tangents x, y and z building a Wehrle-square-number are
x= l - m2 y = l (l - m2) and z =m2 (l +1)
with rational l and m ( l > m2 )
Example: For l=3 and m=1,5 you receive x=3:4 , y=9:4 and z=9.
Multiplied by 4:3 results in x=1, y=3 and z=12, the discrete triangle with sides a=4, b=13 and c=15 and r=1,5.
Specially for m=1 you get only rectangular triangles:
illustration not visible in this excerpt
Example: l=2 yields the minimal discrete one a=3, b=4 and c=5.
The general pythagorean triads are: l2 - m2 , 2lm und l2+ m2
The resulting discrete triangle has the following sides:
illustration not visible in this excerpt
Example: For l=5 and m=2: a = 6, b = 25 and c = 29 with r = 2, A=60.
Every rational triangle consists of two rational rectangular ones!
If it`s a right angled one, the relation of the parts of hypotenuse c devided by the altitude is the relation of squares of cathti: q.p= a : b
Because the area is rational, the altitude (2A/side) is rational too! If one or both area(s) of the rectangular parts is (are) irrational, the sum of the areas A would not be rational. Therefore the sides in question of the rectangular triangles can not be (pure) irrational.
Example: The triangle with the sides 4, 13 and 15 can be split up into the two rectangular with the sides, 2,4, 3,2 and 4 and the other with sides 3,2, 12,6 and 13., The largest side 15 is divided by the height (altitude) into 2,4 and 12,6
Counter- Example: The triangle 2, 5 and 6 has the area
¼√(2+5+6)(-2+5+6)(2-5+6)(2+5-6) = ¼√(13 x 9 x 3 x1) = ¾√39.
The largest side 6 is divided by its altitude into 1,25 and 4,75. But the altitude is not rational: h = ¼√39
### The distance of the centers of in- and circum circle
Very interesting is distance of the centers Mi and Mu, we will note as d = |MiMu|. In rectangular triangle d is always greater the radius r of incircle, cause its the altitude over the hypotenuse c of the triangle, perpendicular to c like the radius of incircle. The middle of c yields Mu and the radius of the circum circle is R= ½. The sides are connected with d through the equation
2(a+b) c = 3c – 4d. As we will still see, this distance d of the center Mi and Mu is not really connected with the sides of the triangle, - and therefore depends not from the concrete triangle itself-, but is only connected with the radii r and R.
You can not construct any triangle into too circles, the one containing the other, without touching it, but only for a certain determined distance of centers. This distance d of centers is very important and has to be he geometrical average of the bigger radius R and the difference of R - 2r:
illustration not visible in this excerpt
Therefore the diameter of the incircle 2r has to be smaller than the radius of the circum circle R, and is at the outside equal, that is only for he regular case, the equilateral triangle, where the radius of circum circle R is just equal to the diameter of the incircle 2r. Because here the perpendiculars in the middle of the sides divide the angles too, and cut each other in the relation 1:2. Cause of R=2r it results a zero distance d=0: Like for every regular polygon, its in- and circum circle are concentrically.
Abbildung in dieser Leseprobe nicht enthalten
For two circles in one another there is
either not a single triangle at all
or there exists an infinite number of triangles !
In case of rational radii and distance of centers, the sides are irrational
(here the catheti are 11+√7 and 11-√7)
The distance condition can be formulated too as:
The square of the distance d has to be the difference of the square of circumradius R and the Wehrle-number!
Abbildung in dieser Leseprobe nicht enthalten
Comparison of the differences
w = A - r with w = R– d (broken lines)
In this special case, there are the cuts of the hypotenuse 2 and 3
with just the product h = A
For rectangular triangles the distance d of the centers of in- and circum circle is always greater than the radius of the incircle r, cause r and d form a rectangular triangle at the hypotenuse c, except for the symmetrical case of an isosceles right angled triangle, - half a square -, where d is perpendicular to the hypotenuse and d = r= a-½c=½ (√2 -1) c.
illustration not visible in this excerpt
The area of a rectangular triangle (half a rectangle) is smaller than the square of circumradius; equality A = R only for the isosceles right angled one (half a square)
The distance d of centers for right-angled triangles is:
illustration not visible in this excerpt
Proving the theorem of WEHRLE!
The Wehrle-number w of the three sides a, b and c of a triangle, is the double product of the radii of the incenter r and circumcenter R
abc : (a+b+c) = 2 rR
and the square root of the differences-Wehrle-number w « of the sides dc = a+b - c , db = a b + c and da = - a + b + c is equal to the incircle-diameter
√ w*= 2r
As you all know, every triangle is definite determined by the length of its three sides, that is, its appearance or form is uniquely described. Therefore all things of the triangle can be calculated by these three sides!
For example the bisector of the side a is[illustration not visible in this excerpt],
the bisector of the angle at he edge A [illustration not visible in this excerpt] : (b+c) or the altitude through A is
[illustration not visible in this excerpt] , which is longer than the incircle-radius[illustration not visible in this excerpt] for the amount of r(b+c):a
The square of a side ca be computed by the square-sum of the other ones, corrected by subtracting the double product of these sides and the cosines of thereby defined angle c2 = a2 +b2 - 2 a b cos γ.. Therefore is
illustration not visible in this excerpt
On the other side, you know sin2 γ + cos2 γ = 1, and sin γ is calculated as:
illustration not visible in this excerpt
and finally
illustration not visible in this excerpt
II.) This sinus-value you can calculate too as sin γ = c : (2R) Equate I. with II. yields
illustration not visible in this excerpt
Therefore the product of the three sides is
illustration not visible in this excerpt
This is proved by simple calculation:
illustration not visible in this excerpt
Because the sum of differences is u = (-a+b+c)+(a-b+c)+(a+b-c), it follows:
illustration not visible in this excerpt
It remains to show, that √w* = 2r .
With I.) put into A = ½ ab sin γ, and on the other hand A = ½ (a+b+c) r
illustration not visible in this excerpt
With
illustration not visible in this excerpt
is now proved:
illustration not visible in this excerpt
qed.
illustration not visible in this excerpt
The sum of the three sinusvalues of the angles of any triangle (left)
illustration not visible in this excerpt
and the sum of the squares of the three sinusvalues z = sin αi of the angles (right)
for the angles x and y from 0 to π
The sum of the sinus-values sin αi is proportional to the sum of the sides ai and the sum of the sinus-squares is is proportional to the sum of the squares of sides. General
illustration not visible in this excerpt
[...]
[1] As we all know, Albert Einstein proved, that the space is not even. But telling this to a working man, that space has a curvature, than he closes the doors suddenly („die Rolläden gehen herunter und der Bordstein wird hochgeklappt“)! And this will be the end of every discussion, with the result, that you will be in his eyes a lunatic all live long!
[2] It`s integer sides are 4, 13 and 15 The thre integer sides are the pythagorean triplets: l2 - m2 , 2lm und l2+ m2 for integer l and m
[3]
## Details
Seiten
174
Jahr
2008
ISBN (eBook)
9783638057837
DOI
10.3239/9783638057837
Dateigröße
8.9 MB
Sprache
Englisch
Katalognummer
v88496
Note
Schlagworte
Beitrag Jahr Mathematik Dreieck Unendlichdimensionaler Raum Viereck Pyramiden Simplices
| 7,239 | 27,451 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.859375 | 4 |
CC-MAIN-2021-25
|
latest
|
en
| 0.612393 |
https://testbook.com/question-answer/two-large-parallel-plates-having-a-gap-of-10-mm-in--5e19860ff60d5d2d6cf17eb8
| 1,632,280,592,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00052.warc.gz
| 594,853,454 | 32,343 |
# Two large parallel plates having a gap of 10 mm in between them are maintained at temperatures T1 = 1000 K and T2 = 400 K. Given emissivity values, ε1 = 0.5, ε2 = 0.25 and Stefan-Boltzmann constant σ = 5.67 × 10-8 W/m2-K4, the heat transfer between the plates (in kW/m2) is ________
This question was previously asked in
PY 11: GATE ME 2016 Official Paper: Shift 3
View all GATE ME Papers >
## Answer (Detailed Solution Below) 10.9 - 11.2
Free
CT 1: Ratio and Proportion
1963
10 Questions 16 Marks 30 Mins
## Detailed Solution
Concept:
Heat transfer between two large parallel plates having a small gap is given by:
$$Q = \frac{{\sigma \left( {T_1^4 - T_2^4} \right)}}{{\frac{1}{{{\varepsilon _1}}} + \frac{1}{{{\varepsilon _2}}} - 1}}$$
Calculation:
Given:
Two large parallel plates having a gap of 10 mm.
T1 (Plate 1 temperature) = 1000 K
ε1 = 0.5
T2 (Plate 2 temperature) = 400 K.
ε2 = 0.25
And σ = 5.67 × 10-8 W/m2K.
To find heat transfer between plates (in kW/m2).
We know that;
$$Q = \frac{{\sigma \left( {T_1^4 - T_2^4} \right)}}{{\frac{1}{{{\varepsilon _1}}} + \frac{1}{{{\varepsilon _2}}} - 1}}$$
$$Q = \frac{{\left( {5.67 \times {{10}^{ - 8}}} \right)\left( {{{1000}^4} - {{400}^4}} \right)}}{{\frac{1}{{0.5}} + \frac{1}{{0.25}} - 1}}$$
Q = 11049.7 W/m2
Q = 11.05 kW/m2
| 495 | 1,302 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.921875 | 4 |
CC-MAIN-2021-39
|
latest
|
en
| 0.797528 |
https://brainmass.com/economics/cost-benefit-analysis/equivalent-annual-worth-economics-engineering-81535
| 1,586,276,233,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-16/segments/1585371803248.90/warc/CC-MAIN-20200407152449-20200407182949-00075.warc.gz
| 379,519,033 | 15,741 |
Explore BrainMass
Share
# Equivalent annual worth for economics engineering
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
1) Consider the following estimates, and use an interest rate of 10% per year. The equivalent annual worth of alternative A is closest to:
A) \$-25,130
B) \$-37,100
C) \$-41,500
D) \$-42,900
2) Consider the following estimates, and use an interest rate of 10% per year. The equivalent annual worth of alternative B is closest to:
A) \$-25,130
B) \$-28,190
C) \$-37,080
D) \$-39,100
3) With an interest rate of 10% per year and given the following estimates, the annual worth of alternative F is closest to:
A) \$32,600
B) \$36,100
C) \$39,020
D) \$43,500
4) Given the following estimates, and with an interest rate of 10% per year, the annual worth of alternative G is closest to:
A) \$10,000
B) \$30,000
C) \$36,000
D) \$40,000
5) The first cost of a fairly large flood control dam is expected to be \$5 million. The maintenance cost will be \$60,000 per year, and a \$100,000 outlay will be required every 5 years. If the dam is expected to last forever, its equivalent annual worth at an interest rate of 10% per year is closest to:
A) \$-576,380
B) \$-591,580
C) \$-630,150
D) \$-691,460
6) An investment of \$50,000 resulted in uniform income of \$10,000 per year for 10 years and a single amount of \$5000 in year 5. The rate of return on the investment was closest to:
A) 10.6% per year
B) 14.2% per year
C) 16.4 % per year
D) 18.6 % per year
7) Five years ago, an alumnus of a small university donated \$50,000 to establish a permanent endowment for scholarships. The first scholarships were awarded 5 years after the money was donated. If the amount awarded each year (i.e., the interest) is \$5000, the rate of return earned on the fund is closest to:
A) 7.5% per year
B) 10% per year
C) 11% per year
D) 14% per year
8) When positive net cash flows are generated before the end of a project, and when these cash flows are reinvested at an interest rate that is less than the internal rate of return,
A) The resulting rate of return is equal to the internal rate of return.
B) The resulting rate of return is less than the internal rate of return.
C) The resulting rate of return is equal to the reinvestment rate of return.
D) The resulting rate of return is greater than the internal rate of return.
9) A \$10,000 municipal bond due in 10 years pays interest of \$400 per year. If an investor purchases the bond now for \$9000 and holds it to maturity, the rate of return received by the investor will be closest to:
A) 3.5% per year
B) 4.2% per year
C) 5.3% per year
D) 6.9% per year
10) The difference between revenue and service alternatives is that service alternatives assume revenues are the same for all alternatives.
A) True
B) False
11) The rate of return for alternative X is 18% and for alternative Y is 16%, with Y requiring a larger initial investment. If a company has a minimum attractive rate of return of 16%,
A) The company should select alternative X.
B) The company should select alternative Y.
C) The company should conduct an incremental analysis between X and Y in order to select the correct alternative.
D) The company should select the do-nothing alternative.
12) When conducting a ROR analysis of mutually exclusive projects that have revenue estimates,
A) All the projects must be compared against the do-nothing alternative.
B) More than one project may be selected.
C) The project with the highest ROR should be selected.
D) An incremental investment analysis maybe necessary to identify the best one.
13) Consider the estimates below. If the alternatives are mutually exclusive and the MARR is 15% per year, the one(s) that should be selected is (are):
A) A
B) D
C) E
D) None of them
14) When a B/C analysis is conducted,
A) The benefits and costs must be expressed in terms of their present worth
B) The benefits and costs must be expressed in terms of their annual worth
C) The benefits and costs must be expressed in terms of their future worth
D) The benefits and costs can be expressed in terms of PW, AW, or FW.
15) An alternative has the following cash flows: benefits = \$50,000 per year; disbenefits = \$27,000 per year; costs = \$25,000 per year. The B/C ratio is closest to:
A) 0.92
B) 0.96
C) 1.04
D) 2.00
16) In evaluating three mutually exclusive alternatives by the B/C method, the alternatives were ranked in terms of increasing total equivalent cost (A,B, and C, respectively), and the following results were obtained for the B/C ratios: 1.1, 0.9, and 1.3. On the basis of these results, you should:
A) Select A
B) Select C
C) Select A and C
D) Compare A and C incrementally.
17) The economic service life of an asset is:
A) The longest time that asset will still perform the function that it was originally purchased for.
B) The length of time that will yield the lowest annual worth of costs.
C) The length of time that will yield the lowest present worth of costs.
D) The time required for its market value to reach the originally estimated salvage value.
18) In a replacement study conducted last year, it was determined that the defender should be kept for 3 more years. Now, however, it is clear that some of the estimates that were made for this year and next year were in error. The proper course of action is to:
A) Replace the existing asset now.
B) Replace the existing asset 2 years from now, as was determined last year.
C) Conduct a new replacement study using the new estimates.
D) Conduct a new replacement study using last year's estimates.
19) When all future cash flows are expressed in constant-value dollars, the rate that should be used in the factor equations is the:
A) Market interest rate.
B) Inflation rate.
C) Inflated interest rate.
D) Real interest rate.
20) An assembly line robot arm with a first cost of \$25,000 in 1985 had a cost of \$29,860 in 1992. If the M&S equipment cost index was 789.6 in 1985 and the robot cost increased exactly in proportion to the index, the value of the index in 1992 was closest to:
A) Less than 800
B) 832.3
C) 914.6
D) More than 925
21) A 50-hp turbine pump was purchased for \$2100. If the exponent in the cost capacity equation has a value of 0.76, a 200-hp turbine could be expected to cost about:
A) Less than \$5000
B) \$5980
C) \$6020
D) More than \$6100
22) A machine with a 10-year life is to be depreciated by the MACRS method. The machine has a first cost of \$30,000 with a \$5000 salvage value. Its annual operating cost is \$7000 per year. The depreciation charge in year 3 is nearest to:
A) \$3600
B) \$4320
C) \$5860
D) \$7120
23) An asset with a first cost of \$50,000 is to be depreciated by the straight-line method over a 5-year period. The asset will have annual operating costs of \$20,000 and a salvage value of \$10,000. According to the straight line method, the book value at the end of year 3 will be closest to:
A) \$8000
B) \$20,000
C) \$24,000
D) \$26,000
24) An asset with a first cost of \$50,000 is depreciated by the straight line method over a 5-year life. Its annual operating cost is \$20,000, and its salvage value is expected to \$10,000. The book value at the end of year 5 will be nearest to:
A) \$0
B) \$8000
C) \$10,000
D) \$14,000
25) An asset had a first cost of \$50,000 an estimated salvage value of \$10,000 and was depreciated by the MACRS method. If its book value at the end of year 3 was \$21,850 and its market value was \$25,850, the amount of depreciation charged against the asset up to that time was closest to:
A) \$18,850
B) \$21,850
C) \$25,850
D) \$28,150
26) Calculate the annual worth (years 1 through 10) of the following series of disbursements. Assume that i = 12% per year.
A) \$-6817
B) \$-5817
C) \$-4817
D) \$-7817
https://brainmass.com/economics/cost-benefit-analysis/equivalent-annual-worth-economics-engineering-81535
#### Solution Preview
Hello!
Question 1 - Answer is B
In order to find the equivalent annual worth, we first find the present worth of machine A:
PV = -50000 - 20000/1.1 - 20000/1.1^2 - 10000/1.1^3 = \$92,223
[notice that the last payment is -\$10,000 rather than -\$20,000, because of thesalvage value of \$10,000]
Now, the equivalent annual worth is the annual payment for a number of periods equal to the duration of the machine (3 periods) such that their present worth is the same as the present worth of the machine.
As before, we proceed first by finding the present worth of \$1 on the next 3 years:
PV = 1/1.1 + 1/1.1^2 + 1/1.1^3 = \$2.48685
Therefore, the annual payment whose PV is \$92,223 is 92223/2.48685 = \$37,084
So that's the equivalent annual worth of this machine.
Question 2 - Answer is A
The idea is the same as in question 1. We first find the present worth of this machine:
PV = -80000 -10000/1.1 - 10000/1.1^2 - 10000/1.1^3 - 10000/1.1^4 - 10000/1.1^5 + 15000/1.1^6
The result is \$109,440.75. We now find the present worth of a \$1 cash flow each year for 6 years:
PV = 1/1.1 + 1/1.1^2 + 1/1.1^3 + 1/1.1^4 + 1/1.1^5 + 1/1.1^6
The result is \$4.35526. Therefore, the equivalent annual worth of machine B is 109440.75/4.35526 = \$25,128.40
Question 3 - Answer is C
The idea is once again the same as in the previous question. We first find the present worth of the machine; then we find the present worth of a \$1 cash flow (for a number of years equal to the duration of the machine), and then we divide the former by ...
#### Solution Summary
The value of an asset at the end of its life is determined.
\$2.19
| 2,693 | 9,570 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.5 | 4 |
CC-MAIN-2020-16
|
latest
|
en
| 0.90974 |
https://www.teacherspayteachers.com/Product/Solving-Two-Step-Equations-with-a-DoUndo-chart-97427
| 1,503,385,001,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-34/segments/1502886110471.85/warc/CC-MAIN-20170822050407-20170822070407-00327.warc.gz
| 982,118,233 | 25,167 |
## Main Categories
Total:
\$0.00
Whoops! Something went wrong.
# Solving Two Step Equations with a Do/Undo chart
Subject
Product Rating
File Type
PDF (Acrobat) Document File
285 KB|2 pages
Share
Product Description
This worksheet is a great way to introduce solving two step equations. It helps the students use inverse operations in the correct order by first looking at what has been done to the variable (the Do part of the chart), and then how to "undo" those steps. The second page has more practice problems, but no chart, so the students have to practice using inverse operations more independently. This is a pdf; if you'd like it in Microsoft Word (for revising), let me know.
Here is an idea of how to use it:
When students are solving two-step equations for the first time, they are often unclear on which number to “undo” first. I’ve tried a variety of tricks, but the do/undo chart works very well for many of the kids. They use the scaffold for a couple days, and then they are able to do the same steps in their heads. What I like about the do/undo chart is that it connects solving equations to the Order of Operations.
This is one way to present it:
1. Start off by reviewing order of operations with various problems. Then, write a problem on the board, like 15(3) + 9 = 54. Ask if it is true or false, and to explain the steps of proving that it is true (multiply 15 times three first, then add 9, which does in fact equal 54).
2. Then, write the equation 12x + 8 = 92. Highlight the variable. Discuss whether or not they can use order of operations to solve it (they can’t because they can’t multiply first, because they don’t know the value of x).
3. I explain that when there is a variable in an equation, you can’t follow the order of operations, so we have to treat it more like a mystery. I emphasize that we have to “undo” everything that should be done (or has been done to the variable). It’s a mystery…we’re finding an unknown, so we have to work backwards to discover what the variable’s solution is.
4. I introduce the DO/UNDO T-chart. I say that it is just a tool that helps us keep track of how to undo the equation which solves the mystery. I tell them that we’ll only use it for a day or two, and that after that it’s optional. We’re still using the 12x + 8 = 92 equation.
5. Starting on the DO side, I ask, “If we could, what would we DO first in the equation (using order of operations). “ Or, another way I’ve presented it is “What has been done to the variable?” The answer is “multiply by 12), which is what I write in the chart under the DO part.
6. I ask, “What would be done next?” or… “What has been done to the variable next?” The answer is ADD 8. I write that in the chart under •12.
7. Now that we have written the two steps in the DO side, I explain that we can solve the mystery by UNDOING these steps. I tell them to put their pencils on the Star…because that is where they need to STARt. They look at the DO side, which says +8. I ask how to undo it, and they say -8, which we write down. That is the first step to solve the equation, so I have them do that step on the actual equation. They subtract 8 from both sides (which they already know from solving one-step equations). Then we move to the next DO/UNDO step. Since it says •12 in the DO side, then need to write /12 in the UNDO side, and then divide by 12 on both sides of the equation. They are left with x = 7, which is the solution.
So, the DO/UNDO chart helps students identify what has been done to the variable, and how to undo it. It helps to explain to them that solving an equation is using Order of operations backwards, because we’re undoing a problem…solving a mystery. It also helps if they know that the order of the steps matters. If they had divided by 12 first, and then subtracted 8, they would have gotten an incorrect answer. The chart helps them know which step to do first until it becomes more automatic.
Finally, it’s good to give them equations with simple math, but also harder math. If all the equations are too simple, they hate having to show their steps. One year I gave them really difficult numbers (big numbers or with decimals) but let them use calculators to do the calculations as long as they showed their steps. It worked well…they were so happy that they didn’t have to do the math, they were willing to show their steps. They also couldn’t use mental math to solve the problem before they started.
Here is an advanced activity where students solve two-step equations with integers:
Solving Two Step Equations: A Riddle
Total Pages
2 pages
N/A
Teaching Duration
N/A
Report this Resource
\$0.75
More products from Shawna H
\$0.75
| 1,131 | 4,692 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.125 | 4 |
CC-MAIN-2017-34
|
longest
|
en
| 0.944176 |
https://dyplasticpallet.com/qa/question-how-do-you-calculate-monthly-occupancy-percentage.html
| 1,611,628,273,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00782.warc.gz
| 300,644,701 | 8,215 |
Question: How Do You Calculate Monthly Occupancy Percentage?
What is average hotel occupancy rate?
What is the average hotel occupancy rate.
For the most part, between 2015 and 2019, global hotel occupancy rates have remained between 50% and 80%, with peaks and troughs in line with seasonality.
However, there have been some occasions where occupancy has drifted outside these margins..
How do you calculate multiple occupancy percentage?
Multiple Occupancy Ratio / Multiple Occupancy Percentage CalculatorSingle Occupancy % (Available Rooms) = (Number of Single Rooms Occupied) / (Total Number of Available rooms) * 100.Double Occupancy % (Availalbe Rooms) = (Number of double Rooms Occupied) / (Total Number of Available rooms) * 100.More items…
How do you calculate the room occupancy percentage and the average daily rate?
Simply multiply your average daily rate (ADR) by your occupancy rate. For example if your hotel is occupied at 70% with an ADR of \$100, your RevPAR will be \$70. The other way to calculate it is by dividing the total number of rooms available in your hotel with the total revenue from the night.
What is a good occupancy rate?
While a 100 percent occupancy rate is desirable, hotel owners may have to lower rates in order to achieve it. Therefore, there could be instances where hotels can actually make more money from an 80 percent occupancy rate than from a 100 percent occupancy rate, if the 80 percent are paying higher prices.
How do you calculate daily rate?
Divide your annual salary by the number of days per year you work to find the daily rate. For this example, if your annual salary equals \$55,900, divide \$55,900 by 260 to get \$215 as your daily rate.
What is RevPAR formula?
Expressed in dollar terms, RevPAR is calculated by multiplying the average daily rate (ADR) by how many rooms are sold (occupancy rate).
How do you increase occupancy rate?
We’ve put together a list of 9 simple and easy-to-implement steps that can help you increase hotel room occupancy.Target the right market. … Customize packages and promotions. … Count on events or cultural festivals. … Discounts, loyalty programs and other perks. … Create a buzz around your locality, not just your property.More items…•
What is occupancy percentage in front office?
Occupancy Percentage is the most commonly used operating ratio in the hotel front office, The Occupancy percentage indicates the proportion of rooms either sold or occupied to the number of rooms available for the selected date or period.
How do you calculate room occupancy?
Calculated your occupancy rate by dividing the total number of rooms occupied by the total number of rooms available times 100, e.g. 75% occupancy.
How is occupancy calculated in BPO?
The most obvious call center occupancy formula would be to divide the time an agent spends on calls by all of their available working time. For instance, if an agent spent 54 minutes on calls during one hour (aka 60 minutes) of work, they would have an occupancy rate of 90 percent (54/60 = 90%).
How do you calculate occupancy per square foot?
How to Determine Occupancy RateDetermine the area to be occupied in square feet (Length X Width) Ex: 30Lx50W=1500 sq ft.Choose appropriate occupancy requirement: … Divide the area by the occupancy requirement for total occupant load.To get 25% occupancy, divide by 4.
What is double occupancy rate?
noun. a type of travel accommodation, as in a hotel, for two persons sharing the same room: The rate is \$35 per person, double occupancy, or \$65, single occupancy.
What is the difference between occupancy and capacity?
Capacity = The number of people the egress system can accommodate safely during an emergency. Occupant Load = The total number of persons that might occupy a building or portion thereof at any one time.
What is the formula of average room rate?
Average Hotel Room Rate (HARR or HADR) = Total Room Revenue / Total Rooms Sold + Comp Rooms.
What is a good occupancy rate for a call center?
It is always important for the managers to set the call center occupancy rate between 85% – 90% to improve both agent productivity as well as a customer service experience.
What is occupancy ratio?
The Allocated Occupancy Ratio is a measure of the size of room requested by Departments compared to the size of room allocated. A figure of 1 would indicate that allocated rooms match exactly the sizes requested.
What is bed occupancy ratio?
The occupancy rate is calculated as the number of beds effectively occupied (bed-days) for curative care (HC. 1 in SHA classification) divided by the number of beds available for curative care multiplied by 365 days, with the ratio multiplied by 100.
Why is occupancy rate important?
Occupancy rates are important to business owners because they can signify success – or failure – of the property in question. If a hotel that has consistently low occupancy rates, for example, it may mean that property has significant problems that make it unattractive to the general public.
| 1,057 | 5,045 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.96875 | 4 |
CC-MAIN-2021-04
|
latest
|
en
| 0.936593 |
https://vincentarelbundock.github.io/marginaleffects/articles/slopes.html
| 1,679,845,902,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-14/segments/1679296945473.69/warc/CC-MAIN-20230326142035-20230326172035-00270.warc.gz
| 681,863,163 | 12,544 |
## Definition
Slopes are defined as:
Partial derivatives of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends.
This vignette follows the econometrics tradition by referring to “slopes” and “marginal effects” interchangeably. In this context, the word “marginal” refers to the idea of a “small change,” in the calculus sense.
A marginal effect measures the association between a change in a regressor $$x$$, and a change in the response $$y$$. Put differently, differently, the marginal effect is the slope of the prediction function, measured at a specific value of the regressor $$x$$.
Marginal effects are extremely useful, because they are intuitive and easy to interpret. They are often the main quantity of interest in an empirical analysis.
In scientific practice, the “Marginal Effect” falls in the same toolbox as the “Contrast.” Both try to answer a counterfactual question: What would happen to $$y$$ if $$x$$ were different? They allow us to model the “effect” of a change/difference in the regressor $$x$$ on the response $$y$$.1
To illustrate the concept, consider this quadratic function:
$y = -x^2$
From the definition above, we know that the marginal effect is the partial derivative of $$y$$ with respect to $$x$$:
$\frac{\partial y}{\partial x} = -2x$
To get intuition about how to interpret this quantity, consider the response of $$y$$ to $$x$$. It looks like this:
When $$x$$ increases, $$y$$ starts to increase. But then, as $$x$$ increases further, $$y$$ creeps back down in negative territory.
A marginal effect is the slope of this response function at a certain value of $$x$$. The next plot adds three tangent lines, highlighting the slopes of the response function for three values of $$x$$. The slopes of these tangents tell us three things:
1. When $$x<0$$, the slope is positive: an increase in $$x$$ is associated with an increase in $$y$$: The marginal effect is positive.
2. When $$x=0$$, the slope is null: a (small) change in $$x$$ is associated with no change in $$y$$. The marginal effect is null.
3. When $$x>0$$, the slope is negative: an increase in $$x$$ is associated with a decrease in $$y$$. The marginal effect is negative.
Below, we show how to reach the same conclusions in an estimation context, with simulated data and the slopes function.
## slopes function
The marginal effect is a unit-level measure of association between changes in a regressor and changes in the response. Except in the simplest linear models, the value of the marginal effect will be different from individual to individual, because it will depend on the values of the other covariates for each individual.
The slopes function thus produces distinct estimates of the marginal effect for each row of the data used to fit the model. The output of marginaleffects is a simple data.frame, which can be inspected with all the usual R commands.
To show this, we load the library, download the Palmer Penguins data from the Rdatasets archive, and estimate a GLM model:
library(marginaleffects)
dat$large_penguin <- ifelse(dat$body_mass_g > median(dat$body_mass_g, na.rm = TRUE), 1, 0) mod <- glm(large_penguin ~ bill_length_mm + flipper_length_mm + species, data = dat, family = binomial) mfx <- slopes(mod) head(mfx) #> #> Term Contrast Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 % #> bill_length_mm dY/dX 0.0176 0.00830 2.12 0.03359 0.00137 0.0339 #> bill_length_mm dY/dX 0.0359 0.01229 2.92 0.00354 0.01176 0.0600 #> bill_length_mm dY/dX 0.0844 0.02108 4.01 < 0.001 0.04312 0.1258 #> bill_length_mm dY/dX 0.0347 0.00642 5.41 < 0.001 0.02214 0.0473 #> bill_length_mm dY/dX 0.0509 0.01350 3.77 < 0.001 0.02444 0.0773 #> bill_length_mm dY/dX 0.0165 0.00770 2.14 0.03202 0.00142 0.0316 #> #> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo, large_penguin, bill_length_mm, flipper_length_mm, species ## The Marginal Effects Zoo A dataset with one marginal effect estimate per unit of observation is a bit unwieldy and difficult to interpret. There are ways to make this information easier to digest, by computing various quantities of interest. In a characteristically excellent blog post, Professor Andrew Heiss introduces many such quantities: • Average Marginal Effects • Group-Average Marginal Effects • Marginal Effects at User-Specified Values (or Representative Values) • Marginal Effects at the Mean • Counterfactual Marginal Effects • Conditional Marginal Effects The rest of this vignette defines each of those quantities and explains how to use the slopes() and plot_slopes() functions to compute them. The main differences between these quantities pertain to (a) the regressor values at which we estimate marginal effects, and (b) the way in which unit-level marginal effects are aggregated. Heiss drew this exceedingly helpful graph which summarizes the information in the rest of this vignette: ## Average Marginal Effect (AME) A dataset with one marginal effect estimate per unit of observation is a bit unwieldy and difficult to interpret. Many analysts like to report the “Average Marginal Effect”, that is, the average of all the observation-specific marginal effects. These are easy to compute based on the full data.frame shown above, but the avg_slopes() function is convenient: avg_slopes(mod) #> #> Term Contrast Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 % #> bill_length_mm dY/dX 0.0276 0.00578 4.772 <0.001 0.01625 0.0389 #> flipper_length_mm dY/dX 0.0106 0.00235 4.509 <0.001 0.00598 0.0152 #> species Chinstrap - Adelie -0.4148 0.05659 -7.330 <0.001 -0.52570 -0.3039 #> species Gentoo - Adelie 0.0617 0.10688 0.577 0.564 -0.14778 0.2712 #> #> Columns: term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high Note that since marginal effects are derivatives, they are only properly defined for continuous numeric variables. When the model also includes categorical regressors, the summary function will try to display relevant (regression-adjusted) contrasts between different categories, as shown above. You can also extract average marginal effects using tidy and glance methods which conform to the broom package specification: tidy(mfx) #> # A tibble: 4 × 8 #> term contrast estimate std.error statistic p.value conf.low conf.high #> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> #> 1 bill_length_mm mean(dY/dX) 0.0276 0.00578 4.77 1.82e- 6 0.0162 0.0389 #> 2 flipper_length_mm mean(dY/dX) 0.0106 0.00235 4.51 6.52e- 6 0.00598 0.0152 #> 3 species mean(Chinstrap) - mean(Adelie) -0.415 0.0566 -7.33 2.31e-13 -0.526 -0.304 #> 4 species mean(Gentoo) - mean(Adelie) 0.0617 0.107 0.577 5.64e- 1 -0.148 0.271 glance(mfx) #> # A tibble: 1 × 7 #> aic bic r2.tjur rmse nobs F logLik #> <dbl> <dbl> <dbl> <dbl> <int> <dbl> <logLik> #> 1 180. 199. 0.695 0.276 342 15.7 -84.92257 ## Group-Average Marginal Effect (G-AME) We can also use the by argument the average marginal effects within different subgroups of the observed data, based on values of the regressors. For example, to compute the average marginal effects of Bill Length for each Species, we do: avg_slopes( mod, by = "species", variables = "bill_length_mm") #> #> Term Contrast species Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 % #> bill_length_mm mean(dY/dX) Adelie 0.04354 0.00879 4.95 <0.001 0.0263 0.06077 #> bill_length_mm mean(dY/dX) Chinstrap 0.03680 0.00976 3.77 <0.001 0.0177 0.05594 #> bill_length_mm mean(dY/dX) Gentoo 0.00287 0.00284 1.01 0.312 -0.0027 0.00844 #> #> Columns: term, contrast, species, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo This is equivalent to manually taking the mean of the observation-level marginal effect for each species sub-group: aggregate( mfx$estimate,
by = list(mfx$species, mfx$term),
FUN = mean)
#> Group.1 Group.2 x
#> 2 Chinstrap bill_length_mm 0.036801185
#> 3 Gentoo bill_length_mm 0.002871562
#> 5 Chinstrap flipper_length_mm 0.014124217
#> 6 Gentoo flipper_length_mm 0.001102231
#> 8 Chinstrap species -0.313337522
#> 9 Gentoo species -0.250726004
Note that marginaleffects follows Stata and the margins package in computing standard errors using the group-wise averaged Jacobian.
## Marginal Effect at User-Specified Values
Sometimes, we are not interested in all the unit-specific marginal effects, but would rather look at the estimated marginal effects for certain “typical” individuals, or for user-specified values of the regressors. The datagrid function helps us build a data grid full of “typical” rows. For example, to generate artificial Adelies and Gentoos with 180mm flippers:
datagrid(flipper_length_mm = 180,
model = mod)
#> large_penguin bill_length_mm flipper_length_mm species
#> 1 0.4853801 43.92193 180 Adelie
#> 2 0.4853801 43.92193 180 Gentoo
The same command can be used (omitting the model argument) to marginaleffects’s newdata argument to compute marginal effects for those (fictional) individuals:
slopes(
mod,
newdata = datagrid(
flipper_length_mm = 180,
#>
#> Term Contrast Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 % bill_length_mm flipper_length_mm species
#> bill_length_mm dY/dX 0.0607 0.03323 1.827 0.0677 -0.00443 0.12581 43.9 180 Adelie
#> bill_length_mm dY/dX 0.0847 0.03939 2.150 0.0316 0.00747 0.16187 43.9 180 Gentoo
#> flipper_length_mm dY/dX 0.0233 0.00550 4.232 <0.001 0.01250 0.03408 43.9 180 Adelie
#> flipper_length_mm dY/dX 0.0325 0.00851 3.817 <0.001 0.01581 0.04918 43.9 180 Gentoo
#> species Chinstrap - Adelie -0.2111 0.10618 -1.988 0.0468 -0.41916 -0.00294 43.9 180 Adelie
#> species Chinstrap - Adelie -0.2111 0.10618 -1.988 0.0468 -0.41916 -0.00294 43.9 180 Gentoo
#> species Gentoo - Adelie 0.1591 0.30242 0.526 0.5988 -0.43361 0.75185 43.9 180 Adelie
#> species Gentoo - Adelie 0.1591 0.30242 0.526 0.5988 -0.43361 0.75185 43.9 180 Gentoo
#>
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo, large_penguin, bill_length_mm, flipper_length_mm, species
When variables are omitted from the datagrid call, they will automatically be set at their mean or mode (depending on variable type).
## Marginal Effect at the Mean (MEM)
The “Marginal Effect at the Mean” is a marginal effect calculated for a hypothetical observation where each regressor is set at its mean or mode. By default, the datagrid function that we used in the previous section sets all regressors to their means or modes. To calculate the MEM, we can set the newdata argument, which determines the values of predictors at which we want to compute marginal effects:
slopes(mod, newdata = "mean")
#>
#> Term Contrast Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 %
#> bill_length_mm dY/dX 0.0502 0.01238 4.059 <0.001 0.02598 0.0745
#> flipper_length_mm dY/dX 0.0193 0.00553 3.487 <0.001 0.00844 0.0301
#> species Chinstrap - Adelie -0.8070 0.07636 -10.569 <0.001 -0.95670 -0.6574
#> species Gentoo - Adelie 0.0829 0.11453 0.723 0.469 -0.14161 0.3073
#>
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo, large_penguin, bill_length_mm, flipper_length_mm, species
## Counterfactual Marginal Effects
The datagrid function allowed us look at completely fictional individuals. Setting the grid_type argument of this function to "counterfactual" lets us compute the marginal effects for the actual observations in our dataset, but with a few manipulated values. For example, this code will create a data.frame twice as long as the original dat, where each observation is repeated with different values of the flipper_length_mm variable:
nd <- datagrid(flipper_length_mm = c(160, 180),
model = mod,
grid_type = "counterfactual")
We see that the rows 1, 2, and 3 of the original dataset have been replicated twice, with different values of the flipper_length_mm variable:
nd[nd$rowid %in% 1:3,] #> rowidcf large_penguin bill_length_mm species flipper_length_mm #> 1 1 0 39.1 Adelie 160 #> 2 2 0 39.5 Adelie 160 #> 3 3 0 40.3 Adelie 160 #> 343 1 0 39.1 Adelie 180 #> 344 2 0 39.5 Adelie 180 #> 345 3 0 40.3 Adelie 180 We can use the observation-level marginal effects to compute average (or median, or anything else) marginal effects over the counterfactual individuals: library(dplyr) slopes(mod, newdata = nd) |> group_by(term) |> summarize(estimate = median(estimate)) #> # A tibble: 3 × 2 #> term estimate #> <chr> <dbl> #> 1 bill_length_mm 0.00985 #> 2 flipper_length_mm 0.00378 #> 3 species 0.0000226 ## Conditional Marginal Effects (Plot) The plot_slopes function can be used to draw “Conditional Marginal Effects.” This is useful when a model includes interaction terms and we want to plot how the marginal effect of a variable changes as the value of a “condition” (or “moderator”) variable changes: mod <- lm(mpg ~ hp * wt + drat, data = mtcars) plot_slopes(mod, variables = "hp", condition = "wt") The marginal effects in the plot above were computed with values of all regressors – except the variables and the condition – held at their means or modes, depending on variable type. Since plot_slopes() produces a ggplot2 object, it is easy to customize. For example: plot_slopes(mod, variables = "hp", condition = "wt") + geom_rug(aes(x = wt), data = mtcars) + theme_classic() ## Example: Quadratic In the “Definition” section of this vignette, we considered how marginal effects can be computed analytically in a simple quadratic equation context. We can now use the slopes function to replicate our analysis of the quadratic function in a regression application. Say you estimate a linear regression model with a quadratic term: $Y = \beta_0 + \beta_1 X^2 + \varepsilon$ and obtain estimates of $$\beta_0=1$$ and $$\beta_1=2$$. Taking the partial derivative with respect to $$X$$ and plugging in our estimates gives us the marginal effect of $$X$$ on $$Y$$: $\partial Y / \partial X = \beta_0 + 2 \cdot \beta_1 X$ $\partial Y / \partial X = 1 + 4X$ This result suggests that the effect of a change in $$X$$ on $$Y$$ depends on the level of $$X$$. When $$X$$ is large and positive, an increase in $$X$$ is associated to a large increase in $$Y$$. When $$X$$ is small and positive, an increase in $$X$$ is associated to a small increase in $$Y$$. When $$X$$ is a large negative value, an increase in $$X$$ is associated with a decrease in $$Y$$. marginaleffects arrives at the same conclusion in simulated data: library(tidyverse) N <- 1e5 quad <- data.frame(x = rnorm(N)) quad$y <- 1 + 1 * quad$x + 2 * quad$x^2 + rnorm(N)
mod <- lm(y ~ x + I(x^2), quad)
slopes(mod, newdata = datagrid(x = -2:2)) |>
mutate(truth = 1 + 4 * x) |>
select(estimate, truth)
#>
#> Estimate
#> -6.995
#> -2.998
#> 0.998
#> 4.995
#> 8.991
#>
#> Columns: estimate, truth
We can plot conditional adjusted predictions with plot_predictions function:
plot_predictions(mod, condition = "x")
We can plot conditional marginal effects with the plot_slopes function (see section below):
plot_slopes(mod, variables = "x", condition = "x")
Again, the conclusion is the same. When $$x<0$$, an increase in $$x$$ is associated with an decrease in $$y$$. When $$x>1/4$$, the marginal effect is positive, which suggests that an increase in $$x$$ is associated with an increase in $$y$$.
## Slopes vs Predictions: A Visual Interpretation
Often, analysts will plot predicted values of the outcome with a best fit line:
library(ggplot2)
mod <- lm(mpg ~ hp * qsec, data = mtcars)
plot_predictions(mod, condition = "hp", vcov = TRUE) +
geom_point(data = mtcars, aes(hp, mpg))
The slope of this line is calculated using the same technique we all learned in grade school: dividing rise over run.
p <- plot_predictions(mod, condition = "hp", vcov = TRUE, draw = FALSE)
plot_predictions(mod, condition = "hp", vcov = TRUE) +
geom_segment(aes(x = p$hp[10], xend = p$hp[10], y = p$estimate[10], yend = p$estimate[20])) +
geom_segment(aes(x = p$hp[10], xend = p$hp[20], y = p$estimate[20], yend = p$estimate[20])) +
annotate("text", label = "Rise", y = 10, x = 140) +
annotate("text", label = "Run", y = 2, x = 200)
Instead of computing this slope manually, we can just call:
avg_slopes(mod, variables = "hp")
#>
#> Term Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 %
#> hp -0.112 0.0126 -8.92 <0.001 -0.137 -0.0874
#>
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
Now, consider the fact that our model includes an interaction between hp and qsec. This means that the slope will actually differ based on the value of the moderator variable qsec:
plot_predictions(mod, condition = list("hp", "qsec" = "quartile"))
We can estimate the slopes of these three fit lines easily:
slopes(
mod,
variables = "hp",
newdata = datagrid(qsec = quantile(mtcars\$qsec, probs = c(.25, .5, .75))))
#>
#> Term Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 % hp qsec
#> hp -0.0934 0.0111 -8.43 <0.001 -0.115 -0.0717 147 16.9
#> hp -0.1093 0.0123 -8.92 <0.001 -0.133 -0.0853 147 17.7
#> hp -0.1325 0.0154 -8.60 <0.001 -0.163 -0.1023 147 18.9
#>
#> Columns: rowid, term, estimate, std.error, statistic, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo, mpg, hp, qsec
As we see in the graph, all three slopes are negative, but the Q3 slope is steepest.
We could then push this one step further, and measure the slope of mpg with respect to hp, for all observed values of qsec. This is achieved with the plot_slopes() function:
plot_slopes(mod, variables = "hp", condition = "qsec") +
geom_hline(yintercept = 0, linetype = 3)
This plot shows that the marginal effect of hp on mpg is always negative (the slope is always below zero), and that this effect becomes even more negative as qsec increases.
## Prediction types
The marginaleffect function takes the derivative of the fitted (or predicted) values of the model, as is typically generated by the predict(model) function. By default, predict produces predictions on the "response" scale, so the marginal effects should be interpreted on that scale. However, users can pass a string or a vector of strings to the type argument, and marginaleffects will consider different outcomes.
Typical values include "response" and "link", but users should refer to the documentation of the predict of the package they used to fit the model to know what values are allowable. documentation.
mod <- glm(am ~ mpg, family = binomial, data = mtcars)
avg_slopes(mod, type = "response")
#>
#> Term Estimate Std. Error z Pr(>|z|) 2.5 % 97.5 %
#> mpg 0.0465 0.00887 5.24 <0.001 0.0291 0.0639
#>
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high
| 5,797 | 19,514 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.859375 | 4 |
CC-MAIN-2023-14
|
latest
|
en
| 0.894668 |
http://catbear.wordpress.com/2009/01/
| 1,386,309,785,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2013-48/segments/1386163049631/warc/CC-MAIN-20131204131729-00062-ip-10-33-133-15.ec2.internal.warc.gz
| 28,557,427 | 12,780 |
The golden ratio
I find maths work a lot easier if I use the printed course books rather than the pdf versions, so for the past few weeks I’ve been eagerly awaiting the delivery of my MS221 materials. However, last night I finally snapped and printed out the first two sections of Chapter A1 from the pdf copy, and today I’ve been working my way through Section 1, which is about the golden ratio.
It’s been fantastic so far – a really fascinating topic, and a very compelling start to the course. The golden ratio is introduced through a rectangle problem:
The Rectangle Problem
Suppose that a rectangle has a square removed from one end, leaving a rectangle the same shape as the original rectangle. What is the ratio of the lengths of the sides of the original rectangle?
Heuristics drive Rob slowly mad
This week I have mostly been bashing my head against M366 TMA01 Q4, the bit about choosing a good heuristic. It took me quite a while just to get my head around what the question is actually asking for, let alone actually answering it! So now after three days of going around and around in circles trying to figure out how to choose a heuristic for this problem, I’ve finally got one – it’s probably not a particularly good one, but hopefully it’s better than the example “poor” heuristic from the previous subsection, and in any case it’s very prettily laid out using Word’s Equation Editor tool, so I’m happy.
I’ve been able to move on to the next bit of the question, at last, which involves heavy use of NetLogo’s BehaviourSpace. I quite enjoyed the BehaviourSpace bit of the M366 Software Guide exercises – it’s great being able to get the simulation data in csv format, and hence into Excel, which feels very much like my home turf since I spend most of my time at work producing spreadsheets. It feels a bit weird to be using Excel stuff for something OU-related, but overall I’m really glad we’re doing this kind of statistical work; I like the emphasis on running simulations several times to build up a reliable picture of the likely outcome, and to be honest, anything that encourages people to use or develop their spreadsheet skills is a good thing in my book!
I have search algorithms coming out of my ears
I finished Unit 2 of M366 Block 2 yesterday, so I spent most of this afternoon writing up my notes on search algorithms and heuristics. And there were tons of them! I didn’t appreciate just how many topics were covered in Unit 2 until I started to summarise them in my M366 tiddlywiki.
One concept that it took me a while to get the hang of was alpha-beta pruning. There’s a great flash demonstration of alpha-beta pruning on the M366 course website (unfortunately only available to M366 students), which helped a lot; but oddly, the thing helped the most was going through the pseudocode example from the course text, rewriting it slightly so that it had a more familiar structure, and adding comments to each line describing what the code was doing. Once I’d done that, and I finally got my head around the recursive back-and-forth calling of the `ValueOfMax` and `ValueOfMin` functions, the exercises in Unit 2 actually made some kind of sense!
M257 TMA01 returned!
I’m really happy to have this TMA back at last, mainly because it means Alex won’t have to put up with me grumbling eleventy-billion times per day about how much I want to see my grade and feedback. I got 95% for this one, which is a little bit worse than the mark I got for M255 TMA01 (96%) and so is a tiny bit disappointing, but overall I’m very happy with the score.
I felt very detached when I submitted TMA01 via the eTMA system, which was quite an unpleasant feeling; since there had been a few weeks between finishing the assignment and actually submitting it, I felt a bit like I didn’t really know what I was submitting (and had to try very hard not to keep unzipping the file and having a look, just to make sure it was all there!). I do keep a record on my study calendar of which questions I’ve finished for each TMA, so I knew I must have fully finished the assignment, but it was still unsettling not to have it fresh in my mind.
So I think for TMA03, and certainly for the assignments in M366 and MS221, I’m going to try to leave the TMA-writing stage until closer to the cut-off date. I definitely don’t want to be feeling unsettled and anxious about three different TMAs at once!
M366 Block 2 computer activities
I’ve been finding the computer activities in Block 2 Unit 2 quite tricky, to the extent that I was beginning to think I’d missed one of the introductory exercises in the Software Guide! Unfortunately I hadn’t missed any, so something must have gone awry with my understanding of how the search algorithms have been implemented in these activities.
I don’t think it’s the NetLogo code itself that’s my problem, I think it’s the details of these particular route-finding programs; oddly, I had no trouble doing the beam-search code in Ex 2.5, but the solution for the previous activities (the depth-limit and iterative-deepening ones) completely eluded me. I’ll probably need to go back through the code for Ex 2.3-2.9, along with the NetLogo user manual, until I can make sense of it. (I guess I know what I’m doing this weekend!)
Nevertheless, I’m enjoying Block 2 a lot more than Block 1, particularly the discussion of various different search algorithms and their advantages/disadvantages. I’m particularly happy to be learning about search trees and tree traversal again, since the Binary Trees unit was one of my favourite parts of M263 (although I liked the bit involving parse trees even more).
Hopefully I’ll be able to get this section finished by the end of this weekend, and then it’s on to the bit about adversarial search and games. I don’t know much about the topic, but it sounds interesting, especially if we’ll be looking at problems where two agents have to react to each others’ moves – and in any case, it’ll certainly be nice to have a break from all the “find the shortest route from Exeter to Norwich/Lincoln/Shrewsbury” problems…
Getting started with M366 TMA01
As I’d hoped, the course website for M366 has opened today, and so I’ve been able to download the TMA01 document and get started on Question 1. This first question is all about describing concepts from the Block 1, so it feels very much like jumping in at the deep end! I’m still not sure whether I’m even thinking in the right way for this course, let alone writing in the right way, but I suppose I won’t know for sure until I get my first assignment feedback (which won’t be for quite some time, since the cut-off date for this is 14/04/09).
On the other hand, I should be able to get a rough idea of how good or bad my understanding of these topics is from the course forums, which are already open. I’m generally a bit reluctant to post on FirstClass forums, but I do quite enjoy reading along with the discussions that go on there. Except during exam revision time, when everything gets a bit frantic!
I’m having some trouble putting together a study schedule now that I’ve got three courses on the go. I’ve had a similar course-load before, back in the summer of 2008, but since those were all programming courses it was fairly easy to organise my study sessions; everything seemed to take roughly the same amount of time, so it was a very regular, steady process. This time around, I’ve got a programming course (M257), a maths course (MS221) and a weird wordy course (M366), so I feel like I’m trying to tessellate three very different shapes – I’m sure they’ll fit together, it’s just a question of how!
Further into the strange and alien world of M366
M366 is proving to be a very different experience to the computing courses I’ve done before, and in some ways I’m feeling out of my element. It’s not that the material is difficult or impenetrable, but that I’m not exactly sure what to do with it. My previous computing courses were mainly about doing and making stuff, but M366 seems to be about understanding and explaining things; for this reason, my unease with the course is giving me a creeping fear that I really am just an oompa-loompa of computing.
First impressions of M366
Unit 1 of M366, featuring some lovely paper wasps
The course materials for M366: Natural and artificial intelligence arrived yesterday, so I’ve finally got my hands on my very first Level 3 textbooks! I’ve read through the first 3 sections of Unit 1, and at the moment I’ve got mixed feelings about the course…
I really enjoyed the introductory NetLogo activities, especially using the BehaviourSpace tool to run a simulation multiple times and output the results to a csv file. I’m really looking forward to making my own simulations, and I’ll be interested to see if we do any kind of statistical analysis of the output of our programs.
I also quite enjoyed the section about Enlightenment automata, particular Vaucanson’s Duck, a copper duck-shaped automaton that appeared to eat, digest and defecate. The course text says that the “digestion” was the result of a chemical plant inside the automaton, but rather disappointingly the Wikipedia page says that this wasn’t the case. I’m not sure who to believe!
| 2,047 | 9,250 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.53125 | 4 |
CC-MAIN-2013-48
|
longest
|
en
| 0.946424 |
https://www.quantamagazine.org/a-new-generation-of-mathematicians-pushes-prime-number-barriers-20231026/
| 1,726,529,564,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-38/segments/1725700651714.51/warc/CC-MAIN-20240916212424-20240917002424-00114.warc.gz
| 885,912,426 | 48,284 |
# A New Generation of Mathematicians Pushes Prime Number Barriers
New work attacks a long-standing barrier to understanding how prime numbers are distributed.
## Introduction
More than 2,000 years ago, the Greek mathematician Eratosthenes came up with a method for finding prime numbers that continues to reverberate through mathematics today. His idea was to identify all the primes up to a given point by gradually “sieving out” the numbers that aren’t prime. His sieve starts by crossing out all the multiples of 2 (except 2 itself), then the multiples of 3 (except 3 itself). The next number, 4, is already crossed out, so the next step is to cross out the multiples of 5, and so on. The only numbers that survive are primes — numbers whose only divisors are 1 and themselves.
Eratosthenes was focused on the full set of primes, but you can use variations on his sieve to hunt for primes with all kinds of special features. Want to find “twin primes,” which are only 2 apart, like 11 and 13 or 599 and 601? There’s a sieve for that. Want to find primes that are 1 bigger than a perfect square, like 17 or 257? There’s a sieve for that too.
Modern sieves have fueled many of the biggest advances in number theory on problems ranging from Fermat’s Last Theorem to the still unproved twin primes conjecture, which says that there are infinitely many pairs of twin primes. Sieve methods, the Hungarian mathematician Paul Erdős wrote in 1965, are “perhaps our most powerful elementary tool in number theory.”
Yet this power is constrained by mathematicians’ limited understanding of how primes are distributed along the number line. It’s simple to carry out a sieve up to some small number, like 100. But mathematicians want to understand the behavior of sieves when numbers get big. They can’t hope to list all the numbers that survive the sieve up to some extremely large stopping point. So instead, they try to estimate how many numbers are on that list.
For the sieve of Eratosthenes, this estimate depends on how often whole numbers are divisible by 2, or 3, or 5, and so on — comparatively easy information to obtain. But for more complicated sieves, such as the ones for twin primes, the crucial information often concerns the remainders that primes leave behind when divided by different numbers. For instance, how often do primes leave a remainder of 1 when divided by 3? Or a remainder of 8 when divided by 15?
As you move out along the number line, these remainders settle into statistically predictable patterns. In 1896, the Belgian mathematician Charles-Jean de la Vallée Poussin proved that remainders gradually even out — for example, if you drop primes into one of two buckets depending on whether their remainder is 1 or 2 when they’re divided by 3, the two buckets will eventually hold roughly the same number of primes. But to extract the full potential from sieve methods, mathematicians need to know not only that the buckets eventually even out, but how soon they do so.
That has proved challenging. After a burst of progress in the 1960s and another in the 1980s, new developments mostly petered out. A notable exception occurred in 2013, when Yitang Zhang published a landmark proof that there are infinitely many pairs of primes closer to each other than some finite bound. But the main body of work developed in the ’80s saw essentially no progress for more than three decades.
Now the subject is enjoying a renaissance, sparked by a series of three papers written by the Oxford mathematician James Maynard in 2020 (two years before he was awarded the Fields Medal, mathematics’ highest honor). Maynard analyzed a number called the “level of distribution” that captures how quickly prime remainders become evenly distributed into buckets (sometimes with reference to particular types of sieves). For many commonly used sieves, he showed that the level of distribution is at least 0.6, beating the previous record of 0.57 from the 1980s.
Maynard’s work and the follow-on studies it has prompted “are breathing new life into analytic number theory,” said John Friedlander of the University of Toronto, who played a large role in the developments in the 1980s. “It’s a real revival.”
In the past few months, three of Maynard’s graduate students have written papers extending both Maynard’s and Zhang’s results; one of these papers, by Jared Duker Lichtman (now a postdoctoral fellow at Stanford University), pushed Maynard’s level of distribution up to about 0.617. Lichtman then used that increase to calculate improved upper bounds on the number of twin primes up to a given stopping point, and the number of “Goldbach representations” — representations of even numbers as the sum of two primes.
“These younger people are following up [on] what’s really the hot topic now,” said Andrew Granville of the University of Montreal.
An increase from 0.6 to 0.617 might seem of small account to people outside number theory. But in sieve theory, Granville said, “sometimes those little wins can have devastating consequences.”
## Including and Excluding
To estimate how many numbers a sieve removes up to some stopping point N, mathematicians use an approach based on something called inclusion/exclusion. To see how this works, consider the sieve of Eratosthenes. This sieve starts by removing all multiples of 2 — that’s about half the numbers up to N. Next the sieve removes all multiples of 3 — about 1/3 of the numbers up to N. So you might think that so far you’ve removed about 1/2 + 1/3 of the numbers up to N.
But this is an overcount, because you’ve double-counted numbers that are multiples of both 2 and 3 (multiples of 6). These are about 1/6 of all the numbers up to N, so to correct for counting them twice, you need to subtract 1/6, bringing the running total for what you are removing to 1/2 + 1/3 − 1/6.
Next you can move on to multiples of 5 — that’s going to add 1/5 to the count, but you have to subtract 1/10 and 1/15 to correct for overcounting numbers that are divisible by both 2 and 5, or both 3 and 5. Even then you’re not quite done — you’ve accidentally corrected twice for the numbers that are divisible by 2, 3 and 5, so to fix that you have to add 1/30 to your count, bringing the running total to 1/2 + 1/3 − 1/6 + 1/5 − 1/10 − 1/15 + 1/30.
As this process continues, the sum gains more and more terms, involving fractions with bigger and bigger denominators. To prevent the small errors in approximations like “about 1/2” and “about 1/3” from piling up too much, number theorists usually stop the addition and subtraction process before they’ve gone through the whole sieve, and content themselves with upper and lower bounds instead of an exact answer.
In theory, a similar process should work for fancier sets of primes, like twin primes. But when it comes to something like twin primes, inclusion/exclusion won’t work unless you know how evenly prime remainders are distributed into buckets.
To see this, think about how a twin prime sieve could work. You can start by using the sieve of Eratosthenes to find all the primes up to N. Then, do a second round of sieving that removes every prime that isn’t part of a twin prime pair. One way to do this is to sieve out a prime if the number that’s two spots to its left is not prime (or you could look two spots to the right — either sieve will work). Using the leftward sieve, you’ll keep primes such as 13, since 11 is also prime, but cross out primes like 23, since 21 is not prime.
You can think of this sieve as first shifting the set of primes two spots to the left on the number line, then crossing out the numbers in the shifted set that aren’t prime (such as 21). In the shifted set, you’ll cross out multiples of 3, then multiples of 5, and so on. (You don’t have to worry about multiples of 2, since the numbers in the shifted set are all odd, except the very first one.)
Next comes inclusion/exclusion, to estimate how many numbers you’ve crossed out. In the sieve of Eratosthenes, crossing out multiples of 3 removes about 1/3 of all numbers. But in the smaller set of shifted primes, it’s harder to predict how many will fall when we cross out multiples of 3.
Any number k in the shifted set is 2 less than some prime. So if k is a multiple of 3, then its corresponding prime, k + 2, has a remainder of 2 when divided by 3. Prime numbers have a remainder of either 1 or 2 when divided by 3 (except for 3 itself), so you might guess that half the primes up to N have a remainder of 1 and half have a remainder of 2. That would mean that in this step of the sieve you’re crossing out approximately half the numbers in the shifted set (instead of 1/3 as in the sieve of Eratosthenes). So you would write a 1/2 term in your inclusion/exclusion sum.
Thanks to de la Vallée Poussin, we know that eventually half of all primes have a remainder of 1 and half have a remainder of 2 when you divide by 3. But to do inclusion/exclusion, it’s not enough to know that the remainder buckets balance out eventually — you need to know that they balance out by N. Otherwise, you can’t have any confidence in the “1/2” in the inclusion/exclusion sum. Perhaps, mathematicians have worried for more than a century, the distribution of primes has weird quirks that undermine some of the counts needed for our inclusion/exclusion sum.
“If you don’t have distribution theorems, you can’t understand what happens when you finish your sieve,” said Terence Tao of the University of California, Los Angeles.
## A Fundamental Waypoint
One prediction about how fast the buckets start to even out was available to number theorists in the form of the most celebrated unsolved problem in number theory — the generalized Riemann hypothesis. This hypothesis, if true, would imply that if we’re looking at all the primes up to some very large number N, then prime remainders are evenly distributed into buckets for any divisor up to about the square root of N. So for example, if you’re looking at the primes less than 1 trillion, you’d expect them to be distributed evenly into remainder buckets when you divide them by 120, or 7,352, or 945,328 — any divisor less than about 1 million (the square root of 1 trillion). Mathematicians say that the generalized Riemann hypothesis predicts that the primes’ level of distribution is at least 1/2, since another way to write the square root of N is as N1/2.
If this hypothesis is correct, that would mean that when you’re sieving up to 1 trillion, you can cross off multiples of 2, then 3, then 5, and keep going until the inclusion/exclusion sum starts to involve divisors over about 1 million — beyond that point, you can’t calculate the terms in your sum. In the mid-1900s, number theorists proved many sieve theorems of the form, “If the generalized Riemann hypothesis is correct, then … ”
But a lot of these results didn’t actually need the full strength of the generalized Riemann hypothesis — it would be enough to know that primes were well distributed into buckets for almost every divisor, instead of every single divisor. In the mid-1960s, Enrico Bombieri and Askold Vinogradov separately managed to prove just that: The primes have a level of distribution of at least 1/2, if we’re content with knowing that the buckets even out for almost every divisor.
The Bombieri-Vinogradov theorem, which is still widely used, instantly proved many of the results that had previously relied on the unproved generalized Riemann hypothesis. “It’s kind of the gold standard of distribution theorems,” Tao said.
But mathematicians have long suspected — and numerical evidence has suggested — that the true level of distribution of the primes is much higher. In the late 1960s, Peter Elliott and Heini Halberstam conjectured that the level of distribution of the primes is just a shade below 1 — in other words, if you’re looking at primes up to some huge number, they should be evenly distributed into buckets even for divisors very close in size to the huge number. And these large divisors matter when you’re doing inclusion/exclusion, since they come up when you’re correcting for overcounts. So the closer mathematicians can get to the level of distribution Elliott and Halberstam predicted, the more terms they can calculate in the inclusion/exclusion sum. Proving the Elliott-Halberstam conjecture, Tao said, is “the dream.”
To this day, however, no one has been able to beat the 1/2 level of distribution in the full degree of generality that the Bombieri-Vinogradov theorem achieves. Mathematicians have taken to calling this stumbling block the “square-root barrier” for prime numbers. This barrier, Lichtman said, is “a fundamental kind of waypoint in our understanding of the primes.”
## New World Records
For many sieve problems, though, you can make progress even with incomplete information about how the primes divide into buckets. Take the twin primes problem: Sieving out a prime if the number two spots to its left is divisible by 3 or 5 or 7 is the same as asking whether the prime itself has a remainder of 2 when divided by 3 or 5 or 7 — in other words, whether the prime falls into the “2” bucket for any of these divisors. So you don’t need to know whether primes are evenly distributed across all the buckets for these divisors — you just need to know whether each “2” bucket holds the number of primes we expect.
In the 1980s, mathematicians started figuring out how to prove distribution theorems that focus on one particular bucket. This work culminated in a 1986 paper by Bombieri, Friedlander and Henryk Iwaniec that pushed the level of distribution up to 4/7 (about 0.57) for single buckets, not for all sieves but for a wide class of them.
As with the Bombieri-Vinogradov theorem, the body of ideas developed in the 1980s found a host of applications. Most notably, it enabled a huge leap in mathematicians’ understanding of Fermat’s Last Theorem, which says that the equation an + bn = cn has no natural-number solutions for any exponent n higher than 2. (This was later proved in 1994 using techniques that didn’t rely on distribution theorems.) After the excitement of the 1980s, however, there was little progress on the level of distribution of the primes for several decades.
Then in 2013, Zhang figured out how to get over the square-root barrier in a different direction from that of Bombieri, Friedlander and Iwaniec. He dug into old, unfashionable methods from the early 1980s to eke out the tiniest of improvements on Bombieri and Vinogradov’s 1/2 level of distribution in a context where you’re sieving only with “smooth” numbers — ones that have no large prime factors. This tiny improvement enabled Zhang to prove the long-standing conjecture that as you go out along the number line, you’ll keep encountering pairs of primes that are closer together than some fixed bound. (Subsequently, Maynard and Tao each separately came up with another proof of this theorem, by using an improved sieve rather than an improved level of distribution.)
Zhang’s result drew on a version of the Riemann hypothesis that lives in the world of algebraic geometry. The work of Bombieri, Friedlander and Iwaniec, meanwhile, relied on what Maynard calls a “somewhat magical connection” to objects called automorphic forms, which have their own version of the Riemann hypothesis. Automorphic forms are highly symmetric objects that, Tao says, belong to “the high-powered end of number theory.”
A few years ago, Maynard became convinced that it should be possible to squeeze more juice out of these two methods by combining their insights. In his series of three papers in 2020, which Granville labeled a “tour de force,” Maynard managed to push the level of distribution up to 3/5, or 0.6, in a slightly narrower context than the one Bombieri, Friedlander and Iwaniec studied.
Now, Maynard’s students are pushing these techniques further. Lichtman recently figured out how to extend Maynard’s level of distribution to about 0.617. He then parlayed this increase into new upper bounds on the counts of both twin primes and Goldbach representations of even numbers as the sum of two primes. For the latter, it’s the first time anyone has been able to use a level of distribution beyond the 1/2 from the classic Bombieri-Vinogradov theorem.
Another of Maynard’s students, Alexandru Pascadi, has matched the 0.617 figure for the level of distribution not of primes but of smooth numbers. Like primes, smooth numbers come up all over number theory, and results about their level of distribution and that of the primes often go hand in hand.
Meanwhile, a third student, Julia Stadlmann, has boosted the level of distribution of primes in the setting that Zhang studied, in which the divisors (instead of the numbers being divided) are smooth numbers. Zhang narrowly beat the square-root barrier in this context, reaching a 0.5017 level of distribution, and then an online collaboration called a Polymath project raised that number to 0.5233; Stadlmann has now raised it to 0.525.
Other mathematicians tease analytic number theorists, Tao said, for their obsession with small numerical advances. But these tiny improvements have a significance beyond the numbers in question. “It’s like the 100-meter dash or something, [where] you shave 3.96 seconds to 3.95 seconds,” he said. Each new world record is “a benchmark for how much your methods have progressed.”
Overall, “the techniques are getting more clear and more unified,” he said. “It’s becoming clear, once you have an advance on one problem, how to adapt it to another problem.”
There’s no bombshell application for these new developments yet, but the new work “definitely changes the way we think,” Granville said. “This isn’t just banging a nail in harder — this is actually getting a more upgraded hammer.”
| 4,057 | 17,852 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.125 | 4 |
CC-MAIN-2024-38
|
latest
|
en
| 0.941393 |
http://mathhelpforum.com/advanced-algebra/137681-prove-q-d-b-d-b-q-field-where-d-positive-integer.html
| 1,519,115,596,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-09/segments/1518891812913.37/warc/CC-MAIN-20180220070423-20180220090423-00166.warc.gz
| 218,692,593 | 12,744 |
# Thread: Prove that Q[√d] = {a + b√d | a, b ϵ Q} is a field where d is a positive integer.
1. ## Prove that Q[√d] = {a + b√d | a, b ϵ Q} is a field where d is a positive integer.
Prove that Q[√d] = {a + b√d | a, b ϵ Q} is a field where d is a positive integer.
Help
2. I hope this helps: in order to prove a ring is field we have to prove that the ring has multiplicative inverse, that is it is a unit.
q(rood (d)) is the elements that contains Q and rood (d)
ie: all the elements of the form a + b*sqrt(d) .now you have to see if there exist another element in this form such that (a+ b*root(d))* (c+ d*root(d))=1
I recomend that you divide 1/(c+d*root(d)) and see if you can find an element in Q(root(d))
Hint: use the fact (a-b)*(a+b)=(a^2)-(b^2)
3. Originally Posted by pila0688
Prove that Q[√d] = {a + b√d | a, b ϵ Q} is a field where d is a positive integer.
Help
Originally Posted by hamidr
I hope this helps: in order to prove a ring is field we have to prove that the ring has multiplicative inverse, that is it is a unit.
q(rood (d)) is the elements that contains Q and rood (d)
ie: all the elements of the form a + b*sqrt(d) .now you have to see if there exist another element in this form such that (a+ b*root(d))* (c+ d*root(d))=1
I recomend that you divide 1/(c+d*root(d)) and see if you can find an element in Q(root(d))
Hint: use the fact (a-b)*(a+b)=(a^2)-(b^2)
Well, he first has to show it's a ring. But, a field is a commutative division ring and so he must not only show that every non-zero element of $\mathbb{Q}[\sqrt{d}]$ has a multiplicative inverse but that the multiplication is commutative (albeit this is trivial)
4. Originally Posted by Drexel28
Well, he first has to show it's a ring. But, a field is a commutative division ring and so he must not only show that every non-zero element of $\mathbb{Q}[\sqrt{d}]$ has a multiplicative inverse but that the multiplication is commutative (albeit this is trivial)
yes, However since Q(root(d)) is isomorphic to complex numbers I thought we could assume that it is an integral domain.
5. Originally Posted by hamidr
yes, However since Q(root(d)) is isomorphic to complex numbers I thought we could assume that it is an integral domain.
What makes you think that $\mathbb{Q}[\sqrt{2}]\cong\mathbb{C}$? Define $\eta:\mathbb{Q}^2\to\mathbb{Q}[\sqrt{2}]$ by $(p,q)\mapsto p+\sqrt{d}q$. Clearly this is a surjection and so since $\mathbb{Q}^2$ is countable it follows from a basic exercise that so is $\mathbb{Q}[\sqrt{d}]$. So, since $\mathbb{C}$ which is equipotent to $\mathbb{R}$ is uncountable they can't be isomorphic.
6. Originally Posted by hamidr
yes, However since Q(root(d)) is isomorphic to complex numbers I thought we could assume that it is an integral domain.
Not only is this false, as Drexel points out, but as false as could be. Even if you meant $\mathbb{Q}[i]$ (which I presume you did), no two quadratic fields $\mathbb{Q}[\sqrt{d}]$ are isomorphic.
7. Originally Posted by hamidr
yes, However since Q(root(d)) is isomorphic to complex numbers I thought we could assume that it is an integral domain.
Originally Posted by Bruno J.
Not only is this false, as Drexel points out, but as false as could be. Even if you meant $\mathbb{Q}[i]$ (which I presume you did), no two quadratic fields $\mathbb{Q}[\sqrt{d}]$ are isomorphic.
Regardless. If you only want it to be an integral domain it shouldn't (and I'm not going to say why!) be hard to show that it is one. In fact, defining a Euclidean valuation on it to make it into a Euclidean domain.
8. Originally Posted by Drexel28
What makes you think that $\mathbb{Q}[\sqrt{2}]\cong\mathbb{C}$? Define $\eta:\mathbb{Q}^2\to\mathbb{Q}[\sqrt{2}]$ by $(p,q)\mapsto p+\sqrt{d}q$. Clearly this is a surjection and so since $\mathbb{Q}^2$ is countable it follows from a basic exercise that so is $\mathbb{Q}[\sqrt{d}]$. So, since $\mathbb{C}$ which is equipotent to $\mathbb{R}$ is uncountable they can't be isomorphic.
it seems that I am so lost in ring isomorphisms.
how is Q(p,q) isomorphic to Q[root(d)] ? we dont know what root(d) is, what if it could be expressed as complex? ie d= -1 then we would have a+bi
though I believe you are right since I am so lost in showing isomorphism relations.
9. Originally Posted by hamidr
it seems that I am so lost in ring isomorphisms.
how is Q(p,q) isomorphic to Q[root(d)] ? we dont know what root(d) is, what if it could be expressed as complex? ie d= -1 then we would have a+bi
though I believe you are right since I am so lost in showing isomorphism relations.
My point was more elementary than that. I didn't even show that they aren't ring isomorphic I showed that they can't be equipotent (have the same cardinal numbers). So, they can't be isomorphic (for they'd have to be equipotent).
Also, think about what you're saying. If $\mathbb{Q}[\sqrt{2}],\mathbb{Q}[\sqrt{1}]\cong\mathbb{C}$ then $\mathbb{Q}[\sqrt{1}]\cong\mathbb{Q}[\sqrt{2}]$. Stand back and take a look that statement.
10. Originally Posted by Drexel28
My point was more elementary than that. I didn't even show that they aren't ring isomorphic I showed that they can't be equipotent (have the same cardinal numbers). So, they can't be isomorphic (for they'd have to be equipotent).
Also, think about what you're saying. If $\mathbb{Q}[\sqrt{2}],\mathbb{Q}[\sqrt{1}]\cong\mathbb{C}$ then $\mathbb{Q}[\sqrt{1}]\cong\mathbb{Q}[\sqrt{2}]$. Stand back and take a look that statement.
lools now I understand my stupidity lolz, " $\mathbb{Q}[\sqrt{1}]\cong\mathbb{Q}[\sqrt{2}]$ " that was funny. I cannot think how stupid my claim was lol
thanks
,
,
,
,
,
# Let d be a positive integer. Prove that Q[sqrt(d)]={a b(sqrt(d)) | a, b [ Q} is a field.
Click on a term to search for related topics.
| 1,735 | 5,768 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 25, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.0625 | 4 |
CC-MAIN-2018-09
|
longest
|
en
| 0.891485 |
https://www.eduzip.com/ask/question/intnbsp-sin23x-cos3x-dx-581948
| 1,623,805,280,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-25/segments/1623487621699.22/warc/CC-MAIN-20210616001810-20210616031810-00527.warc.gz
| 655,342,755 | 8,918 |
Mathematics
$\int sin^{2/3}x cos^{3}x dx$
SOLUTION
$\displaystyle\int \sin^{2/3}x\cos^3xdx$
$=\displaystyle\int \sin^{2/3}x\cos^2x\cos xdx$
$=\displaystyle\int \sin^{2/3}x(1-\sin^2x)\cos xdx$
$=\displaystyle\int \sin^{2/3}x\cos xdx-\displaystyle\int \sin^{2/3+2}x\cos xdx$
$=\displaystyle\int \sin^{2/3}x\cos xdx-\displaystyle\int \sin^{5/2}x\cos xdx$
Let $t=\sin x$
$\Rightarrow dt=\cos xdx$
$=\displaystyle\int t^{2/3}dt-\displaystyle\int t^{5/2}dt$
$=\dfrac{t^{2/3+1}}{2/3+1}-\dfrac{t^{5/2+1}}{5/2+1}+c$
$=\dfrac{t^{5/2}}{5/3}-\dfrac{t^{7/2}}{7/2}+c$ where c is the constant of integration
$=\dfrac{3(\sin x)^{5/3}}{5}-\dfrac{2(\sin x)^{7/2}}{7}+c$
$=\dfrac{3\sin^{5/3}x}{5}-\dfrac{2}{7}\sin^{7/2}x+c$ where $t=\sin x$.
Its FREE, you're just one step away
Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Realted Questions
Q1 Subjective Medium
Solve :
$\int \dfrac{\log x^3}{x}\ dx$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
Q2 Single Correct Medium
$\displaystyle I= \int e^{x}\frac{\left ( 2+\sin 2x \right )}{\left ( 1+\cos 2x \right )}dx.$
• A. $\displaystyle e^{x}\sin x.$
• B. $\displaystyle e^{x}\cos x.$
• C. $\displaystyle e^{x}\cos2x.$
• D. $\displaystyle e^{x}\tan x.$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
Q3 Single Correct Hard
If $a,b$ and $c$ are real numbers then the value of $\mathop {\lim }\limits_{t \to 0} {l_n}\left( {\frac{1}{t}\int_0^1 {{{\left( {1 + a\sin bx} \right)}^{\frac{c}{x}}}dx} } \right)$ equals
• A. $\dfrac{{ab}}{c}$
• B. $\dfrac{{bc}}{a}$
• C. $\dfrac{{ca}}{b}$
• D. $abc$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
Q4 Single Correct Hard
Let $g(x) = \int_{0}^{x} f(t) dt$, where $f$ is such that $\dfrac {1}{2} \leq f(x) \leq 1$ for $t\epsilon [0, 1]$ and $0\leq f(t) \leq \dfrac {1}{2}$ for $t\epsilon [1, 2]$. Then, $g(2)$ satisfies the inequality.
• A. $-\dfrac {3}{2}\leq g(2) < \dfrac {1}{2}$
• B. $0\leq g(2) < 2$
• C. $2 < g (2) < 2$
• D. $\dfrac {1}{2} \leq g(2) \leq \dfrac {3}{2}$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$
Asked in: Mathematics - Integrals
1 Verified Answer | Published on 17th 09, 2020
| 1,152 | 2,660 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.09375 | 4 |
CC-MAIN-2021-25
|
latest
|
en
| 0.524259 |
https://brilliant.org/problems/the-drawing/
| 1,521,699,257,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-13/segments/1521257647777.59/warc/CC-MAIN-20180322053608-20180322073608-00214.warc.gz
| 521,351,233 | 11,013 |
# The drawing
You meet a man who has a jar with 100 slips of paper in it.
One of the slips of paper has "25" written on it.
Four of them have "10" written on them.
Four of them have "5" written on them.
Five of them have "2" written on them.
The rest are blank.
He offers you a chance to play a game. If you pay him 1 dollar, you will be blindfolded and then you can pick one of the slips of paper out of the jar. The blindfold will be removed and if the slip has a number on it, he will pay you that amount in dollars. If you draw a blank slip, you don't get any money.
Given that 100 cents = 1 dollar, what is your mathematical expectation in cents if you play this game one time?
×
| 179 | 693 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.640625 | 4 |
CC-MAIN-2018-13
|
latest
|
en
| 0.980253 |
https://www.physicsforums.com/threads/rcl-circuit-problem-due-at-11-pm-est-tonight-have-been-working-on-it-for-days.297020/#post-2101918
| 1,632,804,734,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780060201.9/warc/CC-MAIN-20210928032425-20210928062425-00437.warc.gz
| 925,476,918 | 15,058 |
# RCL Circuit - Problem due at 11 PM EST tonight, have been working on it for days!
1. Homework Statement
A series RCL circuit contains only a capacitor (C = 8.46 μF), an inductor (L = 6.01 mH), and a generator (peak voltage = 77.4 V, frequency = 2.45 x 103 Hz). When t = 0 s, the instantaneous value of the voltage is zero, and it rises to a maximum one-quarter of a period later. (a) Find the instantaneous value of the voltage across the capacitor/inductor combination when t = 4.32 x 10^-4 s. (b) What is the instantaneous value of the current when t = 4.32 x 10^-4 s?
2. Homework Equations
v(t) = V*sin(2pi*ft)
z= square root(R^2 + (Xl-Xc)^2)
I=V/Z
3. The Attempt at a Solution
the instantaneous voltage in part a = 27.77 V. (this is correct)
my answer to part b is incorrect but I don't know why:
X of L=2pi(2450)(.00601)=92.5 Ohms
X of X=1/(2pi(2450)c) = 7.678 Ohms
Z=square root(0^2 - (92.5 - 7.678)^2) = 84.822
I0=V0/Z = 77.4/84.822 = 0.91249 A
I=I0sin(2pi*ft + (pi/2))
I=(0.91249)sin(2pi(2450)(4.34x10^-4) + (pi/2)) = 0.841 A
| 388 | 1,040 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.71875 | 4 |
CC-MAIN-2021-39
|
latest
|
en
| 0.81872 |
http://www.chegg.com/homework-help/statistics-for-engineering-and-the-sciences-5th-edition-chapter-10.10-solutions-9780131877085
| 1,454,842,883,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-07/segments/1454701148834.81/warc/CC-MAIN-20160205193908-00004-ip-10-236-182-209.ec2.internal.warc.gz
| 336,497,172 | 16,071 |
View more editions
TEXTBOOK SOLUTIONS FOR Statistics for Engineering and the Sciences 5th Edition
• 1253 step-by-step solutions
• Solved by publishers, professors & experts
• iOS, Android, & web
Over 90% of students who use Chegg Study report better grades.
May 2015 Survey of Chegg Study Users
PROBLEM
Chapter: Problem:
SAMPLE SOLUTION
Chapter: Problem:
• Step 1 of 7
The given problem is representing the life tests of the cutting tools. The data were derived from the tests for the two different brands of cutting tools currently used in the production process.
a) Fit the model, to the data for brand A.
Use the MINITAB to get the required results.
Instructions:
1. Type or import the data into the MINITAB work sheet.
2. Go to
3. Specify the Response variable (Hours A) and Predictors (Speed).
4. Click ok.
• Step 2 of 7
By following the MINITAB instructions we get the output as follows:
From the above output the fitted regression line is,
Where, y: useful life
x: cutting speed.
• Step 3 of 7
b) Fit the model, to the data for brand B.
Use the MINITAB to get the required results.
Instructions:
1. Type or import the data into the MINITAB work sheet.
2. Go to
3. Specify the Response variable (Hours B) and Predictors (Speed).
4. Click ok.
• Step 4 of 7
By following the MINITAB instructions we get the output as follows:
From the above output the fitted regression line is,
Where, y: useful life
x: cutting speed.
• Step 5 of 7
c) Find the 90% confidence interval for the estimate of the mean uselife of brand A and Bran B when cutting speed is 45 meters per minute.
Use MINITAB to get the required results.
Instructions:
1. Type or import the data into the MINITAB work sheet.
2. Go to
3. Specify the Response variable (Hours A) and Predictors (Speed).
4. Click on options; specify the value of prediction interval for new observation (45). Click ok.
5. Click ok.
By following the MINITAB instructions we get the output as follows:
From the above output we have the 90% confidence interval is (2.763, 3.937).
• Step 6 of 7
Repeat the same procedure for Brand B, we get the following output.
From the above output we have the 90% confidence interval is (4.169, 4.761).
d) Find the 90% prediction interval for the estimate of the mean uselife of brand A and Bran B when cutting speed is 45 meters per minute.
By following the instructions from part (c) we get the required results.
For Band A:
From the above output we have the 90% prediction interval is (1.127, 5.573).
For Band B:
From the above output we have the 90% prediction interval is (3.345, 5.585).
• Step 7 of 7
e) Find the 90% confidence interval for the estimate of the mean uselife of brand A when cutting speed is 100 meters per minute.
Use MINITAB to get the required results.
Instructions:
1. Type or import the data into the MINITAB work sheet.
2. Go to
3. Specify the Response variable (Hours A) and Predictors (Speed).
4. Click on options; specify the value of prediction interval for new observation (100). Click ok.
5. Click ok.
By following the MINITAB instructions we get the output as follows:
From the above output we have the 90% confidence interval is (-3.128, 1.835).
Corresponding Textbook
Statistics for Engineering and the Sciences | 5th Edition
9780131877085ISBN-13: 0131877089ISBN: Authors:
| 848 | 3,344 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2016-07
|
latest
|
en
| 0.771049 |
http://www.docstoc.com/docs/124461578/ESL-Entry-Math-Assessment
| 1,387,502,393,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2013-48/segments/1387345768632/warc/CC-MAIN-20131218054928-00007-ip-10-33-133-15.ec2.internal.warc.gz
| 368,724,559 | 15,817 |
# ESL Entry Math Assessment
Document Sample
``` ESL Entry Math Assessment
For Teacher Use Only
Name
School Student ID
Raw Score
Date
1. 2. 3.
4 + 2 = 3 7 - 2 =
+ 5
4. 5. Count. 6.
4 How many?
-4 50, 51, 52, , 54
7. Continue. 8. Draw a square. 9. Continue.
What comes next? What comes next?
10, 20, 30, 40,
ESL Entry Math Assessment
10. 11. Write these numbers 12. How long is this
in order from smallest page?
to largest.
+ = ___ 390 317 centimeters
426 304
304, , ,
13. Juan has 29 fish. He 14. 15.
sells 23. How many 64 80
does he have now?
- 13 - 57
16. 17. 18.
55 + 13 + 12 = 35 1 day = hours
+ 46
19. 20. 21.
+ 8 = 16 175 703
+ 354 - 526
22. 23. How many fish did we catch in June?
3 x 4 x 2 = fish
May
June
July
August
0 10 20 30
2
ESL Entry Math Assessment
24. 25. 26.
40
30
20
10 40 4 =
0 1 year = months
-10
-20
What is the
temperature?
27. Complete. 28. There are 85 girls and 29. Write <, >, or = in the
33 boys in the circle.
115, 112, 109, , , cafeteria. How many
more are girls? 4,639 4,936
, 97, 94
30. How many thousands 31. 32.
in 63,612? A
92 7 =
-3 0 +3
A=
33. Estimate how high 34. Round 269 to the 35.
this ceiling is in nearest hundred.
meters. 57
x 27
meters
3
ESL Entry Math Assessment
36. One kilogram of 37. 38. What is the perimeter
candy costs \$2.50. of this figure?
Mai buys 5 kilograms. 2.8 + 1.67 = 20
How much money
does she spend? 10 10
20
39. Raul puts some bread 40. Write >, <, or = in 41. Draw 2 lines that are
in the oven at 9:10 the circle. parallel.
a.m. It needs to bake
85 minutes. When will 4 6
42. Nadia has 10 meters 43. 44. How many millions in
of rope. She cuts a 14, 683, 257?
piece 3.5 meters long. 7.09
How much rope is x 0.7
left?
45. 46. 47.
13 5.68 8 = 4 hours =
4 =
minutes
4
ESL Entry Math Assessment
48. 49. 50.
6 + 2 x 7 = 32 x 4 = 1 2
4 + 8 =
\$1600 each month, for \$0.35 each. How
how much do they much change does 63 0.9 =
spend on food? she get from \$4.00?
10% clothes
10% other
food
25%
5% car
rent
50%
54. 55. 56.
H
4 2
7 6 3 x 3
=
1 O F
-5 2
E
G
Which line shows
the diameter?
5
ESL Entry Math Assessment
57. What is the area of 58. 59.
this rectangle?
0.03 = % 30% of 300 =
13
7 7
13
60. 61. 62.
(-7) + 3 + (-3) = (-5) x (-2) = 30 - (-8) =
63. B 64. 65.
<B= 6N + 2 - 3N = 1
4 2 =
3
o
A 40 C
66. 67. If A = 4, then 68.
6A – 5 =
x 12 x
3 = 9 5 = 4
X =
X =
6
ESL Entry Math Assessment
69. 70. 71.
x + 4 = 9 8 10 3x
5 + 6 = 12
x =
N x =
N =
72. 73. 74. Solve for x.
G15 4K – 3 = K + 15 4x > 12
=
G5
K=
Courtesy Fairfax County, VA Public Schools
7
```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views: 13 posted: 7/18/2012 language: pages: 7
How are you planning on using Docstoc?
| 1,286 | 5,369 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2013-48
|
latest
|
en
| 0.811316 |
https://hyperleap.com/topic/Law_of_large_numbers
| 1,601,339,444,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-40/segments/1600401617641.86/warc/CC-MAIN-20200928234043-20200929024043-00563.warc.gz
| 440,766,435 | 34,085 |
# Law of large numbers
strong law of large numbersweak law of large numbersBernoulli's Golden TheoremLaws of large numbers approachesapproaches one-halfexpectationlaw of averageslong runPoisson's law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.wikipedia
209 Related Articles
### Expected value
expectationexpectedmean
According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer to the expected value as more trials are performed. Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
For example, the expected value of rolling a six-sided die is 3.5, because the average of all the numbers that come up converges to 3.5 as the number of rolls approaches infinity (see for details).
### Probability theory
theory of probabilityprobabilityprobability theorist
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.
Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
### Monte Carlo method
Monte CarloMonte Carlo simulationMonte Carlo methods
Another good example about LLN is Monte Carlo method.
By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable.
### Ars Conjectandi
1713
It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713.
The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject.
### Theorem
theoremspropositionconverse
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.
### Almost surely
almost alwaysalmost surezero probability
In particular, the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity.
Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion.
### Jacob Bernoulli
Jakob BernoulliBernoulliJames Bernoulli
A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.
However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.
### Cauchy distribution
LorentzianCauchyLorentzian distribution
For instance, the average of the results from Cauchy distribution or some Pareto distribution (α
Various results in probability theory about expected values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.
### Variance
sample variancepopulation variancevariability
Based on the assumption of finite variance (for all i) and no correlation between random variables, the variance of the average of n random variables Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.
### Chebyshev's inequality
Bienaymé–Chebyshev inequalityChebyshev inequalityAn inequality on location and scale parameters
Chebyshev's inequality.
For example, it can be used to prove the weak law of large numbers.
### Pafnuty Chebyshev
ChebyshevP. L. ChebyshevChebychev
After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli and Kolmogorov and Khinchin.
The Chebyshev inequality is used to prove the weak law of large numbers.
### Asymptotic equipartition property
Shannon–McMillan–Breiman theorem
(This is a consequence of the law of large numbers and ergodic theory.) Although there are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set.
### Convergence of random variables
convergence in distributionconverges in distributionconvergence in probability
For interpretation of these modes, see Convergence of random variables.
This result is known as the weak law of large numbers.
### Central limit theorem
Lyapunov's central limit theoremlimit theoremscentral limit
By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as
### Law of averages
averaging theory
While there is a real theorem that a random variable will reflect its underlying probability over a very large sample, the law of averages typically assumes that unnatural short-term "balance" must occur.
### Characteristic function (probability theory)
characteristic functioncharacteristic functionscharacteristic function:
By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as
This theorem is frequently used to prove the law of large numbers, and the central limit theorem.
### Law of the iterated logarithm
The law of iterated logarithms operates “in between” the law of large numbers and the central limit theorem.
### Random variable
random variablesrandom variationrandom
Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.
### Regression toward the mean
regression to the meanRegression towards the meanmean regression
Similarly, the law of large numbers states that in the long term, the average will tend towards the expected value, but makes no statement about individual trials.
### Average
Rushing averageReceiving averagemean
According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer to the expected value as more trials are performed.
### Roulette
roulette wheelAmerican roulettebetting wheel
For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins.
### Gambler's fallacy
D'Alembert systemexampleLe Grande Casino
There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy).
### Probability
probabilisticprobabilitieschance
For example, a single roll of a fair, six-sided dice produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability.
### Sample mean and covariance
sample meansample covariancesample covariance matrix
: According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the precision increasing as more dice are rolled.
| 1,618 | 7,697 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.875 | 4 |
CC-MAIN-2020-40
|
latest
|
en
| 0.936586 |
https://mathexamination.com/class/fake-projective-plane.php
| 1,618,475,114,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-17/segments/1618038084601.32/warc/CC-MAIN-20210415065312-20210415095312-00118.warc.gz
| 494,465,865 | 7,019 |
## Take My Fake Projective Plane Class
A "Fake Projective Plane Class" QE" is a basic mathematical term for a generalized continuous expression which is utilized to fix differential equations and has solutions which are routine. In differential Class solving, a Fake Projective Plane function, or "quad" is utilized.
The Fake Projective Plane Class in Class form can be revealed as: Q( x) = -kx2, where Q( x) are the Fake Projective Plane Class and it is an essential term. The q part of the Class is the Fake Projective Plane constant, whereas the x part is the Fake Projective Plane function.
There are 4 Fake Projective Plane functions with proper option: K4, K7, K3, and L4. We will now look at these Fake Projective Plane functions and how they are fixed.
K4 - The K part of a Fake Projective Plane Class is the Fake Projective Plane function. This Fake Projective Plane function can likewise be written in partial portions such as: (x2 - y2)/( x+ y). To fix for K4 we increase it by the right Fake Projective Plane function: k( x) = x2, y2, or x-y.
K7 - The K7 Fake Projective Plane Class has a service of the type: x4y2 - y4x3 = 0. The Fake Projective Plane function is then increased by x to get: x2 + y2 = 0. We then need to multiply the Fake Projective Plane function with k to get: k( x) = x2 and y2.
K3 - The Fake Projective Plane function Class is K3 + K2 = 0. We then multiply by k for K3.
K3( t) - The Fake Projective Plane function equationis K3( t) + K2( t). We multiply by k for K3( t). Now we increase by the Fake Projective Plane function which provides: K2( t) = K( t) times k.
The Fake Projective Plane function is also called "K4" because of the initials of the letters K and 4. K indicates Fake Projective Plane, and the word "quad" is noticable as "kah-rab".
The Fake Projective Plane Class is one of the primary techniques of solving differential equations. In the Fake Projective Plane function Class, the Fake Projective Plane function is first increased by the proper Fake Projective Plane function, which will offer the Fake Projective Plane function.
The Fake Projective Plane function is then divided by the Fake Projective Plane function which will divide the Fake Projective Plane function into a real part and an imaginary part. This offers the Fake Projective Plane term.
Finally, the Fake Projective Plane term will be divided by the numerator and the denominator to get the ratio. We are left with the right-hand man side and the term "q".
The Fake Projective Plane Class is an essential principle to understand when resolving a differential Class. The Fake Projective Plane function is simply one approach to solve a Fake Projective Plane Class. The approaches for resolving Fake Projective Plane formulas consist of: singular value decomposition, factorization, optimum algorithm, numerical solution or the Fake Projective Plane function approximation.
## Hire Someone To Do Your Fake Projective Plane Class
If you would like to become acquainted with the Quartic Class, then you require to very first start by checking out the online Quartic page. This page will reveal you how to use the Class by using your keyboard. The description will also show you how to develop your own algebra formulas to help you study for your classes.
Before you can understand how to study for a Fake Projective Plane Class, you need to initially comprehend making use of your keyboard. You will find out how to click on the function keys on your keyboard, as well as how to type the letters. There are 3 rows of function keys on your keyboard. Each row has four functions: Alt, F1, F2, and F3.
By pressing Alt and F2, you can increase and divide the worth by another number, such as the number 6. By pressing Alt and F3, you can use the 3rd power.
When you push Alt and F3, you will key in the number you are trying to increase and divide. To multiply a number by itself, you will push Alt and X, where X is the number you want to increase. When you press Alt and F3, you will enter the number you are attempting to divide.
This works the exact same with the number 6, other than you will just key in the two digits that are 6 apart. Finally, when you push Alt and F3, you will utilize the 4th power. However, when you push Alt and F4, you will utilize the real power that you have found to be the most suitable for your problem.
By utilizing the Alt and F function keys, you can multiply, divide, and after that use the formula for the 3rd power. If you need to multiply an odd variety of x's, then you will need to enter an even number.
This is not the case if you are trying to do something complex, such as increasing 2 even numbers. For instance, if you want to multiply an odd variety of x's, then you will require to enter odd numbers. This is particularly true if you are trying to figure out the answer of a Fake Projective Plane Class.
If you want to convert an odd number into an even number, then you will need to press Alt and F4. If you do not know how to increase by numbers on their own, then you will need to use the letters x, a b, c, and d.
While you can multiply and divide by use of the numbers, they are a lot easier to utilize when you can take a look at the power tables for the numbers. You will need to do some research when you initially begin to use the numbers, however after a while, it will be force of habit. After you have actually developed your own algebra equations, you will be able to develop your own multiplication tables.
The Fake Projective Plane Solution is not the only way to fix Fake Projective Plane formulas. It is necessary to discover trigonometry, which uses the Pythagorean theorem, and then use Fake Projective Plane solutions to resolve issues. With this method, you can learn about angles and how to fix issues without needing to take another algebra class.
It is essential to attempt and type as quickly as possible, since typing will assist you know about the speed you are typing. This will help you compose your responses much faster.
## Hire Someone To Take My Fake Projective Plane Class
A Fake Projective Plane Class is a generalization of a direct Class. For example, when you plug in x=a+b for a given Class, you get the worth of x. When you plug in x=a for the Class y=c, you acquire the worths of x and y, which provide you a result of c. By using this standard principle to all the equations that we have attempted, we can now resolve Fake Projective Plane equations for all the values of x, and we can do it quickly and effectively.
There are lots of online resources offered that supply free or budget-friendly Fake Projective Plane formulas to resolve for all the values of x, consisting of the expense of time for you to be able to make the most of their Fake Projective Plane Class assignment assistance service. These resources typically do not need a subscription cost or any type of financial investment.
The responses provided are the result of complex-variable Fake Projective Plane formulas that have been fixed. This is also the case when the variable utilized is an unidentified number.
The Fake Projective Plane Class is a term that is an extension of a direct Class. One benefit of using Fake Projective Plane equations is that they are more general than the linear formulas. They are simpler to resolve for all the worths of x.
When the variable utilized in the Fake Projective Plane Class is of the kind x=a+b, it is easier to solve the Fake Projective Plane Class since there are no unknowns. As a result, there are less points on the line specified by x and a consistent variable.
For a right-angle triangle whose base points to the right and whose hypotenuse indicate the left, the right-angle tangent and curve graph will form a Fake Projective Plane Class. This Class has one unknown that can be discovered with the Fake Projective Plane formula. For a Fake Projective Plane Class, the point on the line defined by the x variable and a continuous term are called the axis.
The existence of such an axis is called the vertex. Given that the axis, vertex, and tangent, in a Fake Projective Plane Class, are an offered, we can find all the worths of x and they will sum to the offered worths. This is accomplished when we use the Fake Projective Plane formula.
The aspect of being a continuous factor is called the system of equations in Fake Projective Plane equations. This is in some cases called the main Class.
Fake Projective Plane equations can be resolved for other worths of x. One way to solve Fake Projective Plane equations for other worths of x is to divide the x variable into its aspect part.
If the variable is given as a positive number, it can be divided into its factor parts to get the normal part of the variable. This variable has a magnitude that is equal to the part of the x variable that is a consistent. In such a case, the formula is a third-order Fake Projective Plane Class.
If the variable x is unfavorable, it can be divided into the very same part of the x variable to get the part of the x variable that is increased by the denominator. In such a case, the formula is a second-order Fake Projective Plane Class.
Option aid service in fixing Fake Projective Plane formulas. When using an online service for fixing Fake Projective Plane equations, the Class will be fixed immediately.
| 2,065 | 9,361 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.96875 | 4 |
CC-MAIN-2021-17
|
latest
|
en
| 0.903257 |
https://www.askiitians.com/forums/Trigonometry/in-triangle-abc-sin-3-acos-3-b-c-sin-3-b-cos-3_131633.htm
| 1,713,869,586,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-18/segments/1712296818474.95/warc/CC-MAIN-20240423095619-20240423125619-00041.warc.gz
| 586,022,302 | 43,169 |
# in triangle ABC,sin^3 Acos^3 (B-C)+sin^3 B cos^3 (C-A)+sin^3C cos^3 (A-B)=
Anoopam Mishra
126 Points
8 years ago
$sin^3(A)cos^3(B-C) = sin^3(\pi - A)cos^3(B-C) = sin^3(B+C)cos^3(B-C) = (cos^2B - sin^2C)^3$
So the given expression simplifies to
$(cos^2B-sin^2C)^3 + (cos^2C-sin^2A)^3 + (cos^2A-sin^2B)^3$
$a^3 + b^3 + c^3 = 3abc \ , \ if \ a+b+c=0$
$3(cos^2B-sin^2C)(cos^2C-sin^2A)(cos^2A-sin^2B)$
Lab Bhattacharjee
121 Points
8 years ago
$\text{HINT:}\\ \cos^2x-\sin^2y=\cos(x-y)\cos(x+y) \\ 2\sin(x+y)\cos(x-y)=\sin2x+\sin2y$
72 Points
8 years ago
$sin^3(A)cos^3(B-C) = sin^3(\pi - A)cos^3(B-C) = sin^3(B+C)cos^3(B-C) = (cos^2B - sin^2C)^3$
So the given expression simplifies to
$(cos^2B-sin^2C)^3 + (cos^2C-sin^2A)^3 + (cos^2A-sin^2B)^3$
$a^3 + b^3 + c^3 = 3abc \ , \ if \ a+b+c=0$
$3(cos^2B-sin^2C)(cos^2C-sin^2A)(cos^2A-sin^2B)$
| 439 | 836 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.15625 | 4 |
CC-MAIN-2024-18
|
latest
|
en
| 0.401196 |
https://web2.0calc.com/questions/triangles_63
| 1,701,693,525,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-50/segments/1700679100529.8/warc/CC-MAIN-20231204115419-20231204145419-00458.warc.gz
| 701,907,189 | 6,054 |
+0
# Triangles
0
223
2
In ABC, sin A : sin B : sin C = 5:6:7 and the perimeter is 18. Find the area of triangle ABC.
$$\phantom{\cos A}$$
May 4, 2022
#1
+125697
+1
A = (5/18)*180 = 50°
B = (6/18) * 180 = 60°
C = (7/18)* 180 = 70°
By the Law of Sines
a /sin 50 = b / sin 60 c / sin 70 = b / sin 60
a / b = sin 50 /sqrt(3)/2 c/b = sin 70 / sqrt (3)/2
a = b [ 2sin 50 / sqrt 3] c = b [ 2sin 70 / sqrt 3 ]
a + b + c = 180
b* 2sin 50 /sqrt 3 + b + b [ 2sin 70 /sqrt 3 ] = 18
b [ 2sin 50 /sqrt 3 + 1 + 2sin70/sqrt 3 ] = 18
b = 18 / [ 2sin50/sqrt 3 + 1 + 2sin 70 / sqrt 3 ] ≈ 6.06
a = 6.06 [ 2 sin 50 / sqrt 3] ≈ 5.36
c = 6.06 [2 sin 70 /sqrt 3 ] ≈ 6.58
Area = sqrt [ 9 (9 - 6.06) (9 - 5.36) (9 - 6.58) ] ≈ 15.267 units
May 5, 2022
#2
0
The sine rule says that
$$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C} = 2R,$$
(where R is the radius of the circumscribing circle), so
$$\displaystyle a=2R\sin A,\\b = 2R \sin B,\\ c = 2R\sin C,$$
and since
$$\displaystyle \sin A :\sin B :\sin C$$
are in the ratio 5 : 6 : 7, it follows that a, b, and c are also in the ratio 5 to 6 to 7.
Since a + b + c = 18 it then follows that a = 5, b = 6 and c = 7.
From that, s (the semi-perimeter) = 9, so the area of the triangle is
$$\displaystyle \sqrt{9(9-5)(9-6)(9-7)}=6\sqrt{6}.$$
n.b. the angles, (in degrees to 3dp) are 44.415, 57.122 and 78.463.
May 5, 2022
| 726 | 1,490 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.40625 | 4 |
CC-MAIN-2023-50
|
latest
|
en
| 0.410138 |
https://glasp.co/youtube/p/integrating-multivariable-functions-with-respect-to-one-variable-calculus-3
| 1,719,156,463,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-26/segments/1718198862474.84/warc/CC-MAIN-20240623131446-20240623161446-00792.warc.gz
| 231,011,974 | 82,339 |
# Integrating Multivariable Functions with Respect to One Variable - Calculus 3 | Summary and Q&A
1.6K views
July 19, 2019
by
The Math Sorcerer
Integrating Multivariable Functions with Respect to One Variable - Calculus 3
## TL;DR
Integrating multivariable functions with respect to one variable involves adding an unknown function of the other variable.
## Key Insights
• 😀 In calculus 1, the plus C term is added to integrals of functions of one variable to represent the infinitely many possible solutions.
• 🫡 When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added to maintain the partial derivative.
• 🫡 When integrating a function of two variables with respect to the other variable, an unknown function of the first variable must be added to maintain the partial derivative.
• 🪜 This concept of adding an unknown function of the other variable also applies to potential functions and solving exact equations in differential equations.
## Transcript
hi everyone in this video I want to talk about integrating multivariable functions with respect to one variable so let's look at some examples so first let's go back to like calc one for a second let's say you have x squared DX so we have a function of one variable x squared and we're integrating with respect to X so you use the power rule so you a... Read More
### Q: Why is the plus C added when integrating a function of one variable?
The plus C is added because the derivative of any constant is 0. Since the constant term can be any number, it must be included to represent the infinitely many possible solutions.
### Q: How does integrating a function of two variables differ from integrating a function of one variable?
When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added. This is necessary to ensure that the partial derivative of the integrated function with respect to the variable being held constant results in the original function.
### Q: What happens when integrating a function of two variables with respect to the other variable?
When integrating a function of two variables with respect to the other variable, an unknown function of the first variable must be added. This is done to ensure that the partial derivative of the integrated function with respect to the variable being held constant results in the original function.
### Q: In what other areas of mathematics does this concept of adding an unknown function of the other variable arise?
This concept is important in potential functions and solving exact equations in differential equations. It is used to account for unknown functions that arise during the integration process.
## Summary & Key Takeaways
• In calculus 1, when integrating a function of one variable, the power rule is used, resulting in an additional constant term.
• When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added.
• The same rule applies when integrating with respect to the other variable, adding an unknown function of the first variable.
| 630 | 3,203 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.3125 | 4 |
CC-MAIN-2024-26
|
latest
|
en
| 0.885876 |
https://mathematica.stackexchange.com/questions/152403/partial-derivative-understanding-output
| 1,716,122,025,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-22/segments/1715971057786.92/warc/CC-MAIN-20240519101339-20240519131339-00247.warc.gz
| 333,350,185 | 38,969 |
# Partial derivative: Understanding output
I have a simple double sum as function:
u[m_, n_] = Sum[m^M/M! * n^N/N!, {M, 0, Infinity}, {N, 0, M - 1}]
and would like to compute the partial derivatives.
D[u[m, n], m]
I can't get the output neatly here, but it basically gives me (with correct indices on the summations)
$$\left(\sum \sum \frac{m^{M-1}M n^N}{M! N!}\right)[m, n] + \left(\sum \sum \frac{m^{M} n^N}{M! N!}\right)^{(1, 0)}[m, n]$$
I would have thought that the first term would be the correct answer. I looked up the $(1, 0)$ notation and it apparently means "partial derivative w.r.t. the first input" - but isn't that exactly what I've put in here?
How am I supposed to understand this result?
• Might be better to differentiate the summand instead of the sum. Also, avoid using N since it is a built-in function. Jul 27, 2017 at 11:52
• I only seem to get the first term (11.0.1.0 OSX) Jul 27, 2017 at 11:55
I think you got the wrong result because you evaluated some code that is different from what included in the question. I think you used:
u = Sum[m^M/M! * n^N/N!, {M, 0, Infinity}, {N, 0, M - 1}]
and forgot to Clear[u].
To differentiate appropriately you need Inactive[Sum] and replace the built in N by e.g. esc N esc.
Clear[u]
u[m_, n_] = Sum[m^M/M!*n^Ν/Ν!, {M, 0, Infinity}, {Ν, 0, M - 1}]
uSym[m_, n_] = Inactive[Sum][m^M/M!*n^Ν/Ν!, {M, 0, Infinity}, {Ν, 0, M - 1}]
uDm[m_, n_] = Activate[D[uSym[m, n], m]];
$\sum _{M=0}^{\infty } \sum _{Ν=0}^{M-1} \frac{M m^{M-1} n^Ν}{M! Ν!}$
| 533 | 1,520 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.53125 | 4 |
CC-MAIN-2024-22
|
latest
|
en
| 0.921209 |
https://pdfcoffee.com/prepinsta-com-tcs-coding-questions-pdf-free.html
| 1,709,529,552,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-10/segments/1707947476413.82/warc/CC-MAIN-20240304033910-20240304063910-00829.warc.gz
| 455,139,417 | 9,037 |
# Prepinsta Com Tcs Coding Questions
##### Citation preview
TCS Coding Questions with Answers July 3, 2019 by
TCS Programming Questions with Answers TCS Programming Test Questions are not like a general Coding Round Questions with Solutions it is all together different from C programming. We have analysed over 100+ TCS Programming Questions they use Command Line Programming in the Programming Section of the test. Below you will find TCS Programming Round Questions, TCS Coding Questions that are asked constantly in TCS Placement Papers Programming, programming Test of TCS Placement Coding Questions, TCS C Coding Questions, TCS C Programming Questions, TCS Coding Section, TCS C Programming Questions and Answers.
Ch eck Qu esti o n s Here
Important Note TCS has just introduced new test pattern for 2020 Batch pass-outs students. The pattern for 2019 pass-outs being the same. Below are the tables with detailed analysis on the same. Make sure you do not refer any other paper pattern as below one is the most updated paper pattern for both 2019 and 2020 Passouts. For 2020 Passouts : No Of Questions:- 1 Question Time - 30 mins For 2019 Passouts : No Of Questions:- 1 Question Time - 20 mins Get all the questions asked Today in TCS here - TCS Ninja Live Questions Dashboard TCS Placement Papers Programming
Online Classes 1. Free Paid Materials also given 2. Click here to join Online Live Class
TCS Paid Material – In Online class you will get preparation paid materials access also, but if you just want materials then you can buy the ones below – TCS Aptitude Preparation Paid Materials Basic Pro TCS Coding Preparation Paid Materials Basic Pro TCS Verbal English Preparation Paid Materials Basic Pro TCS Programming MCQ Preparation Paid Materials Basic Pro or 1. Click here to join Online Live Class
31
37
MIN UTES
S EC ON D S
Free Materials Checking if a given year is leap year or not Explanation: To check whether a year is leap or not Step 1: We first divide the year by 4. If it is not divisible by 4 then it is not a leap year. If it is divisible by 4 leaving remainder 0 Step 2: We divide the year by 100 If it is not divisible by 100 then it is a leap year. If it is divisible by 100 leaving remainder 0 Step 3: We divide the year by 400 If it is not divisible by 400 then it is a leap year. If it is divisible by 400 leaving remainder 0 Then it is a leap year
C
#include int leap(int year) { if(year%4 == 0) { if( year%100 == 0) { if ( year%400 == 0) printf("%d is a leap year.", year); else printf("%d is not a leap year.", year); } else printf("%d is a leap year.", year ); } else printf("%d is not a leap year.", year); return 0; } int main()
{ int yr, a; printf("Enter year : "); //enter the year to check scanf("%d",&yr); a = leap(yr); return 0; }
C++ Java Python
Prime Numbers with a Twist Ques. Write a code to check whether no is prime or not. Condition use function check() to find whether entered no is positive or negative ,if negative then enter the no, And if yes pas no as a parameter to prime() and check whether no is prime or not? Whether the number is positive or not, if it is negative then print the message “please enter the positive number” It is positive then call the function prime and check whether the take positive number is prime or not.
C
#include void prime(int n) { int c=0; for(int i=2;i=1) printf("%d is not a prime number",n); else printf("%d is a prime number",n); } void main() { int n; printf("Enter no : "); //enter the number scanf("%d",&n); if(n
| 966 | 3,847 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.625 | 4 |
CC-MAIN-2024-10
|
latest
|
en
| 0.795846 |
https://dangoldner.wordpress.com/2011/03/24/thats-the-game/
| 1,670,649,691,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-49/segments/1669446711712.26/warc/CC-MAIN-20221210042021-20221210072021-00149.warc.gz
| 232,000,234 | 26,490 |
Yesterday: a student tells me he solved a 3×3 system by letting x, y & z be (3, 1, 1) in the first equation, (2, 2, 1) in the second, and (1, 2, 2) in the third. It’s cool, I say, that he found solutions for each equation, but we’re looking for x, y & z that are the same in all three equations and still work. “That’s the game,” I say, as his eyes widen and he really takes in the nature of the puzzle he’s facing.
Today: the puzzle is to find the equation of a parabola that passes through three given points. The aim is for them to discover a strategy. The minutes tick by. Many students, not knowing how to do this, have given up. I call them together, we go through finding the equation of a line. “So, how are we going to find b?” “Plug in x and y,” says J. “Good. Does that help with our problem?”
I tour the groups. It has not helped. Energy in the room: zero. Less than 10 minutes to go. Call them back together. “Let’s try J’s solution on our problem. If I plug in the first point, what do I get?” Two or three voices chant out: “0 equals a times 1 squared plus b times 1 plus c”. We continue and soon there’s a 3×3 system on the board. How do we solve that? “Like we learned yesterday.” “Ok, go to it.” Energy is still zero – there isn’t a problem anymore. Assiduous students plug away. The rest wait for the bell.
In inquiry classes, what do you do when they stop looking? Leaving them stuck deflates them. Revealing the answer deflates them. What’s the right thing in between, and what do you do if you don’t have it? That’s the game.
| 415 | 1,550 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.65625 | 4 |
CC-MAIN-2022-49
|
latest
|
en
| 0.963542 |
https://www.hsslive.co.in/2022/05/ap-board-class-10th-maths-chapter-9-tangents-and-secants-to-a-circle-intext-questions-textbook-solutions-pdf.html
| 1,716,229,796,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-22/segments/1715971058293.53/warc/CC-MAIN-20240520173148-20240520203148-00134.warc.gz
| 717,857,639 | 46,023 |
# HSSlive: Plus One & Plus Two Notes & Solutions for Kerala State Board
## AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions PDF: Download Andhra Pradesh Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Book Answers
AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions PDF: Download Andhra Pradesh Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Book Answers
## Andhra Pradesh State Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Books Solutions
Board AP Board Materials Textbook Solutions/Guide Format DOC/PDF Class 10th Subject Maths Chapters Maths Chapter 9 Tangents and Secants to a Circle InText Questions Provider Hsslive
## How to download Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions Answers PDF Online?
2. Click on the Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Answers.
3. Look for your Andhra Pradesh Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks PDF.
4. Now download or read the Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions for PDF Free.
## AP Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks Solutions with Answer PDF Download
Find below the list of all AP Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions for PDF’s for you to download and prepare for the upcoming exams:
### 10th Class Maths 9th Lesson Tangents and Secants to a Circle InText Questions and Answers
Do this
Question 1.
Draw a circle with any radius. Draw four tangents at different points. How many tangents can you draw to this circle? (Page No. 226)
Let ‘O’ be the centre of the circle with radius OA.
l, m, n, p and q be the tangents to the circles at A, B, C, D and E. We can draw a tangent at each point on the circle, i.e., infinitely many tangents can be drawn to a circle.
Question 2.
How many tangents you can draw to circle from a point away from it? (Page No. 226)
We can draw only two tangents from an exterior point.
Question 3.
In the below figure which are tangents to the given circles? (Page No. 226)
P and M are the tangents to the given circles.
Question 4.
Draw a circle and a secant PQ of the circle on a paper as shown below. Draw various lines parallel to the secant on both sides of it. What happens to the length of chord coming closer and closer to the centre of the circle? (Page No. 227)
The length of the chord increases as it comes closer to the centre of the circle.
Question 5.
What is the longest chord? (Page No. 227)
Diameter is the longest of all chords.
Question 6.
How many tangents can you draw to a circle, which are parallel to each other? (Page No. 227)
Only one tangent can be drawn parallel to a given tangent.
To a circle, we can draw infinitely many pairs of parallel tangents.
Try this
(Page No. 228)
How can you prove the converse of the above theorem.
Question 1.
“If a line in the plane of a circle is perpendicular to the radius at its end point on the circle, then the line is tangent to the circle”.
Given: Circle with centre ‘O’, a point A on the circle and the line AT perpendicular to OA.
R.T.P: AT is a tangent to the circle at A.
Construction:
Suppose AT is not a tangent then AT produced either way if necessary, will meet the circle again. Let it do so at P, join OP.
Proof: Since OA = OP (radii)
∴ ∠OAP = ∠OPA But ∠OPA = 90°
∴ Two angles of a triangle are right angles which is impossible.
∴ Our supposition is false.
∴ Hence AT is a tangent.
We can find some more results using the above theorem.
i) Since there can be only one perpendicular OP at the point P, it follows that one and only one tangent can be drawn to a circle at a given point on the circumference.
ii) Since there can be only one perpendicular to XY at the point P, it follows that the perpendicular to a tangent at its point of contact passes through the centre.
Question 2.
How can you draw the tangent to a circle at a given point when the centre of the circle is not known? (Page No. 229)
Steps of Construction:
1. Take a point P and draw a chord PR through P.
2. Construct ∠PRQ and measure it.
3. Construct ∠QPX at P equal to ∠PRQ.
4. Extend PX on other side. XY is the required tangent at P.
Note: Angle between a tangent and chord is equal to angle in the alternate segment.
Hint: Draw equal angles ∠QPX and ∠PRQ. Explain the construction.
Steps of construction:
1. Draw any two chords AB and AC in the given circle.
2. Draw the perpendicular bisectors to AB and AC, they meet at the centre of the circle.
3. Tet O be the centre, join OP.
4. Draw a perpendicular to OP at P and extend it on either sides which forms a tangent to the circle at ‘P’.
Try this
Question 1.
Use Pythagoras theorem and write proof of above theorem “the lengths of tangents drawn from an external point to a circle are equal.” (Page No. 231)
Given: Two tangents PA and PB to a circle with centre O, from an exterior point P.
R.T.P: PA = PB
Proof: In △OAP; ∠OAP = 90°
∴ AP2 = OP2 – OA2
[∵ Square of the hypotenuse is equal to the sum of squares on the other two sides – Pythagoras theorem]
[∵ OA = OB, radii of the same circle]
= BP2 [∵ In AOBP; OB2 + BP2 = OP2
⇒ BP2 – OP2 – OB2]
⇒ AP2 – BP2
⇒ PA – PB Hence proved.
Question 2.
Draw a pair of radii OA and OB such that ∠BOA = 120°. Draw the bisector of ∠BOA and draw lines perpendiculars to OA and OB at A and B. These lines meet on the bisector of ∠BOA at a point which is the external point and the perpendicular lines are the required tangents. Construct and justify. (Page No. 235)
Justification:
OA ⊥ PA
OB ⊥ PB
Also in △OAP, △OBP
OA = OB
∠OAP = ∠OBP
OP – OP
∴ △OAP ≅ △OBP
∴ PA = PB. [Q.E.D.]
Do this
Question 1.
Shankar made the following pictures also with washbasin.
What shapes can they be broken into that we can find area easily? (Page No. 237)
Question 2.
Make some more pictures and think of the shapes they can be divided into different parts. (Page No. 237)
Question 3.
Find the area of sector, whose radius is 7 cm. with the given angles. (Page No. 239)
i) 60° ii) 30° iii) 72° iv) 90° v) 120°
Question 4.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 10 minutes. (Page No. 239)
Angle made by minute hand in 1 m = 360∘60 = 6°
Angle made by minute hand in 10m = 10 × 6 = 60°
The area swept by minute hand is in the shape of a sector with radius r = 14 cm and angle x = 60°
Area swept by the minute hand in 10 minutes = 102.66 cm2.
Try this
Question 1.
How can you find the area of major segment using area of minor segment? (Page No. 239)
Area of the major segment = Area of the circle – Area of the minor segment.
## Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks for Exam Preparations
Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions can be of great help in your Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions exam preparation. The AP Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks study material, used with the English medium textbooks, can help you complete the entire Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Books State Board syllabus with maximum efficiency.
## FAQs Regarding Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions
#### Can we get a Andhra Pradesh State Board Book PDF for all Classes?
Yes you can get Andhra Pradesh Board Text Book PDF for all classes using the links provided in the above article.
## Important Terms
Andhra Pradesh Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions, AP Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks, Andhra Pradesh State Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions, Andhra Pradesh State Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook solutions, AP Board Class 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks Solutions, Andhra Pradesh Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions, AP Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks, Andhra Pradesh State Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions, Andhra Pradesh State Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook solutions, AP Board STD 10th Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbooks Solutions,
Share:
| 2,364 | 9,131 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.859375 | 4 |
CC-MAIN-2024-22
|
latest
|
en
| 0.796974 |
http://study91.co.in/Exam/PostQuestionComment?QuestionId=26417
| 1,582,089,363,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-10/segments/1581875144027.33/warc/CC-MAIN-20200219030731-20200219060731-00282.warc.gz
| 137,919,672 | 16,637 |
India’s No.1 Educational Platform For UPSC,PSC And All Competitive Exam
• 0
• Donate Now
# The length of a rectangle is halved, while its breadth is tripled. What is the percentage change in area?
Option :
Explanation:
Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{original}\phantom{\rule{thinmathspace}{0ex}}\text{length}=x\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\\ & \text{Original}\phantom{\rule{thinmathspace}{0ex}}\text{breadth}=y\\ & \text{Original}\phantom{\rule{thinmathspace}{0ex}}\text{area}=xy\\ & \text{New}\phantom{\rule{thinmathspace}{0ex}}\text{length}=\frac{x}{2}\\ & \text{New}\phantom{\rule{thinmathspace}{0ex}}\text{breadth}=3y\\ & \text{New}\phantom{\rule{thinmathspace}{0ex}}\text{area}=\left(\frac{x}{2}×3y\right)=\frac{3}{2}xy\\ & \therefore \text{Increase}\phantom{\rule{thinmathspace}{0ex}}\mathrm{%}=\left(\frac{1}{2}xy×\frac{1}{xy}×100\right)\\ & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=50\mathrm{%}\end{array}$
| 816 | 2,238 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.71875 | 5 |
CC-MAIN-2020-10
|
latest
|
en
| 0.340037 |
https://www.shaalaa.com/question-bank-solutions/solutions-quadratic-equations-factorization-if-sin-cos-are-roots-equations-ax2-bx-c-0-then-b2_63577
| 1,563,534,394,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-30/segments/1563195526210.32/warc/CC-MAIN-20190719095313-20190719121313-00242.warc.gz
| 814,305,586 | 11,525 |
Share
Books Shortlist
# If Sin α and Cos α Are the Roots of the Equations Ax2 + Bx + C = 0, Then B2 = - CBSE Class 10 - Mathematics
ConceptSolutions of Quadratic Equations by Factorization
#### Question
If sin α and cos α are the roots of the equations ax2 + bx + c = 0, then b2 =
• a2 − 2ac
• a2 + 2ac
• a2 − ac
• a2 + ac
#### Solution
The given quadric equation is ax^2 + bx + c = 0, and sin alpha and cos beta are roots of given equation.
And, a = a,b = b and, c = c
Then, as we know that sum of the roots
sin alpha + cos beta - (-b)/a…. (1)
And the product of the roots
sin alpha .cos beta =c/a…. (2)
Squaring both sides of equation (1) we get
(sin alpha + cos beta)^2 = ((-b)/a)^2
sin^2 alpha + cos^2 beta + 2 sin alpha cos beta = b^2/a^2
Putting the value of sin^2 alpha + cos^2 beta = 1, we get
1 + 2 sin alpha cos beta = b^2/a^2
a^2 (1+2 sin alpha cos beta) = b^2
Putting the value ofsin alpha.cos beta = c/a , we get
a^2 (1 + 2 c/a) = b^2
a^2 ((a+2c)/a) = b^2
a^2 + 2ac =b^2
Therefore, the value of b^2 = a^2 + 2ac.
Is there an error in this question or solution?
#### APPEARS IN
Solution If Sin α and Cos α Are the Roots of the Equations Ax2 + Bx + C = 0, Then B2 = Concept: Solutions of Quadratic Equations by Factorization.
S
| 461 | 1,269 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.1875 | 4 |
CC-MAIN-2019-30
|
longest
|
en
| 0.662564 |
https://www.mvorganizing.org/how-do-you-report-a-mean-and-standard-deviation-2/
| 1,638,452,250,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-49/segments/1637964362219.5/warc/CC-MAIN-20211202114856-20211202144856-00022.warc.gz
| 946,027,069 | 24,668 |
Uncategorized
# How do you report a mean and standard deviation?
## How do you report a mean and standard deviation?
Also, with the exception of some p values, most statistics should be rounded to two decimal places. Mean and Standard Deviation are most clearly presented in parentheses: The sample as a whole was relatively young (M = 19.22, SD = 3.45). The average age of students was 19.22 years (SD = 3.45).
## How do you describe the standard deviation of a report?
A low standard deviation indicates that the data points tend to be very close to the mean; a high standard deviation indicates that the data points are spread out over a large range of values. A useful property of standard deviation is that, unlike variance, it is expressed in the same units as the data.
## Should I use SEM or SD?
SEM quantifies uncertainty in estimate of the mean whereas SD indicates dispersion of the data from mean. As readers are generally interested in knowing the variability within sample, descriptive data should be precisely summarized with SD.
## How do you find standard deviation from a table?
1. The standard deviation formula may look confusing, but it will make sense after we break it down.
2. Step 1: Find the mean.
3. Step 2: For each data point, find the square of its distance to the mean.
4. Step 3: Sum the values from Step 2.
5. Step 4: Divide by the number of data points.
6. Step 5: Take the square root.
## How do you find the standard deviation of a stock?
The calculation steps are as follows:
1. Calculate the average (mean) price for the number of periods or observations.
2. Determine each period’s deviation (close less average price).
3. Square each period’s deviation.
4. Sum the squared deviations.
5. Divide this sum by the number of observations.
## What is a good standard deviation for stocks?
When stocks are following a normal distribution pattern, their individual values will place either one standard deviation below or above the mean at least 68% of the time. A stock’s value will fall within two standard deviations, above or below, at least 95% of the time.
## What does standard deviation mean for stocks?
Standard deviation is the statistical measure of market volatility, measuring how widely prices are dispersed from the average price. If prices trade in a narrow trading range, the standard deviation will return a low value that indicates low volatility.
## How is standard deviation used in forecasting?
Method 2 – Standard Deviation
1. Find the mean of the data set.
2. Find the distance from each data point to the mean, and square the result.
3. Find the sum of those values.
4. Divide the sum by the number of data points.
5. Take the square root of that answer.
## How do you trade with standard deviation?
The standard deviation calculation is based on a couple of steps:
1. Find the average closing price (mean) for the periods under consideration (the default setting is 20 periods)
2. Find the deviation for each period (closing price minus average price)
3. Find the square for each deviation.
## What does higher standard deviation mean?
A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
## What is the standard deviation of a portfolio?
Portfolio Standard Deviation is the standard deviation of the rate of return on an investment portfolio and is used to measure the inherent volatility of an investment. It measures the investment’s risk and helps in analyzing the stability of returns of a portfolio.
## Can you have a standard deviation of 0?
A standard deviation can range from 0 to infinity. A standard deviation of 0 means that a list of numbers are all equal -they don’t lie apart to any extent at all.
## Why is the mean 0 and the standard deviation 1?
The mean of 0 and standard deviation of 1 usually applies to the standard normal distribution, often called the bell curve. The most likely value is the mean and it falls off as you get farther away. The simple answer for z-scores is that they are your scores scaled as if your mean were 0 and standard deviation were 1.
## What must be true of a data set if its standard deviation is 0?
When the standard deviation is zero, there is no spread; that is, the all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean.
## What does it mean if standard deviation is less than 1?
Popular Answers (1) This means that distributions with a coefficient of variation higher than 1 are considered to be high variance whereas those with a CV lower than 1 are considered to be low-variance. Remember, standard deviations aren’t “good” or “bad”. They are indicators of how spread out your data is.
## How do you tell if a standard deviation is high or low?
Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out. A standard deviation close to zero indicates that data points are close to the mean, whereas a high or low standard deviation indicates data points are respectively above or below the mean.
## What is the relationship between mean and standard deviation?
Standard deviation is basically used for the variability of data and frequently use to know the volatility of the stock. A mean is basically the average of a set of two or more numbers. Mean is basically the simple average of data. Standard deviation is used to measure the volatility of a stock.
## What if standard deviation is higher than 1?
The answer is yes. (1) Both the population or sample MEAN can be negative or non-negative while the SD must be a non-negative real number. A smaller standard deviation indicates that more of the data is clustered about the mean while A larger one indicates the data are more spread out.
## Is the standard deviation always positive?
The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero.
## When mean and standard deviation are equal?
One situation in which the mean is equal to the standard deviation is with the exponential distribution whose probability density is f(x)={1θe−x/θif x>0,0if x<0. for all positive numbers x and y.
## What is a good standard error?
Thus 68% of all sample means will be within one standard error of the population mean (and 95% within two standard errors). The smaller the standard error, the less the spread and the more likely it is that any sample mean is close to the population mean. A small standard error is thus a Good Thing.
## How do you interpret standard error?
The standard error tells you how accurate the mean of any given sample from that population is likely to be compared to the true population mean. When the standard error increases, i.e. the means are more spread out, it becomes more likely that any given mean is an inaccurate representation of the true population mean.
## How do I interpret standard error?
The Standard Error (“Std Err” or “SE”), is an indication of the reliability of the mean. A small SE is an indication that the sample mean is a more accurate reflection of the actual population mean. A larger sample size will normally result in a smaller SE (while SD is not directly affected by sample size).
## What does a standard error of 2 mean?
The standard deviation tells us how much variation we can expect in a population. We know from the empirical rule that 95% of values will fall within 2 standard deviations of the mean. 95% would fall within 2 standard errors and about 99.7% of the sample means will be within 3 standard errors of the population mean.
## What is considered a small standard error?
The Standard Error (“Std Err” or “SE”), is an indication of the reliability of the mean. A small SE is an indication that the sample mean is a more accurate reflection of the actual population mean. If the mean value for a rating attribute was 3.2 for one sample, it might be 3.4 for a second sample of the same size.
## How do you interpret standard error in regression?
The standard error of the regression provides the absolute measure of the typical distance that the data points fall from the regression line. S is in the units of the dependent variable. R-squared provides the relative measure of the percentage of the dependent variable variance that the model explains.
## What is the difference between sampling error and standard error?
Generally, sampling error is the difference in size between a sample estimate and the population parameter. The standard error of the mean (SEM), sometimes shortened to standard error (SE), provided a measure of the accuracy of the sample mean as an estimate of the population parameter (c is true).
Category: Uncategorized
Begin typing your search term above and press enter to search. Press ESC to cancel.
| 1,888 | 9,080 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.5625 | 5 |
CC-MAIN-2021-49
|
latest
|
en
| 0.888565 |
https://newpathworksheets.com/math/kindergarten/comparing-and-ordering-2121/massachusetts-standards
| 1,571,318,655,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-43/segments/1570986675316.51/warc/CC-MAIN-20191017122657-20191017150157-00010.warc.gz
| 606,262,600 | 7,953 |
◂Math Worksheets and Study Guides Kindergarten. Comparing, and ordering
The resources above correspond to the standards listed below:
Massachusetts Standards
MA.CC.K.CC. Counting and Cardinality
Compare numbers.
K.CC.6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
Count to tell the number of objects.
K.CC.4. Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.4.c. Understand that each successive number name refers to a quantity that is one larger.
Know number names and the count sequence.
K.CC.1. Count to 100 by ones and by tens.
K.CC.2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
MA.CC.K.OA. Operations and Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
K.OA.2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
K.OA.5. Fluently add and subtract within 5.
| 313 | 1,369 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.875 | 4 |
CC-MAIN-2019-43
|
latest
|
en
| 0.903298 |
https://infinitylearn.com/surge/question/mathematics/how-many-three-quarters-parts-of--can-pancakes-be-made/
| 1,716,215,250,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-22/segments/1715971058291.13/warc/CC-MAIN-20240520142329-20240520172329-00415.warc.gz
| 265,807,913 | 191,370 |
How many three-quarters parts of can pancakes be made?
# How many three-quarters parts of can pancakes be made?
1. A
18
2. B
36
3. C
32
4. D
16
Fill Out the Form for Expert Academic Guidance!l
+91
Live ClassesBooksTest SeriesSelf Learning
Verify OTP Code (required)
### Solution:
Assume that the number of three-quarters pancakes that can be made from pancakes is . Now, calculate the total amount of three-quarters of pancakes by multiplying and .
Since three-quarters of pancakes have been made from full pancakes, the total amount of three-quarter pancakes must be equal to pancakes. Now, make equal to and calculate the value of .
They are given pancakes and we have to figure out the number of the three-quarters part that can be made from pancakes.
The total number of pancakes =
First of all, assume that the number of three-quarters pancakes that can be made from 24 pancakes is Every pancake is three-fourth of its original size and the total number of three-fourth pancakes is . Also, three-quarters of pancakes have been made from full pancakes. So, the total amount of three-quarter pancakes must be equal pancakes
The total amount of pancakes required to make three-quarters of pancakes =
Now, from equation (1), equation (3), and equation (4), we get
In equation (1), we have assumed that is the number of three-quarters of pancakes made from pancakes.
Therefore, the number of three-quarters of pancakes that can be made from pancakes is
## Related content
Matrices and Determinants_mathematics Critical Points Solved Examples Type of relations_mathematics
+91
Live ClassesBooksTest SeriesSelf Learning
Verify OTP Code (required)
| 368 | 1,669 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.703125 | 4 |
CC-MAIN-2024-22
|
latest
|
en
| 0.933711 |
http://gmatclub.com/forum/x-y-and-z-are-three-different-prime-numbers-the-product-106043.html?fl=similar
| 1,484,988,770,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-04/segments/1484560281001.53/warc/CC-MAIN-20170116095121-00207-ip-10-171-10-70.ec2.internal.warc.gz
| 121,374,050 | 60,651 |
X, Y, and Z are three different Prime numbers, the product : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 21 Jan 2017, 00:52
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# X, Y, and Z are three different Prime numbers, the product
Author Message
TAGS:
### Hide Tags
Manager
Joined: 13 Jul 2010
Posts: 169
Followers: 1
Kudos [?]: 73 [2] , given: 7
X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
09 Dec 2010, 21:03
2
KUDOS
5
This post was
BOOKMARKED
00:00
Difficulty:
25% (medium)
Question Stats:
67% (01:47) correct 33% (00:38) wrong based on 236 sessions
### HideShow timer Statistics
X, Y, and Z are three different Prime numbers, the product XYZ is divisible by how many different positive numbers?
A. 4
B. 6
C. 8
D. 9
E. 12
[Reveal] Spoiler: OA
Manager
Joined: 30 Aug 2010
Posts: 91
Location: Bangalore, India
Followers: 5
Kudos [?]: 160 [2] , given: 27
### Show Tags
09 Dec 2010, 22:08
2
KUDOS
1
This post was
BOOKMARKED
gettinit wrote:
X, Y, and Z are three different Prime numbers, the product XYZ is divisible by how many different positive numbers?
4
6
8
9
12
PRIME # is the # that has only 2 factors: One is 1 and another is the # itself.
infer that
FOR any given # 1 and the # iteself are the definite factors.
Knowing above conepts:
product of X,Y, and Z = XYZ
divisible by 1, xyz, x, y, z, xy,yz,xz ==> total 8
if question has 5 constants a,b,c,d,e, we do not have to count in the above way
Basically we are selecting, from the product, one constant, set of two constants, set of 3 constants ....and so on set of all the # of constants.
so if 5 varibales are given, total # ways to select is 5C1+5C2+5C3+5C4+5C5 = 5+10+10+5+1 = 31
And answer will be 31+1 =32 (as "1" is a factor for every #)
Regards,
Murali.
Kudos?
Math Expert
Joined: 02 Sep 2009
Posts: 36583
Followers: 7087
Kudos [?]: 93298 [3] , given: 10555
### Show Tags
09 Dec 2010, 23:46
3
KUDOS
Expert's post
6
This post was
BOOKMARKED
gettinit wrote:
X, Y, and Z are three different Prime numbers, the product XYZ is divisible by how many different positive numbers?
4
6
8
9
12
MUST KNOW FOR GMAT:
Finding the Number of Factors of an Integer
First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.
The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.
Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$
Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.
BACK TO THE ORIGINAL QUESTION:
$$n=xyz$$ (n=x^1y^1z^1) where $$x$$, $$y$$, and $$z$$ are different prime factors will have $$(1+1)(1+1)(1+1)=8$$ different positive factors including 1 and xyz itself.
For more on number properties check: math-number-theory-88376.html
Hope it helps.
_________________
Senior Manager
Status: Bring the Rain
Joined: 17 Aug 2010
Posts: 406
Location: United States (MD)
Concentration: Strategy, Marketing
Schools: Michigan (Ross) - Class of 2014
GMAT 1: 730 Q49 V39
GPA: 3.13
WE: Corporate Finance (Aerospace and Defense)
Followers: 7
Kudos [?]: 45 [0], given: 46
### Show Tags
10 Dec 2010, 06:52
Thanks for the equation
_________________
Manager
Joined: 13 Jul 2010
Posts: 169
Followers: 1
Kudos [?]: 73 [0], given: 7
### Show Tags
10 Dec 2010, 11:55
Thanks Bunuel very helpful, I thought I could do a quick calculation here to find out the number of factors but fell for the trap I guess:
2*3*5=30 so 30 is divisible by 1,30,15,6,5,3,2,10 but in my haste forgot to use 10,and 1 in here. One should just stick to the formula for a sure shot.
Manager
Joined: 30 Aug 2010
Posts: 91
Location: Bangalore, India
Followers: 5
Kudos [?]: 160 [0], given: 27
### Show Tags
10 Dec 2010, 23:14
Friends,
The approach specified by Bunuel is an efficient, a quick and a standard one. Use the same. Ignore the one specified by me as it is a bit time consuming when compared to that given by Bunuel.
Regards,
Murali.
Manager
Joined: 25 May 2011
Posts: 156
Followers: 2
Kudos [?]: 60 [1] , given: 71
### Show Tags
20 Oct 2011, 06:16
1
KUDOS
1- x- y- z- xy- xz- yz- xyz
Intern
Joined: 19 Oct 2011
Posts: 2
Location: United States
Concentration: Finance, General Management
GMAT Date: 01-06-2012
GPA: 2.6
Followers: 0
Kudos [?]: 6 [1] , given: 8
### Show Tags
20 Oct 2011, 10:07
1
KUDOS
Ans 8
1
x
y
z
xy
yz
zx
xyz
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13471
Followers: 575
Kudos [?]: 163 [0], given: 0
Re: X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
21 Jan 2014, 22:03
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Senior Manager
Affiliations: CrackVerbal
Joined: 03 Oct 2013
Posts: 356
Location: India
GMAT 1: 780 Q51 V46
Followers: 69
Kudos [?]: 309 [0], given: 5
Re: X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
21 Jan 2014, 22:33
gettinit wrote:
X, Y, and Z are three different Prime numbers, the product XYZ is divisible by how many different positive numbers?
A. 4
B. 6
C. 8
D. 9
E. 12
Let us say X = 2, Y = 3 and Z = 5. Then XYZ = 30 - it is divisible by
1, 2, 3, 5, 6, 10, 15, 30 - 8 different numbers
_________________
For 5 common traps on CR check here :
http://gmatonline.crackverbal.com/p/gmat-critical-reasoning
For sentence correction videos check here -
http://gmatonline.crackverbal.com/p/gmat-sentence-correction
Learn all PS and DS strategies here-
http://gmatonline.crackverbal.com/p/mastering-quant-on-gmat
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13471
Followers: 575
Kudos [?]: 163 [0], given: 0
Re: X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
24 Apr 2015, 03:50
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13471
Followers: 575
Kudos [?]: 163 [0], given: 0
Re: X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
15 May 2016, 08:45
Hello from the GMAT Club BumpBot!
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 1901
Followers: 49
Kudos [?]: 367 [0], given: 454
Re: X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
15 Jan 2017, 23:03
Since X,Y,Z are different primes=>
Number of factors of X*Y*Z => 2*2*2=> 8
Hence C.
_________________
Mock Test -1 (Integer Properties Basic Quiz) ---> http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1676182
Mock Test -2 (Integer Properties Advanced Quiz) --->http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1765951
Mock Test -2 (Evensand Odds Basic Quiz) --->http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1768023
Give me a hell yeah ...!!!!!
Intern
Joined: 06 Oct 2016
Posts: 1
Followers: 0
Kudos [?]: 0 [0], given: 3
X, Y, and Z are three different Prime numbers, the product [#permalink]
### Show Tags
20 Jan 2017, 00:51
X, Y, and Z are three different Prime numbers, the product XYZ is divisible by how many different positive numbers?
A. 4
B. 6
C. 8
D. 9
E. 12
Suppose
X = 2,
Y = 3 and
Z = 5.
Then XYZ = 30 - it is divisible by
1, 2, 3, 5, 6, 10, 15, 30 -
so ther are 8 different numbers
X, Y, and Z are three different Prime numbers, the product [#permalink] 20 Jan 2017, 00:51
Similar topics Replies Last post
Similar
Topics:
2 If x and y are different prime numbers, each greater than 2, which of 2 17 Feb 2016, 03:38
8 If three positive real numbers x, y, z satisfy y – x = z – y and x y z 6 07 Nov 2015, 23:42
20 If x, y, and z are 3 different prime numbers, which of the 22 25 May 2013, 02:45
If x and y are two different prime numbers, which of the 6 08 Jun 2012, 08:14
23 For a three-digit number xyz, where x, y, and z are the 10 29 Jan 2008, 12:44
Display posts from previous: Sort by
| 3,114 | 9,751 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4 | 4 |
CC-MAIN-2017-04
|
latest
|
en
| 0.894146 |
http://www.gradesaver.com/textbooks/math/calculus/calculus-10th-edition/chapter-2-differentiation-2-5-exercises-page-145/31
| 1,481,028,101,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-50/segments/1480698541905.26/warc/CC-MAIN-20161202170901-00259-ip-10-31-129-80.ec2.internal.warc.gz
| 486,804,303 | 38,737 |
## Calculus 10th Edition
$\dfrac{dy}{dx}=0.$
$\dfrac{d}{dx}((x^2+y^2)^2)=\dfrac{d}{dx}(4x^2y)\rightarrow$ Using the Chain Rule with $u=x^2+y^2\rightarrow\dfrac{du}{dx}=2x+\dfrac{dy}{dx}(2y)\rightarrow$ $\dfrac{d}{dx}((x^2+y^2)^2)=2(x^2+y^2)(2x+\dfrac{dy}{dx}(2y)).$ $2(x^2+y^2)(2x+\dfrac{dy}{dx}(2y))=\dfrac{dy}{dx}(4x^2)+8xy\rightarrow$ $\dfrac{dy}{dx}(x^2-x^2y-y^3)=xy^2-2xy+x^3\rightarrow$ $\dfrac{dy}{dx}=\dfrac{xy^2-2xy+x^3}{x^2-x^2y-y^3}.$ At $(1, 1)\rightarrow\dfrac{dy}{dx}=\dfrac{(1)(1^2)-2(1)(1)+(1)^3}{1^2-(1^2)(1)-1^3}=0.$
| 287 | 535 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.21875 | 4 |
CC-MAIN-2016-50
|
longest
|
en
| 0.274304 |
http://avitzur.hax.com/2007/01/the_library_of_babel_function.html
| 1,708,869,430,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-10/segments/1707947474617.27/warc/CC-MAIN-20240225135334-20240225165334-00475.warc.gz
| 4,034,240 | 4,527 |
### The Library of Babel function
MathWorld calls this Tupper's Self-Referential Formula because a graph of that inequality in the domain 0<x<105, N<y<N+16 for a particular 541-digit number N shows an image of the function itself.
Closer investigation reveals that MathWorld does not even begin to do justice to the function, for higher up on the graph of this function one can find the complete works of Shakespeare, the contents of the lost library of Alexandria, and perhaps even the solution to unifying quantum mechanic with general relativity! At y values nearer the origin, you can find tomorrow's winning lottery number! If that sounds unbelievable, read on....
Consider rearranging the function slightly, as above, with k=17. Then note that in the domain used, N<y<N+k-1 the numerator, floor(y/k) is a constant. Let's call that constant B. As a computer graphics programmer, the structure of the denominator of that fraction jumps out to me eye. The image appears in a rectangle 105 pixels wide by 17 pixels tall. Let's call the integer parts of x and y, ix=floor(x) and iy=floor(y). The term mod(iy,k) counts from 0 to 16. The exponent k*ix + mod(iy,k) just counts from 0 to 105*17-1, sequentially, moving first up each column, and starting over at the bottom of the next column. So let's label index=k*ix + mod(iy,k) as an index pointing to the pixel at (ix, iy). As an old C programmer, I see division by 2^index and recognize that as a bit-shift, moving the bits in B to the right by index. Taking mod 2 of an integer just asks if it is even or odd, or thinking of it another way, asks the value of the least significant bit in the binary representation of that number. Performing a bit-shift to the right, and then taking that value mod 2, is a way of asking
What is the value of the bit at position index in the binary representation of B?
The graph of the formula contains every possible black and white bitmap of height k, in order, as you move up along the y-axis from the origin!
The particular 541 digit number chosen is, essentially, the integer representation of the bitmap. To see this directly, you can graph this function using the Unix utility bc by issuing the following commands in Terminal on Mac OS X, for example:
setenv BC_LINE_LENGTH 19
This tells bc to format its output with 17 characters to the line (not counting the \ and return character at the end of each line.)
bc
obase=2
That tells bc to display output results in binary, base 2.
```
print"\n",0,0,(960939379918958884971672962127852754715004339660129306651505519271702802395266424689642842174350718121267153782770623355993237280874144307891325963941337723487857735749823926629715517173716995165232890538221612403238855866184013235585136048828693337902491454229288667081096184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610003652351370343874461848378737238198224849863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143786841806593422227898388722980000748404719/17)
```
To get the 105 x 17 pixel rectangle to line up, the line above tells bc to print two leading zeros. If you then turn your head sideways and squint, or copy the output to a text editor and replace 0 with space and 1 with * to squint slightly less, you can see the bitmap image of the function.
This Graphing Calculator document graphs the case for k=1 enumerating all 1-pixel high bitmaps as you travel up the y-axis, which is just like counting in binary.
This Graphing Calculator document constructs the magic number B to find the first image of a "+" sign along the y-axis for the case k=3.
```Number the pixels in a 3x3 bitmap like this:
2 5 8
1 4 7
0 3 6
.*.
***
.*.
```
Then look at 558<y<561 since 3*(2+8+16+32+128)=558.
| 978 | 3,787 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2024-10
|
latest
|
en
| 0.886154 |
http://www.jiskha.com/display.cgi?id=1394507534
| 1,496,018,693,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-22/segments/1495463612003.36/warc/CC-MAIN-20170528235437-20170529015437-00015.warc.gz
| 684,571,432 | 3,889 |
# Algebra II
posted by on .
Simplify.
√(-24 - 10i)
• Algebra II - ,
oh, and also, if it is not too much too ask, please include steps and explanations. ty
• Algebra II - ,
√(-24 - 10i)
If you want to find a perfect square which equals -24-10i, it will have to include -5i, so
(a-5i)^2 = -24-10i
a^2 - 10ai - 25 = -24-10i
Hmmm. Looks like a=1
(1-5i)^2 = 1 - 10i - 25 = -24-10i
So,
√(-24 - 10i) = 1-5i or -1+5i
Or, if you use polar form, you can see that
r=26
tanθ = 5/12
the square root has
r = √26
tanθ = (1-(-12/13))/(-5/13) = -5
so, we get the same result.
| 236 | 568 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.96875 | 4 |
CC-MAIN-2017-22
|
latest
|
en
| 0.829631 |
https://socratic.org/questions/how-do-you-solve-7-1-3-8-n
| 1,597,014,288,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-34/segments/1596439738595.30/warc/CC-MAIN-20200809222112-20200810012112-00327.warc.gz
| 482,216,808 | 6,140 |
# How do you solve 7.1= 3.8+ n?
Mar 18, 2018
$n = 3.3$
#### Explanation:
Given: $7.1 = 3.8 + n$
Don't like decimals so lets get rid of them. Multiply everything on both sides by 10
$71 = 38 + 10 n$
As per convention put the unknown on the left. So turn everything round the other way
$10 n + 38 = 71 \leftarrow \text{ Means exactly the same thing}$
Subtract $\textcolor{red}{38}$ from both sides
$\textcolor{g r e e n}{10 n + 38 = 71 \textcolor{w h i t e}{\text{ddd")->color(white)("ddd")10n+38color(red)(-38)color(white)("d")=color(white)("d}} 71 \textcolor{red}{- 38}}$
$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddd.d")->color(white)("ddd")10n + color(white)("dd")0color(white)("ddd.")=color(white)("ddd}} 33}$
Divide both sides by $\textcolor{red}{10}$
$\textcolor{g r e e n}{10 n = 33 \textcolor{w h i t e}{\text{ddddddd")->color(white)("ddd")10/color(red)(10)ncolor(white)("d")=color(white)("d}} \frac{33}{\textcolor{red}{10}}}$
But $\frac{10}{10} = 1 \mathmr{and} 1 \times n = n$ giving:
$\textcolor{g r e e n}{n = 3.3}$
| 401 | 1,060 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.59375 | 5 |
CC-MAIN-2020-34
|
latest
|
en
| 0.629544 |
https://www.slideserve.com/tacey/related-rates-problems
| 1,643,178,423,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-05/segments/1642320304915.53/warc/CC-MAIN-20220126041016-20220126071016-00588.warc.gz
| 1,035,825,181 | 18,895 |
Related Rates Problems
# Related Rates Problems
## Related Rates Problems
- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
##### Presentation Transcript
1. Related Rates Problems Do Now: How fast is the Circumference of a circle changing compared to the rate of change of its radius?
2. Problem #1 • How fast is the area of a square increasing at the instant when the sides are 4 feet long if the sides increase at a rate of 1.5 ft/min?
3. Solution #1 • (units?)
4. Problem #2 • Suppose . At the instant when x=1 and y=3, x is decreasing at 2 units/sec and y is increasing at 2 units/sec. How fast is z changing at this instant? Increasing or decreasing?
5. Solution #2 • units? • Units/sec
6. Problem #3 • Let be the acute angle of a right triangle with legs of length x and y. At a certain instant, x=4 and is increasing at 2 units/second, while y=3 and is decreasing at ½ units/sec. How fast is changing at that instant? Is it increasing or decreasing?
7. Solution #3
8. Problem #4 • Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 60 ft?
9. Solution #4
10. Problem #5 • Suppose that a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How fast is the radius of the balloon increasing when the radius is 5 centimeters?
11. Solution #5
12. Problem #6 • One end of a 13-foot ladder is on the floor and the other rests on a vertical wall. If the bottom end is drawn away from the wall at 3 feet per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?
13. Solution #6 • (Where did I get 12?)
14. Problem #7 • One end of a 13-foot ladder is on the floor and the other rests on a vertical wall. If the bottom end is drawn away from the wall at 3 feet per second, how fast is the angle of elevation of the ladder changing when the bottom of the ladder is 5 feet from the wall?
15. Solution #7
16. Problem #8 • A baseball diamond is a square whose sides are 90 ft long. Suppose that a player running from second base to third base has a speed of 30 ft/sec at the instant when he is 20 feet from third base. At what rate is the player’s distance from home plate changing at that instant?
17. Solution #8 • At this instant
18. Problem #9 • A rocket is rising vertically at 880 ft/sec. At the instant when the rocket is 4000 feet high, how fast must a camera’s elevation angle change to keep the rocket in sight if the camera is 3000 ft away from the launching pad?
19. Solution #9 • . • At this moment, the hypotenuse is .
| 714 | 2,758 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.96875 | 4 |
CC-MAIN-2022-05
|
latest
|
en
| 0.87298 |
http://www.chegg.com/homework-help/applied-differential-equations-3rd-edition-chapter-7.2.2-solutions-9780130400970
| 1,472,384,457,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-36/segments/1471982937576.65/warc/CC-MAIN-20160823200857-00197-ip-10-153-172-175.ec2.internal.warc.gz
| 358,739,810 | 18,143 |
View more editions
# Applied Differential Equations (3rd Edition)Solutions for Chapter 7.2.2
• 1531 step-by-step solutions
• Solved by publishers, professors & experts
• iOS, Android, & web
Looking for the textbook?
Over 90% of students who use Chegg Study report better grades.
May 2015 Survey of Chegg Study Users
Chapter: Problem:
SAMPLE SOLUTION
Chapter: Problem:
• Step 1 of 61
Consider the differential equation
To find the Frobenius-solution of the differential equation, determine the general solution and discuss the convergence.
• Step 2 of 61
a) The differential equation is
…… (1)
• Step 3 of 61
Let,
be the Frobenius-solution of (1)
By differentiation
And
Substitute these values in (1), to get
• Step 4 of 61
Thus,
Now replace by in the first and second summation and by in the third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 5 of 61
Case 1:
That is,
Put to get
and so on
• Step 6 of 61
Therefore, the general solution is
And the series converges for all
Hence, the general solution is,
b) The differential equation is
…… (2)
• Step 7 of 61
Let,
be the Frobenius-solution of (2)
By differentiation
And
Substitute these values in (2), to get
Thus,
• Step 8 of 61
Now replace by in the first and second summation and by in the third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 9 of 61
Case 1:
That is,
Put to get
and so on
• Step 10 of 61
Case 2:
That is,
Put to get
and so on
• Step 11 of 61
Therefore, the general solution is,
all x
c) The differential equation is
…… (3)
• Step 12 of 61
Let,
be the Frobenius-solution of (3)
By differentiation
And
Substitute these values in (3), to get
• Step 13 of 61
Now replace by in the first and third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 14 of 61
Case 1:
That is,
Put to get
• Step 15 of 61
Case 2:
That is,
Put to get
• Step 16 of 61
Therefore, the general solution is,
all
• Step 17 of 61
d) The differential equation is
…… (4)
• Step 18 of 61
Let,
be the Frobenius-solution of (4)
By differentiation
And
Substitute these values in (4), to get
• Step 19 of 61
Now replace by in the first summation, such that all summation involves
Therefore,
Putting,
So,
• Step 20 of 61
Case 1:
That is,
Put to get
And so on
• Step 21 of 61
Case 2:
That is,
Put to get
• Step 22 of 61
Therefore, the general solution is,
and series converges for all
• Step 23 of 61
e) The differential equation is
…… (5)
• Step 24 of 61
Let,
be the Frobenius-solution of (5)
By differentiation
And
Substitute these values in (5), to get
Thus,
Now replace by in the first summation, such that all summation involves
Therefore,
• Step 25 of 61
Putting,
So,
Case 1:
That is,
Put to get
and so on
Then
• Step 26 of 61
Similarly for
• Step 27 of 61
Therefore, the general solution is,
,
where
And
,
also series converges for all x
• Step 28 of 61
f) The differential equation is
…… (6)
• Step 29 of 61
Let,
be the Frobenius-solution of (6)
By differentiation
And
Substitute these values in (6), to get
• Step 30 of 61
Thus,
Now replace by in the third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 31 of 61
Case 1:
That is,
Put to get
and so on
• Step 32 of 61
Then
Similarly for
That is,
Put to get
and so on
Then
• Step 33 of 61
Therefore, the general solution is,
,
and series converges for all
• Step 34 of 61
g) The differential equation is
…… (7)
• Step 35 of 61
Let,
be the Frobenius-solution of (7)
By differentiation
And
Substitute these values in (7), to get
• Step 36 of 61
Thus,
Now replace by in the first and second summation, such that all summation involves
Therefore,
Putting,
So,
Thus
And
• Step 37 of 61
Therefore, the general solution is,
,
and series converges for all
• Step 38 of 61
h) The differential equation is
…… (8)
Let,
be the Frobenius-solution of (8)
• Step 39 of 61
By differentiation
And
Substitute these values in (8), to get
Thus,
• Step 40 of 61
Now replace by in the first and second summation and by in the third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 41 of 61
Case 1:
That is,
Put to get
and so on
• Step 42 of 61
Case 2:
That is,
Put to get
and so on
• Step 43 of 61
Therefore, the general solution is
And the series converges for all
Hence, the general solution is, , and series converges for all
• Step 44 of 61
i) The differential equation is
…… (9)
Let,
be the Frobenius-solution of (9)
By differentiation
And
Substitute these values in (9), to get
Thus,
• Step 45 of 61
Now replace by in the fourth summation, such that all summation involves
Therefore,
Putting,
So,
• Step 46 of 61
Case 1:
That is,
Put to get
and so on
Then
• Step 47 of 61
Similarly for
That is,
Put to get
and so on
Then
• Step 48 of 61
Therefore, the general solution is,
,
and converges for all
j) The differential equation is
…… (10)
• Step 49 of 61
Let,
be the Frobenius-solution of (10)
By differentiation
And
Substitute these values in (10), to get
• Step 50 of 61
Thus,
Now replace by in the first and second summation, such that all summation involves
Therefore,
Putting,
So,
• Step 51 of 61
Case 1:
That is,
Put to get
and so on
So,
And
• Step 52 of 61
Therefore, the general solution is,
,
where
And
,
and series converges for all
• Step 53 of 61
k) The differential equation is
…… (11)
• Step 54 of 61
Let,
be the Frobenius-solution of (11)
By differentiation
And
Substitute these values in (11), to get
• Step 55 of 61
Thus,
Now replace by in the first and fourth summation and replace by in the second and third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 56 of 61
Case 1:
That is,
Put to get
and so on
So,
And
• Step 57 of 61
Therefore, the general solution is,
,
where
And
,
and series converges for all
• Step 58 of 61
l) The differential equation is
…… (12)
• Step 59 of 61
Let,
be the Frobenius-solution of (12)
By differentiation
And
Substitute these values in (12), to get
Thus,
Now replace by in the third summation, such that all summation involves
Therefore,
Putting,
So,
• Step 60 of 61
Case 1:
That is,
Put to get
Then
And
• Step 61 of 61
Therefore, the general solution is,
,
where
And
,
and series converges for all
Corresponding Textbook
Applied Differential Equations | 3rd Edition
9780130400970ISBN-13: 0130400971ISBN: Murray R SpiegelAuthors:
| 1,935 | 6,639 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.25 | 4 |
CC-MAIN-2016-36
|
latest
|
en
| 0.839697 |
http://openstudy.com/updates/4fb523e5e4b05565342ae74a
| 1,448,491,020,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2015-48/segments/1448398446218.95/warc/CC-MAIN-20151124205406-00266-ip-10-71-132-137.ec2.internal.warc.gz
| 172,551,286 | 10,187 |
## momotay13 3 years ago What are the slope and the y-intercept of y = –2x + 5?
1. shivam_bhalla
y = mx+ c m-->slope c--->y-intercept
2. jazy
y = mx + b m=slope b=y-intercept Figure it out
3. chocolategirl
-2x is the slope and 5 is the y intercept
4. shivam_bhalla
@chocolategirl , your slope is wrong.
5. jazy
drop the x since m is what equals the slope. So that -2, instead of -2x, would be the slope
6. shivam_bhalla
@jazy is right
| 155 | 446 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.5625 | 4 |
CC-MAIN-2015-48
|
longest
|
en
| 0.82994 |
https://assignmentgrade.com/question/690584/
| 1,600,413,946,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-40/segments/1600400187354.1/warc/CC-MAIN-20200918061627-20200918091627-00380.warc.gz
| 381,331,173 | 7,213 |
Find the product: 8/19 × 12/20 = ? - AssignmentGrade.com
# Find the product: 8/19 × 12/20 = ?
QUESTION POSTED AT 01/06/2020 - 04:05 PM
reduce 12/20 to 3/5
multiply both top numbers and then both bottom numbers
8 x 3 = 24
19 x 5 = 95
## Related questions
### Antoine and Tess have a disagreement over how to compute a 15% gratuity on \$46.00. Tess says, “It is easy to find 10% of 46 by moving the decimal point one place to the left to get \$4.60. Do that twice. Then add the two amounts to get \$4.60 + \$4.60 = \$9.20 for the 15% gratuity.” How should Antoine respond to Tess’s method?
QUESTION POSTED AT 02/06/2020 - 01:43 AM
### Find the six arithmetic means between 1 and 29.
QUESTION POSTED AT 02/06/2020 - 01:36 AM
### Find an irrational number that is between 7.7 and 7.9. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth. (3 points)
QUESTION POSTED AT 02/06/2020 - 01:29 AM
### Find the differential dy of y=1/x+1 Evaluate dy for x=1 , dx=−.01
QUESTION POSTED AT 02/06/2020 - 01:25 AM
### Let y=4x^2 find the differential dy when x=5 and dx=0.4
QUESTION POSTED AT 02/06/2020 - 01:22 AM
### A Pentagon has three angles that are congruent and two other angles that are supplementary to each other. Find the measure of each of the three congruent angles in the Pentagon
QUESTION POSTED AT 02/06/2020 - 01:21 AM
### M is the midpoint of JK. the coordinates of J are (6,3) and the coordinates of M are negative (3,4) , find the coordinates of K.
QUESTION POSTED AT 02/06/2020 - 01:21 AM
### In a triangle, the measure of the first angle is three times the measure of the second angle. The measure of the third angle is 55 more than the measure of the second angle. Use the fact that the sum of the measures of the three angles is 180 to find the measure of each angle to find the measure of each angle.
QUESTION POSTED AT 02/06/2020 - 01:21 AM
### Find the derivative of the following functions f(x) = x^7 - 2x^3 - 9x
QUESTION POSTED AT 02/06/2020 - 01:20 AM
### _____ 20. Brandon needs \$480 to buy a TV and stereo system for his room. He received \$60 in cash for birthday presents. He plans to save \$30 per week from his part-time job. To find how many weeks w it will take to have \$480, solve 60 + 30w = 480. Type your answer in the blank to the left. A. 16 weeks B. 13 weeks C. 15 weeks D. 14 weeks
QUESTION POSTED AT 01/06/2020 - 04:55 PM
### Three-sevenths of a number is 21. Find the number
QUESTION POSTED AT 01/06/2020 - 04:54 PM
### 14. Tell whether the sequence 1 3 , 0, 1, −2 … is arithmetic, geometric, or neither. Find the next three terms of the sequence. Type your answer in the blank to the left. A. neither; 7, -20, 61 B. geometric;7, -20, 61 C. arithmetic; − 1 3 , 1 1 3 , 3 D. geometric;−3 1 3 , −5 5 9 , −9 7 27
QUESTION POSTED AT 01/06/2020 - 04:49 PM
### Points B, D, and F are midpoints of the sides of ace EC = 44 and BF = 2x - 6. Find the value of x.
QUESTION POSTED AT 01/06/2020 - 04:49 PM
### Find the circumference of a circle that has a diameter of 2 ft. use 3.14 for pi
QUESTION POSTED AT 01/06/2020 - 04:47 PM
### In a right triangle if one acute angle has a measure of 10 x + 3 and the other has a measure of 3x + 9 find the larger of these two angles
QUESTION POSTED AT 01/06/2020 - 04:47 PM
QUESTION POSTED AT 01/06/2020 - 04:46 PM
### Find the greatest common factor. 3x4y2 + 12x2y3 + 6x3y4
QUESTION POSTED AT 01/06/2020 - 04:45 PM
### Geometry help ASAP 15 points Need help to find A and B
QUESTION POSTED AT 01/06/2020 - 04:45 PM
### Find the greatest common factor. 2x3y + x2y2 + 4xy3
QUESTION POSTED AT 01/06/2020 - 04:44 PM
### Use the Pythagorean theorem to find the measure of the hypotenuse of right triangle of the measures of the legs are 30 meters and 72 meters
QUESTION POSTED AT 01/06/2020 - 04:39 PM
### Find the value of x and y. (1 point) I need help ASAP
QUESTION POSTED AT 01/06/2020 - 04:37 PM
### Jim works at a furniture shop. He can assemble 88 chairs in 22 hours. If he can assemble c chairs in 1 hour, identify the equation that can be used to find the value of c.
QUESTION POSTED AT 01/06/2020 - 04:36 PM
### Use the continuous compound interest formula to find the indicated value. A=90,000; P=65,452; r=9.1%; t=? t=years (Do not round until the final answer. Then round to two decimal places as needed.)
QUESTION POSTED AT 01/06/2020 - 04:36 PM
### Find m∠Ym∠Y if m∠Ym∠Y is six more than three times its complement.
QUESTION POSTED AT 01/06/2020 - 04:35 PM
### A ball is thrown from initial height of 2 feet with an initial upward velocity of 35 ft./s. The ball's height H (in feet) after T seconds is given by the following. H=2+35T-16T^2 Find all values of T for which the ball's height is 20 feet. Round your answer(s) to the nearest hundredth.
QUESTION POSTED AT 01/06/2020 - 04:33 PM
### Given the equation 2x2 - 5x + 7 = 0. Find the discriminant. Describe the roots and explain your reasoning.
QUESTION POSTED AT 01/06/2020 - 04:32 PM
### ABCD is a rectangle. Find the length of each diagonal. AC=3y/5 BD=3y-4
QUESTION POSTED AT 01/06/2020 - 04:28 PM
### The hypotenuse of a 30 degree, -60 degree, -90 degree triangle is 24.2-ft. Explain how to find the lengths of the legs of the triangle.
QUESTION POSTED AT 01/06/2020 - 04:27 PM
### How could you use Descartes' Rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial, as well as find the number of possible positive and negative real roots to a polynomial?
QUESTION POSTED AT 01/06/2020 - 04:25 PM
### Productive use of machinery on a farm made it possible to decrease the number of tractors by 12. How many tractors does the farm have now if it used to have 1.5 times as many? Team one had one-fourth as many people as team two. after 6 quit team two and 12 people were transferred from team two to team one, both teams became equal. How many people were originally on team one?
QUESTION POSTED AT 01/06/2020 - 04:23 PM
| 1,892 | 6,028 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.03125 | 4 |
CC-MAIN-2020-40
|
latest
|
en
| 0.863491 |
https://www.nagwa.com/en/videos/674125814907/
| 1,550,854,346,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-09/segments/1550247518497.90/warc/CC-MAIN-20190222155556-20190222181556-00083.warc.gz
| 910,114,764 | 7,245 |
# Video: Evaluating Algebraic Expressions
Kathryn Kingham
If 𝑎 = 3 and 𝑏 = 8, evaluate 𝑎 ⋅ 𝑏.
01:41
### Video Transcript
If 𝑎 equals three and 𝑏 equals eight, evaluate 𝑎 times 𝑏.
Here’s what we know, that 𝑎 is equal to three and 𝑏 is equal to eight. We want to evaluate 𝑎 times 𝑏.
Before we evaluate, I wanna take a second and note this symbol. When you see a dot between two variables, or between two numbers, it means multiplication. Both the dot and the ⨯ can be used as a representation of multiplication.
Okay, now that that’s settled, back to evaluating 𝑎 times 𝑏. We were told that 𝑎 equals three. So, we substitute the number three for the letter 𝑎 in our problem.
The letter 𝑏 is equal to eight. So we substitute the number eight for the letter 𝑏. We remember that symbol represents multiplication. And now we multiply three times eight equals 24.
Under these conditions, 𝑎 times 𝑏 equals 24.
| 259 | 911 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.28125 | 4 |
CC-MAIN-2019-09
|
longest
|
en
| 0.906378 |
https://byjus.com/ncert-solutions-for-class-10-maths-chapter-3-linear-equations-in-two-variables-ex-3-4/
| 1,632,609,707,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780057775.50/warc/CC-MAIN-20210925202717-20210925232717-00646.warc.gz
| 205,449,945 | 154,345 |
# NCERT Solutions for Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables Exercise 3.4
NCERT Solutions are very beneficial for the students while preparing for board examinations. NCERT Solutions Class 10 Maths Chapter 3- Pair of Linear Variables is an important topic for the first term examinations and should be dealt with in complete detail. The solutions to the problems provided in the chapter are given in the NCERT Solutions Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables Exercise 3.4.
The NCERT Solutions Class 10 Maths contains stepwise solutions to all the Maths problems provided in the textbook or otherwise. The students should practise the exercises repeatedly to score well in the first term examinations.
NCERT Solutions contains detailed study material including all the important topics. The subject experts have come up with accurate solutions after a lot of research and brainstorming.
## Access other exercise solutions of Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables
Exercise 3.1 Solutions– 3 Questions
Exercise 3.2 Solutions– 7 Questions
Exercise 3.3 Solutions– 3 Questions
Exercise 3.5 Solutions– 4 Questions
Exercise 3.6 Solutions– 2 Questions
Exercise 3.7 Solutions– 8 Questions
### Access Answers of Maths NCERT Class 10 Chapter 3- Pair of Linear Equations in Two Variables Exercise 3.4
1. Solve the following pair of linear equations by the elimination method and the substitution method:
(i) x + y = 5 and 2x – 3y = 4
(ii) 3x + 4y = 10 and 2x – 2y = 2
(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7
(iv) x/2+ 2y/3 = -1 and x-y/3 = 3
Solutions:
(i) x + y = 5 and 2x – 3y = 4
By the method of elimination.
x + y = 5 ……………………………….. (i)
2x – 3y = 4 ……………………………..(ii)
When the equation (i) is multiplied by 2, we get
2x + 2y = 10 ……………………………(iii)
When the equation (ii) is subtracted from (iii) we get,
5y = 6
y = 6/5 ………………………………………(iv)
Substituting the value of y in eq. (i) we get,
x=5−6/5 = 19/5
∴x = 19/5 , y = 6/5
By the method of substitution.
From the equation (i), we get:
x = 5 – y………………………………….. (v)
When the value is put in equation (ii) we get,
2(5 – y) – 3y = 4
-5y = -6
y =Â 6/5
When the values are substituted in equation (v), we get:
x =5− 6/5 = 19/5
∴x = 19/5 ,y = 6/5
Â
(ii) 3x + 4y = 10 and 2x – 2y = 2
By the method of elimination.
3x + 4y = 10……………………….(i)
2x – 2y = 2 ………………………. (ii)
When the equation (i) and (ii) is multiplied by 2, we get:
4x – 4y = 4 ………………………..(iii)
When the Equation (i) and (iii) are added, we get:
7x = 14
x = 2 ……………………………….(iv)
Substituting equation (iv) in (i) we get,
6 + 4y = 10
4y = 4
y = 1
Hence, x = 2 and y = 1
By the method of Substitution
From equation (ii) we get,
x = 1 + y……………………………… (v)
Substituting equation (v) in equation (i) we get,
3(1 + y) + 4y = 10
7y = 7
y = 1
When y = 1 is substituted in equation (v) we get,
A = 1 + 1 = 2
Therefore, A = 2 and B = 1
(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7
By the method of elimination:
3x – 5y – 4 = 0 ………………………………… (i)
9x = 2y + 7
9x – 2y – 7 = 0 …………………………………(ii)
When the equation (i) is multiplied by 3 we get,
9x – 15y – 12 = 0 ………………………………(iii)
When the equation (iii) is subtracted from equation (ii) we get,
13y = -5
y = -5/13 ………………………………………….(iv)
When equation (iv) is substituted in equation (i) we get,
3x +25/13 −4=0
3x = 27/13
x =9/13
∴x = 9/13 and y = -5/13Â
By the method of Substitution:
From the equation (i) we get,
x = (5y+4)/3 …………………………………………… (v)
Putting the value (v) in equation (ii) we get,
9(5y+4)/3 −2y −7=0
13y = -5
y = -5/13
Substituting this value in equation (v) we get,
x = (5(-5/13)+4)/3
x = 9/13
∴x = 9/13, y = -5/13
(iv) x/2 + 2y/3 = -1 and x-y/3 = 3
By the method of Elimination.
3x + 4y = -6 …………………………. (i)
x-y/3 = 3
3x – y = 9 ……………………………. (ii)
When the equation (ii) is subtracted from equation (i) we get,
-5y = -15
y = 3 ………………………………….(iii)
When the equation (iii) is substituted in (i) we get,
3x – 12 = -6
3x = 6
x = 2
Hence, x = 2 , y = -3
By the method of Substitution:
From the equation (ii) we get,
x = (y+9)/3…………………………………(v)
Putting the value obtained from equation (v) in equation (i) we get,
3(y+9)/3 +4y =−6
5y = -15
y = -3
When y = -3 is substituted in equation (v) we get,
x = (-3+9)/3 = 2
Therefore, x = 2 and y = -3
2. Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes if we only add 1 to the denominator. What is the fraction?
Solution:
Let the fraction be a/b
According to the given information,
(a+1)/(b-1) = 1
=> a – b = -2 ………………………………..(i)
a/(b+1) = 1/2
=> 2a-b = 1…………………………………(ii)
When equation (i) is subtracted from equation (ii) we get,
a = 3 …………………………………………………..(iii)
When a = 3 is substituted in equation (i) we get,
3 – b = -2
-b = -5
b = 5
Hence, the fraction is 3/5.
(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
Solution:
Let us assume, present age of Nuri is x
And present age of Sonu is y.
According to the given condition, we can write as;
x – 5 = 3(y – 5)
x – 3y = -10…………………………………..(1)
Now,
x + 10 = 2(y +10)
x – 2y = 10…………………………………….(2)
Subtract eq. 1 from 2, to get,
y = 20 ………………………………………….(3)
Substituting the value of y in eq.1, we get,
x – 3.20 = -10
x – 60 = -10
x = 50
Therefore,
Age of Nuri is 50 years
Age of Sonu is 20 years.
(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Solution:
Let the unit digit and tens digit of a number be x and y respectively.
Then, Number (n) = 10B + A
N after reversing order of the digits = 10A + B
According to the given information, A + B = 9…………………….(i)
9(10B + A) = 2(10A + B)
88 B – 11 A = 0
-A + 8B = 0 ………………………………………………………….. (ii)
Adding the equations (i) and (ii) we get,
9B = 9
B = 1……………………………………………………………………….(3)
Substituting this value of B, in the equation (i) we get A= 8
Hence the number (N) is 10B + A = 10 x 1 +8 = 18
(iv) Meena went to a bank to withdraw Rs.2000. She asked the cashier to give her Rs.50 and Rs.100 notes only. Meena got 25 notes in all. Find how many notes of Rs.50 and Rs.100 she received.
Solution:
Let the number of Rs.50 notes be A and the number of Rs.100 notes be B
According to the given information,
A + B = 25 ……………………………………………………………………….. (i)
50A + 100B = 2000 ………………………………………………………………(ii)
When equation (i) is multiplied with (ii) we get,
50A + 50B = 1250 …………………………………………………………………..(iii)
Subtracting the equation (iii) from the equation (ii) we get,
50B = 750
B = 15
Substituting in the equation (i) we get,
A = 10
Hence, Meena has 10 notes of Rs.50 and 15 notes of Rs.100.
(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs.27 for a book kept for seven days, while Susy paid Rs.21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
Solution:
Let the fixed charge for the first three days be Rs.A and the charge for each day extra be Rs.B.
According to the information given,
A + 4B = 27 …………………………………….…………………………. (i)
A + 2B = 21 ……………………………………………………………….. (ii)
When equation (ii) is subtracted from equation (i) we get,
2B = 6
B = 3 …………………………………………………………………………(iii)
Substituting B = 3 in equation (i) we get,
A + 12 = 27
A = 15
Hence, the fixed charge is Rs.15
And the Charge per day is Rs.3
This is the fourth exercise of the NCERT Solutions Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables. It contains 2 questions the solutions to which are provided in the NCERT Solutions Class 10 Maths. The students can refer to these solutions in case of any doubts.
### Key Features of NCERT Solutions Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables Exercise 3.4
• The answers to the questions are accurate.
• The solutions to all the questions in the textbook are provided here.
• The questions are solved by the subject experts.
• NCERT Solutions for Class 10 Maths Chapter 3- Pair of Linear Equations in Two Variables Exercise 3.4 can help the students to score well in the first term examinations.
| 4,321 | 9,649 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.53125 | 5 |
CC-MAIN-2021-39
|
longest
|
en
| 0.845652 |
https://rnasystemsbiology.org/and-pdf/1709-shear-force-and-bending-moment-equations-pdf-849-825.php
| 1,632,127,662,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00495.warc.gz
| 517,727,502 | 7,115 |
# Shear force and bending moment equations pdf
File Name: shear force and bending moment equations .zip
Size: 25559Kb
Published: 12.04.2021
## Bending moment
Given below are solved examples for calculation of shear force and bending moment and plotting of the diagrams for different load conditions of simply supported beam, cantilever and overhanging beam. All the steps of these examples are very nicely explained and will help the students to develop their problem solving skills. Moment of Inertia Calculator Calculate moment of inertia of plane sections e. Reinforced Concrete Calculator Calculate the strength of Reinforced concrete beam. Fixed Beam Calculator Calculation tool for beanding moment and shear force for Fixed Beam for many load cases.
### Things to keep in mind:
In solid mechanics , a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The diagram shows a beam which is simply supported free to rotate and therefore lacking bending moments at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed; therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever , which is fixed at one end and is free at the other end neither simple or fixed. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.
A cantilever beam is subjected to various loads as shown in figure. Draw the shear force diagram and bending moment diagram for the beam. Bending moment between C and A;. The sign of bending moment is taken to be negative because the load creates hogging. Draw the shear force and bending moment diagrams for the beam.
A Beam is defined as a structural member subjected to transverse shear loads during its functionality. Due to those transverse shear loads, beams are subjected to variable shear force and variable bending moment. Shear force at a cross section of beam is the sum of all the vertical forces either at the left side or at the right side of that cross section. Bending moment at a cross section of beam is the sum of all the moments either at the left side or at the right side of that cross section. A beam is said to be statically determinate if all its reaction components can be calculated by applying three conditions of static equilibrium.
Bending Moment Equations for Beams Bending Moment Equations offer a quick and easy analysis to determine the maximum bending moment in a beam. Below is a concise table that shows the bending moment equations for different beam setups. A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend. The diagram shows a beam which is simply supported at both ends. Simply supported means. Bending Moment for different loading cases Bending moment is required for design of beam and also for the calculation of slope and deflection of beam. The following examples will illustrate how to write bending moment equation for different types of load.
Your guide to SkyCiv software - tutorials, how-to guides and technical articles. In this tutorial, we will look at calculating the shear force diagram of a simple beam. This is an important concept to understand, as shear force is something a beam will need to be checked for, for a safe design. Firstly, what is a shear force? A shearing force occurs when a perpendicular force is applied to static material in this case a beam.
### Bending moment
The support reactions a and c have been computed, and their values are shown in fig.
Determining shear and moment diagrams is an essential skill for any engineer. This is a problem. Shear force and bending moment diagrams tell us about the underlying state of stress in the structure. The quickest way to tell a great CV writer from a great graduate engineer is to ask them to sketch a qualitative bending moment diagram for a given structure and load combination!
#### Lesson 19. SOLVED EXAMPLES BASED ON SHEAR FORCE AND BENDING MOMENT DIAGRAMS
Мы успеем выспаться перед поездкой на север. Дэвид грустно вздохнул: - Потому-то я и звоню. Речь идет о нашей поездке. Нам придется ее отложить. - Что-о? - Сьюзан окончательно проснулась. - Прости.
• #### Astrid Q. 14.04.2021 at 12:17
The three types of beams are statically determinate because the support reactions can be found from the equilibrium equations. () g g. () g g. (c)
• #### Tilo R. 19.04.2021 at 07:01
Speechless aladdin sheet music pdf free power system stability and control leonard l grigsby pdf
• #### Helena R. 20.04.2021 at 13:52
Draw the free body diagram for the beam. Step 2: Apply equilibrium equations. In X direction. ∑ FX = 0. ⇒ RAX = 0. In.
#### Romana A.
Он не мог отказаться.
• #### Intro to quantum physics by french and taylor pdf
18.09.2020 at 01:30
• #### Freedom of speech and expression in indian constitution pdf
04.06.2021 at 03:31
• #### Fundamental of design and manufacturing pdf
18.09.2020 at 01:30
• #### Physics glencoe principles and problems pdf
21.08.2020 at 15:15
• #### Quantum physics of atoms molecules solids nuclei and particles pdf
18.09.2020 at 01:30
| 1,188 | 5,346 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.9375 | 4 |
CC-MAIN-2021-39
|
longest
|
en
| 0.917061 |
http://math.stackexchange.com/questions/119514/how-to-prove-phimn-phim-phin-if-m-n-ne-1?answertab=oldest
| 1,462,433,765,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-18/segments/1461860125897.68/warc/CC-MAIN-20160428161525-00062-ip-10-239-7-51.ec2.internal.warc.gz
| 192,499,323 | 18,052 |
# How to prove $\phi(mn) > \phi(m)\phi(n)$ if $(m,n) \ne 1$
I need to prove that
$$\phi(mn) > \phi(m)\phi(n)$$
if $m$ and $n$ have a common factor greater than 1.
I have read up on the case where $m$ and $n$ are relatively prime, then $\phi(mn)=\phi(m)\phi(n)$.
-
Write both $m,n$ as product of power of primes, and observe that $$\varphi(p^k) = p^{k-1}(p-1) > \varphi(p)^k = (p-1)^{k-1}(p-1)$$ – user2468 Mar 13 '12 at 1:46
J.D., this is pretty much the answer OP is looking for. You should post this. – Patrick Da Silva Mar 13 '12 at 1:48
It'd be nice to show explicitly that the natural map $U_{mn}\to U_m \times U_n$ is surjective but not injective, if it is true. – lhf Mar 13 '12 at 1:50
Ah, it seems that the natural map $U_{mn}\to U_m \times U_n$ has a kernel of size $d$ and an image of index $\phi(d)$ and so my idea above does not seem to work... – lhf Mar 13 '12 at 2:20
@lhf: Your idea works well enough...it leads to Dane's answer. (By the way, I had the same thought of expressing things in terms of this natural homomorphism. I think it would be a positive contribution if you left this as answer.) – Pete L. Clark Mar 13 '12 at 2:40
Hint: Write the prime factorization of both $m,n$ and observe that $$\varphi(p^k) = p^{k-1}(p-1) > \varphi(p)^k = (p-1)^{k-1}(p-1).$$
I'm sorry I'm too lazy this moment to work a whole proof. But here is a simple case. If $$m = pq, n = ph$$ for primes $p,q,h$. Then $$\varphi(m)\varphi(n) = \varphi(p)^2 \varphi(q)\varphi(h)$$ and $$\varphi(mn) = \varphi(p^2)\varphi(q)\varphi(h)$$ Result follows since $\varphi(p^2) > \varphi(p)^2$ as above.
According this recently asked question, $$\phi(mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} ,$$ where $d = \gcd(m,n)$. Your question follows from the fact that $\varphi(d) < d$ whenever $d>1$.
| 632 | 1,786 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.71875 | 4 |
CC-MAIN-2016-18
|
latest
|
en
| 0.878371 |
https://www.jiskha.com/questions/1556072/have-you-ever-been-on-an-airplane-and-heard-the-pilot-say-that-the-plane-would-be-a-little
| 1,620,719,998,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00501.warc.gz
| 861,178,152 | 4,965 |
# Math
Have you ever been on an airplane and heard the pilot say that the plane would be a little late because it would be flying into a strong headwind or that even though the plane was taking off a bit late, you would be making up time because you would be flying with a tailwind? This problem asks you to analyze such a situation. You have the following data: A plane flying at its maximum speed can go 210 miles per hour with a tailwind or 170 miles per hour into a headwind.
What is the wind speed?
What would be the maximum speed of the plane if there were no wind?
1. 👍
2. 👎
3. 👁
1. Assuming both winds are the same velocity.
210 -170 = 40, therefore wind is
20mph
Therefore plane's unhindred speed would be 210-20 or 170+2 = 190mph
1. 👍
2. 👎
## Similar Questions
1. ### Math
An airplane pilot over the Pacific sights an atoll at an angle of depression of 7°. At this time, the horizontal distance from the airplane to the atoll is 3,729 meters. What is the height of the plane to the nearest meter? 458
2. ### physics
An airplane is flying in a horizontal circle at a speed of 104 m/s. The 84.0 kg pilot does not want the centripetal acceleration to exceed 6.98 times free-fall acceleration. Find the minimum radius of the plane’s path. The
3. ### science
The pilot of an airplane traveling 180km/h wants to drop supplies to flood victims isolated on a patch of land 160m below. The supplies should be dropped how many seconds before the plane is directly overhead?
4. ### Physics
An airplane is flying in a horizontal circle at a speed of 105 m/s. The 76.0 kg pilot does not want the centripetal acceleration to exceed 6.35 times free-fall acceleration. a) Find the minimum radius of the plane’s path. Answer
1. ### Precalculus
An airplane is flying with a constant altitude at a speed of 450 mph in a direction 10° north of west directly towards its final destination. The airplane then encounters wind blowing at a steady 34 mph in a direction 85° north
2. ### Physics
An airplane flies at airspeed (relative to the air) of 430 km/h . The pilot wishes to fly due North (relative to the ground) but there is a 43 km/h wind blowing Southwest (direction 225◦). In what direction should the pilot
3. ### Math
An airplane needs to head due north, but there is a wind blowing from the southwest at 85 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly
4. ### Calculus
The pilot of an airplane that flies with an airspeed of 800 km/h wishes to travel to a city 800 km due east. There is a 80 km/h wind FROM the northeast a) what should the plane's heading be? b) How long will the trip take?
1. ### phys
An airplane is heading due south at a speed of 590 km/h. If a wind begins blowing from the southwest at a speed of 65.0 km/h (average). Calculate magnitude of the plane's velocity, relative to the ground. Calculate direction of
2. ### Physics
The piolt of an airplane wishes to fly due north, but there is a 65-km/h wind blowing toward the east. in what direction should the pilot head her plane if its speed relative to the air is 340 km/h?
3. ### Pre-Calc
A pilot of an airplane flying at 12,000 feet sights a water tower. The angle of depression to the base of the tower is 25°. What is the length of the line of sight from the plane to the tower?
4. ### Physics
The pilot of an airplane traveling 160 km/h wants to drop supplies to flood victims isolated on a patch of land 160m below. The supplies should be dropped how many seconds before the plane is directly overhead?
| 900 | 3,603 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.734375 | 4 |
CC-MAIN-2021-21
|
latest
|
en
| 0.914006 |
http://www.ck12.org/geometry/Negative-Statements/lesson/Negative-Statements/r20/
| 1,448,850,598,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2015-48/segments/1448398460519.28/warc/CC-MAIN-20151124205420-00244-ip-10-71-132-137.ec2.internal.warc.gz
| 343,672,401 | 34,720 |
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.
Negative Statements
Truth values for negation of conditional statements, conjunctions, and disjunctions.
0%
Progress
Practice Negative Statements
Progress
0%
Negative Statements
In everyday speech, negative statements are often ambiguous or unclear. Mathematically, you need a precise way to negate statements so that you can accurately determine whether statements are true or false. When is the negation of the following sentence true?
If I am not cold then it is not snowing.
Watch This
http://www.youtube.com/watch?v=hfz1gAoNd1w James Sousa: Showing Statements are Equivalent
Guidance
While in everyday language the opposite of “dog” might be “cat”, in mathematics the opposite of “dog” is “not a dog”. Using the word “not” is the basic way to negate an atomic sentence. An atomic sentence is a logical statement without logical connectives that has a truth value.
• Original sentence \begin{align*}(D)\end{align*}: That thing is a dog.
• Negation of sentence \begin{align*}(\sim D)\end{align*}: That thing is not a dog.
The box below represents the universe of all things. This universe can be separated into things that are dogs and things that are not dogs.
To negate complex statements that involve logical connectives like or, and, or if-then, you should start by constructing a truth table and noting that negation completely switches the truth value.
The negation of a conditional statement is only true when the original if-then statement is false.
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}P \rightarrow Q\end{align*} \begin{align*}\sim(P \rightarrow Q)\end{align*} T T T F T F F T F T T F F F T F
The negation of a conjunction is only false when the original two statements are both true.
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}P \land Q\end{align*} \begin{align*}\sim(P \land Q)\end{align*} T T T F T F F T F T F T F F F T
The negation of a disjunction is only true when both of the original statements are false.
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}P \lor Q\end{align*} \begin{align*}\sim(P \lor Q)\end{align*} T T T F T F T F F T T F F F F T
As mathematical sentences become more complex with additional connectives, truth tables and set theory circles are good ways to interpret when the statements are true and when the statements are false.
Example A
Use set theory circles to interpret the negation of a disjunction and explain how the negation of a disjunction can be written in a different way.
Solution: The shaded portion in the box represents the area that is within \begin{align*}P\end{align*} or within \begin{align*}Q\end{align*}. Recall that in mathematical logic, this is written as \begin{align*}P \lor Q\end{align*}. In set theory this area is represented similarly as \begin{align*}P \cup Q\end{align*} where the symbol \begin{align*}\cup\end{align*} stands for union. While the notation is slightly different, the reasoning about the relationships and implications is identical.
When you negate the statement, you completely switch what is shaded.
A different way to think about this shaded region is that it is the region that is not in \begin{align*}P\end{align*} and also not in \begin{align*}Q\end{align*}. This means that \begin{align*}\sim (P \lor Q)\end{align*} is equivalent to \begin{align*}\sim P \land \sim Q\end{align*}.
\begin{align*}\sim(P \lor Q) \Leftrightarrow \ \sim P \land \sim Q\end{align*}
The symbol “\begin{align*}\Leftrightarrow\end{align*}” works in mathematical logic and set theory the same way “\begin{align*}=\end{align*}” works in arithmetic and algebra. In this case, the negative appears to distribute throughout the or statement by negating each statement individually and changes the or statement to an and statement. This is called De Morgan’s Law
Example B
Use set theory circles to interpret the negation of a conjunction and explain how the negation of a conjunction can be written in a different way.
Solution: The shaded portion in the box represents the area that is within \begin{align*}P\end{align*} and within \begin{align*}Q\end{align*}. In mathematical logic this is written as \begin{align*}P \land Q\end{align*}. In set theory this area is represented similarly as \begin{align*}P \cap Q\end{align*} where the symbol \begin{align*}\cap\end{align*} stands for intersection. As before, the notation between mathematical logic and set theory is slightly different. However, the reasoning about the relationships and the logical implications are identical.
When you negate the statement, you completely switch what is shaded.
A different way to think about this shaded region is to consider that it represents everything that isn’t in \begin{align*}P\end{align*} or isn’t in \begin{align*}Q\end{align*}.
\begin{align*}\sim(P \land Q) \Leftrightarrow \ \sim P \lor \sim Q\end{align*}
This is a second representation of De Morgan’s Law.
Example C
Use set theory circles to interpret the negation of a conditional statement and explain how the negation of a conditional can be written in a different way.
Solution: The shaded portion in the box represents the area that makes the following statement true.
Notice that there are four different regions in the Venn Diagram which correspond to the four different rows of the conditional truth table. The only space that does not make the statement true is when \begin{align*}P\end{align*} is true (inside circle \begin{align*}P\end{align*}) and \begin{align*}Q\end{align*} is false (outside circle \begin{align*}Q\end{align*}).
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}P \rightarrow Q\end{align*} T T T T F F F T T F F T
To negate this statement, you switch the values in the truth table and switch the shaded region in the Venn Diagram.
A different way to think about this shaded region is to see it as the space that is in \begin{align*}P\end{align*} but not in \begin{align*}Q\end{align*}.
\begin{align*}\sim(P \rightarrow Q) \Leftrightarrow P \land \sim Q\end{align*}
Concept Problem Revisited
The statement already has several negative parts, so it is incorrect to simply switch one or both of the negations.
If I am not cold, then it is not snowing.
• \begin{align*}P = I \ am \ cold\end{align*}.
• \begin{align*}Q = It \ is \ snowing\end{align*}.
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}\sim P\end{align*} \begin{align*}\sim Q\end{align*} \begin{align*}\sim P \rightarrow \ \sim Q\end{align*} \begin{align*}\sim(\sim P \rightarrow \ \sim Q)\end{align*} T T F F T F T F F T T F F T T F F T F F T T T F
Start by building up to the original statement. Then, add a column that completely negates the original statement. Notice that there is only one row where the final negated statement is true. That is when \begin{align*}P\end{align*} is false and \begin{align*}Q\end{align*} is true. Therefore, the negation of the original sentence is true when “I am not cold” and “it is snowing”.
Vocabulary
De Morgan’s Law transforms a conjunction into a disjunction using negation.
A tautology is a logical statement that is always true. A tautology is a type of basic theorem.
Guided Practice
1. Show that the following statements are equivalent in a truth table. The symbol ≡ means equivalent. To be equivalent in this case means to be true at the same time and false at the same time.
\begin{align*}P \land Q \equiv \ \sim(\sim P \lor \sim Q)\end{align*}
2. Write a sentence in two different ways illustrating the mathematical statement in Guided Practice #1.
3. Demonstrate the following tautology in a truth table.
\begin{align*}B \rightarrow(A \lor \sim A)\end{align*}
1. While a truth table is not a proof, it can help you recognize when two statements have the same truth values.
\begin{align*}P\end{align*} \begin{align*}Q\end{align*} \begin{align*}\sim P\end{align*} \begin{align*}\sim Q\end{align*} \begin{align*}\sim P \lor \sim Q\end{align*} \begin{align*}\sim(\sim P \lor \sim Q)\end{align*} \begin{align*}P \land Q\end{align*} T T F F F T T T F F T T F F F T T F T F F F F T T T F F
Notice that the final two columns are identical.
2. \begin{align*}P \land Q\end{align*}: I like movies and I like TV. \begin{align*}\sim(\sim P \lor \sim Q)\end{align*}: It is not the case that either I don’t like movies or I don’t like TV.
3. A tautology is a logical statement that is always true.
\begin{align*}A\end{align*} \begin{align*}\sim A\end{align*} \begin{align*}B\end{align*} \begin{align*}A \lor \sim A\end{align*} \begin{align*}B \rightarrow (A \lor \sim A)\end{align*} T F F T T T F T T T F T T T T F T F T T
Practice
I’m either going to go skiing or snowboarding next weekend.
1. Identify the atomic statements in the above sentence and use logical connectives to rewrite the sentence with symbols.
2. Write the negation of the sentence with symbols and write the negation of the sentence in words in a natural way.
Mike and John both ate lunch with me.
3. Identify the atomic statements in the above sentence and use logical connectives to rewrite the sentence with symbols.
4. Write the negation of the sentence with symbols and write the negation of the sentence in words in a natural way.
Neither my brother nor my sister wants to play with me.
5. Identify the atomic statements in the above sentence and use logical connectives to rewrite the sentence with symbols.
6. Write the negation of the sentence with symbols and write the negation of the sentence in words in a natural way.
Write negations for the following statements.
7. All dogs go to heaven.
8. My teacher is seldom wrong.
9. Everyone likes pizza.
Make truth tables for each of the following.
10. \begin{align*}(P \land Q) \lor \sim Q \end{align*}
11. \begin{align*}P \land(Q \lor \sim Q)\end{align*}
12. \begin{align*}(P \lor Q)\lor \sim R \end{align*}
13. \begin{align*}(\sim P \land \sim Q)\lor \sim R\end{align*}
14. What is the simplest statement that is equivalent to #11: \begin{align*}P \land(Q \lor \sim Q)\end{align*}?
15. Use De Morgan’s Law to find a statement equivalent to the following statement: \begin{align*}\sim(Q \lor \sim Q)\end{align*}
16. Use De Morgan’s Law to find a statement equivalent to the following statement: \begin{align*}\sim (P \lor Q)\lor \sim R\end{align*}
Vocabulary Language: English
atomic statement
atomic statement
An atomic statement is a declarative statement without logical connectives that has a truth value.
conjunction
conjunction
A conjunction is an "and" statement, which is a statement that combines two logical statements and is only true when both statements are true. The symbol for “and” is “$\land$”.
De Morgan's law
De Morgan's law
De Morgan’s law transforms a conjunction into a disjunction using negation.
disjunction
disjunction
A disjunction is an "or" statement that combines two logical statements, and is only false when both statements are false. The symbol for “or” is “ $\lor>/math>”.$
negation
negation
The negation of a statement is the opposite of the statement. If the original statement is $D$, then the negation of the statement is represented by $\sim D$. A statement and its negation will always have opposite truth values.
tautology
tautology
A tautology is a logical statement that is always true. A tautology is a type of basic theorem.
| 3,169 | 11,581 |
{"found_math": true, "script_math_tex": 77, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 3, "texerror": 0}
| 4.4375 | 4 |
CC-MAIN-2015-48
|
longest
|
en
| 0.786457 |
https://docplayer.net/21874913-Sample-size-and-power-in-clinical-trials.html
| 1,716,071,365,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-22/segments/1715971057516.1/warc/CC-MAIN-20240518214304-20240519004304-00086.warc.gz
| 175,561,884 | 26,271 |
# Sample Size and Power in Clinical Trials
Size: px
Start display at page:
Transcription
1 Sample Size and Power in Clinical Trials Version 1.0 May Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance Level 5. One Sided and Two Sided Tests
2 Statistical Power The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is false i.e. it will not make a Type II error, or a false negative decision. As the power increases, the chances of making a Type II error decrease. The probability of a Type II error is referred to as the false negative (β). Therefore power is equal to 1 β, which is also known as the sensitivity. Uses of Power Analysis A power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size. It can also be used to calculate the minimum effect size that is likely to be detected in a study using a given sample size. Additionally, the concept of power can be used in making comparisons between different statistical testing procedures: for example, between a parametric and a nonparametric test of the same hypothesis. The two sample problem Statistical tests use data from samples to assess, or make inferences about a population. In the concrete setting of a two sample comparison, the goal is to assess whether the mean values of some attribute obtained for individuals in two sub populations differ. For example, to test the null hypothesis that the mean heart rate of smokers and non smokers do not differ, samples of smokers and non smokers are drawn and their heart rate is recorded. The mean heart rate of one group is compared to that of the other group using a statistical test such as the twosample t test. The power of the test is the probability that the test will find a statistically significant difference between smokers and non smokers, as a function of the size of the true difference between those two populations. Conceptually, power is the probability of finding a difference that does exist, as opposed to the likelihood of declaring a difference that does not exist (which is known as a Type I error or false positive ).
3 Factors Influencing Power There are many factors that affect the power of a study. In part power depends upon the test statistic chosen at analysis (that why we always aim to chose the best statistic for any given situation). The power of the test is also dependent on the amount of variation present; the greater the variability the lower the power. This can be partly offset by consideration of the best design and a careful consideration about the design of the study can lead to increased power. In general, large differences are easier to find than small differences. The relative size of the difference is known as the effect size (difference) and in general we have increasing power with increasing effect size. It also stands to reason that we are more likely to make a correct inference with a big representative sample. Accordingly sample size affects power. Finally the power of a test is dependent of the nominal significance level chosen as this is quite key in determining the decision rule. The power of a study can be greatly affected by the study design (which can alter the effect size). We will consider each of these aspects qualitatively. 1) An appropriate Test Statistic There are many ways of analysing a given set of data. In any one situation we may be able to argue that one particular method or one particular technique or one particular approach is better than another. For instance, suppose we are going to compare two independent samples for differences in location. Possible methods of analysis might include the two sample independent t test or the Mann Whitney test or you might even invent your own test. If the underpinning assumptions for the two sample independent t test are satisfied (or are not grossly violated) then we would argue that the t test is the best one to apply. Note that by best we really mean the most powerful test i.e. the one that is most likely to allow us to claim a statistically significant difference when in fact the null hypothesis is false. Theory shows that when the assumptions underpinning the t test are true then the t test is the most powerful test to apply. Likewise when the assumptions underpinning the two samples t test are slightly violated then simulation studies show that the t test still retains relatively high power compared with other general techniques (such as the Mann Whitney test). However when the assumptions underpinning the two samples t test are grossly violated then the Mann Whitney test can have greater power.
4 Power depends upon the test statistic chosen. An appropriate test statistic permits a defendable way of analysis data in any particular situation and it impacts on the probability of obtaining a statistically significant result. For this reason we devote considerable effort in deciding on the best test statistic to use. Power also depends on the approach. A modern branch of statistics is the computer intensive approach known as the bootstrap; for non normal populations the bootstrapping approach can have greater power than corresponding parametric or non parametric techniques. Power depends upon the data analysis approach. 1) Variation Statistical inference is concerned with identifying a message in the data. The message (e.g. a difference in means) can be obscured by variation in the data. If the variation in data can be reduced then it is easier to obtain a clear message (i.e. we would ideally want a high signal tonoise ratio). Studies should be designed to minimize error variance relative to the systematic variance and/or to partition out (remove or account for) some aspects of variation in the analysis phase. For instance in a before and after study we would have data on experimental units (e.g. participants) at the outset and then later on. At the outset the experimental units would, most likely, differ. These initial differences might be large and would add to the variation in the data and this between experimental unit variation could cloud the message in the data. To reduce the effect of the variation we would look at the changes between the before and after pairs. Doing this means that we are focussing directly on the phenomenon of interest (the change between before and after) and are removing some of the variation due to initial differences in the experimental units. In general if we are designing a study and if we have the choice of generating paired data or unpaired data then we would probably want to opt for the paired design as it gives [1] a direct focus on the phenomenon of interest and [] at the analysis phase we can remove or account for some variation. These two benefits means that a Blocked or repeated measures design tends to be more powerful than the corresponding independent or between subjects design.
5 Alternatively suppose that a researcher is looking to see if differences in performance exist between those that ingest caffeine and those that do not. If such effects do exist then it more likely that the effects would be apparent in a study comparing an intake of 500 mg of caffeine against 0 mg of caffeine than a study comparing 5 mg of caffeine against 0 mg of caffeine. In general, if the variation is relatively large then the power of the test would be relatively low (all other things remaining equal). Careful thought about the design of the study can lead to a more powerful test. ) Effect Size If population mean values are widely separated relative to the variation then we would anticipate this being reflected in the data. The greater the true separation the easier it will be to detect the difference using a sample. Consequently we have increasing power with increasing differences between means (all other things being equal). For comparing two means the effect size (ES) is defined as the difference between means relative to the standard deviation (note standard deviation and not standard error of the means). See Section 3 for comparing two means. Similarly consider a correlation study. If two variables are strongly correlated then we anticipate this strong correlation being reflected in a sample. The greater the true correlation the easier it will be to detect the correlation in the sample. Consequently we have increasing power with increasing strength of population correlation. In a correlation analysis the correlation coefficient is the measure of effect size. More generally we have increasing power with increasing size of effect.
6 3) Sample Size Generally speaking the precision of a statistic increases with increasing sample size. By way of example suppose we consider the sample mean x which is used to estimate a population mean. For different samples each of size n the sample means (say x 1, x, x 3, x 4,. ) will not all have the same value. If we are dealing with random samples then the variance of the sample estimates is given by / n where is the variance of the population. In any practical setting the population variance is a fixed positive number. The ratio / n decreases as n increases (i.e. n, / n 0 ). Because the error variance of the statistic diminishes with increasing sample size we are more likely to make a correct inference. More likely to make a correct inference indicates that the power of the test increases with increasing sample size (all other things being equal). Increasing sample size is typically associated with an increase in power. 4) Significance Level Contemporary practice is to perform hypothesis tests using a nominal significance level of The significance level indicates the probability of committing a Type I error (i.e. the probability of incorrectly rejecting the null hypothesis when in fact it is true). In some situations the consequences of making a Type I error may be judged as not being too serious and in these cases we may choose to work with a bigger nominal significance level (e.g ). The bigger the significance level the easier it is to reject the null hypothesis. If the null hypothesis is false and the criterion to reject the null hypothesis has been relaxed then the test will be more powerful than it would have been if the criterion had not been relaxed. Sometimes the consequences of making a Type I error may be viewed as being very serious. In these cases we may require compelling evidence before rejecting the null hypothesis and opt to use a more stringent (lower) significance level (e.g ). Making it harder to reject the null hypothesis means that the power of the test will be reduced (all other things being equal).
7 In general we have decreasing power with decreasing nominal significance level. 5) One Sided or Two Sided Test In some very rare instances theory might exist so that some possibilities concerning the direction of an effect can be eliminated. In these cases a researcher might carry out a onesided test. In one sided tests there is a focus in one direction only (e.g. a positive difference) and all other things being equal then one sided tests will be more powerful than two sided tests. Standards for Power Although there are no formal standards for power, most researchers assess the power of their tests using 0.80 (80%) as a standard for adequacy. This convention implies a four to one trade off between β risk and α risk. (β is the probability of a Type II error, α is the probability of a Type I error, 0. = and 0.05 are conventional values for β and α). However, there will be times when this 4 to 1 weighting is inappropriate. In medicine, for example, tests are often designed in such a way that no false negatives (Type II errors) will be produced. But this inevitably raises the risk of obtaining a false positive (a Type I error). The rationale is that it is better to tell a healthy person we may have found something let s test further, than to tell a diseased person all is well. Power analysis is appropriate when the concern is with the correct rejection, or not, of a null hypothesis. In many contexts, the issue is about determining if there is or is not a difference but rather with getting the more refined estimate of the population effect size. For example, if we were expecting a population correlation between height and weight of around 0.50, a sample size of 0 will give us approximately 80% power (α = 0.05, two sided) to reject the null hypothesis of zero correlation. However, in doing this study we are probably more interested in knowing whether the correlation is 0.30 or 0.60 or In this context we would need a much larger sample size in order to reduce the confidence interval of our estimate to a range that is acceptable for our purposes. Techniques similar to those employed in a traditional power analysis can be used to determine the sample size required for the width of a confidence interval to be less than a given value.
8 An Example Power Analysis A clinician wants to select the better of two treatments for Tinea Capitis (a fungal infection of the scalp or ringworm of the scalp) and the primary outcome of interest is time to complete disappearance of the fungus on the scalp measured in days. We will assume that the two sample t test will be used; that a mean difference of size 1 is anticipated and that the population standard deviation is estimated to be 1. We will assume that a researcher can take a sample of size 0 per group and then wonder what the power of the test would be. The example information has been put into Minitab as shown Under Options we have selected to perform our analysis at the standard 0.05 significance level.
9 Output from running the power routine using the above parameters is as follows: -Sample t Test Testing mean 1 = mean (versus not =) Calculating power for mean 1 = mean + difference Alpha = 0.05 Assumed standard deviation = 1 Sample Difference Size Power The sample size is for each group From the above output we conclude that if the null hypothesis is false and the true mean difference is +1 and if the population standard deviation is also equal to 1 then if we obtain a sample of size n = 0 per group we have an 87% chance of correctly rejecting the null hypothesis (again assuming that the underpinning assumptions of the two sample t test would not be grossly violated).
10 Applying for Grants Funding agencies, ethics boards and research review panels frequently request that a researcher perform a power analysis, for example to determine the minimum number of test subjects needed for an experiment to be informative. This is because and underpowered study is unlikely to allow one to choose between hypotheses at the desired significance level. Power remains the most convenient measure of how much a given experiment size can be expected to refine prior beliefs. A study with low power is unlikely to lead to a large change in prior beliefs.
### II. DISTRIBUTIONS distribution normal distribution. standard scores
Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,
### Statistical tests for SPSS
Statistical tests for SPSS Paolo Coletti A.Y. 2010/11 Free University of Bolzano Bozen Premise This book is a very quick, rough and fast description of statistical tests and their usage. It is explicitly
### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing
### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS
QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.
### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses
Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the
### Two-Sample T-Tests Assuming Equal Variance (Enter Means)
Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of
### Descriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
### Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)
Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption
### NCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the
### Tutorial 5: Hypothesis Testing
Tutorial 5: Hypothesis Testing Rob Nicholls [email protected] MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................
### Simple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
### Confidence Intervals for One Standard Deviation Using Standard Deviation
Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from
### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES
SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR
### HYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
### Difference tests (2): nonparametric
NST 1B Experimental Psychology Statistics practical 3 Difference tests (): nonparametric Rudolf Cardinal & Mike Aitken 10 / 11 February 005; Department of Experimental Psychology University of Cambridge
### WHAT IS A JOURNAL CLUB?
WHAT IS A JOURNAL CLUB? With its September 2002 issue, the American Journal of Critical Care debuts a new feature, the AJCC Journal Club. Each issue of the journal will now feature an AJCC Journal Club
### "Statistical methods are objective methods by which group trends are abstracted from observations on many separate individuals." 1
BASIC STATISTICAL THEORY / 3 CHAPTER ONE BASIC STATISTICAL THEORY "Statistical methods are objective methods by which group trends are abstracted from observations on many separate individuals." 1 Medicine
### Hypothesis testing - Steps
Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
### Skewed Data and Non-parametric Methods
0 2 4 6 8 10 12 14 Skewed Data and Non-parametric Methods Comparing two groups: t-test assumes data are: 1. Normally distributed, and 2. both samples have the same SD (i.e. one sample is simply shifted
### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
### Study Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
### Hypothesis testing. c 2014, Jeffrey S. Simonoff 1
Hypothesis testing So far, we ve talked about inference from the point of estimation. We ve tried to answer questions like What is a good estimate for a typical value? or How much variability is there
### Introduction to Hypothesis Testing OPRE 6301
Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about
### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance
### Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
### Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples
Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The
### UNDERSTANDING THE TWO-WAY ANOVA
UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables
### Correlational Research
Correlational Research Chapter Fifteen Correlational Research Chapter Fifteen Bring folder of readings The Nature of Correlational Research Correlational Research is also known as Associational Research.
### Research Methods & Experimental Design
Research Methods & Experimental Design 16.422 Human Supervisory Control April 2004 Research Methods Qualitative vs. quantitative Understanding the relationship between objectives (research question) and
### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)
CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.
### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
### Fairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
### NAG C Library Chapter Introduction. g08 Nonparametric Statistics
g08 Nonparametric Statistics Introduction g08 NAG C Library Chapter Introduction g08 Nonparametric Statistics Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Parametric and Nonparametric
### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
### C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
### Non-Inferiority Tests for Two Means using Differences
Chapter 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous
### Introduction to Fixed Effects Methods
Introduction to Fixed Effects Methods 1 1.1 The Promise of Fixed Effects for Nonexperimental Research... 1 1.2 The Paired-Comparisons t-test as a Fixed Effects Method... 2 1.3 Costs and Benefits of Fixed
### Statistics 2014 Scoring Guidelines
AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home
### Using Excel for inferential statistics
FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied
### Hypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes
### Data Analysis Tools. Tools for Summarizing Data
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
### CHAPTER 14 NONPARAMETRIC TESTS
CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences
### DATA COLLECTION AND ANALYSIS
DATA COLLECTION AND ANALYSIS Quality Education for Minorities (QEM) Network HBCU-UP Fundamentals of Education Research Workshop Gerunda B. Hughes, Ph.D. August 23, 2013 Objectives of the Discussion 2 Discuss
### Module 5: Multiple Regression Analysis
Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College
### Confidence Intervals for Cpk
Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process
### The Null Hypothesis. Geoffrey R. Loftus University of Washington
The Null Hypothesis Geoffrey R. Loftus University of Washington Send correspondence to: Geoffrey R. Loftus Department of Psychology, Box 351525 University of Washington Seattle, WA 98195-1525 [email protected]
### Unit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
### Introduction to Quantitative Methods
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
### Part 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
### DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript
DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript JENNIFER ANN MORROW: Welcome to "Introduction to Hypothesis Testing." My name is Dr. Jennifer Ann Morrow. In today's demonstration,
### Chapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS
Chapter Seven Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Section : An introduction to multiple regression WHAT IS MULTIPLE REGRESSION? Multiple
### CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
### T test as a parametric statistic
KJA Statistical Round pissn 2005-619 eissn 2005-7563 T test as a parametric statistic Korean Journal of Anesthesiology Department of Anesthesia and Pain Medicine, Pusan National University School of Medicine,
### Week 4: Standard Error and Confidence Intervals
Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.
### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
### Tests for Two Proportions
Chapter 200 Tests for Two Proportions Introduction This module computes power and sample size for hypothesis tests of the difference, ratio, or odds ratio of two independent proportions. The test statistics
### NCSS Statistical Software
Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the
### The correlation coefficient
The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative
### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)
NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions
### Confidence Intervals for Cp
Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process
### Non-Inferiority Tests for One Mean
Chapter 45 Non-Inferiority ests for One Mean Introduction his module computes power and sample size for non-inferiority tests in one-sample designs in which the outcome is distributed as a normal random
### Chapter G08 Nonparametric Statistics
G08 Nonparametric Statistics Chapter G08 Nonparametric Statistics Contents 1 Scope of the Chapter 2 2 Background to the Problems 2 2.1 Parametric and Nonparametric Hypothesis Testing......................
### UNIVERSITY OF NAIROBI
UNIVERSITY OF NAIROBI MASTERS IN PROJECT PLANNING AND MANAGEMENT NAME: SARU CAROLYNN ELIZABETH REGISTRATION NO: L50/61646/2013 COURSE CODE: LDP 603 COURSE TITLE: RESEARCH METHODS LECTURER: GAKUU CHRISTOPHER
### Inclusion and Exclusion Criteria
Inclusion and Exclusion Criteria Inclusion criteria = attributes of subjects that are essential for their selection to participate. Inclusion criteria function remove the influence of specific confounding
### CALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents
### Non-Inferiority Tests for Two Proportions
Chapter 0 Non-Inferiority Tests for Two Proportions Introduction This module provides power analysis and sample size calculation for non-inferiority and superiority tests in twosample designs in which
### Two-sample inference: Continuous data
Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As
### HYPOTHESIS TESTING WITH SPSS:
HYPOTHESIS TESTING WITH SPSS: A NON-STATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER
### Selecting Research Participants
C H A P T E R 6 Selecting Research Participants OBJECTIVES After studying this chapter, students should be able to Define the term sampling frame Describe the difference between random sampling and random
### Principles of Hypothesis Testing for Public Health
Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine [email protected] Fall 2011 Answers to Questions
### Nonparametric statistics and model selection
Chapter 5 Nonparametric statistics and model selection In Chapter, we learned about the t-test and its variations. These were designed to compare sample means, and relied heavily on assumptions of normality.
### DATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
### Association Between Variables
Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi
### Projects Involving Statistics (& SPSS)
Projects Involving Statistics (& SPSS) Academic Skills Advice Starting a project which involves using statistics can feel confusing as there seems to be many different things you can do (charts, graphs,
### Independent samples t-test. Dr. Tom Pierce Radford University
Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of
### Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS
Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS About Omega Statistics Private practice consultancy based in Southern California, Medical and Clinical
### SIMULATION STUDIES IN STATISTICS WHAT IS A SIMULATION STUDY, AND WHY DO ONE? What is a (Monte Carlo) simulation study, and why do one?
SIMULATION STUDIES IN STATISTICS WHAT IS A SIMULATION STUDY, AND WHY DO ONE? What is a (Monte Carlo) simulation study, and why do one? Simulations for properties of estimators Simulations for properties
### Topic #6: Hypothesis. Usage
Topic #6: Hypothesis A hypothesis is a suggested explanation of a phenomenon or reasoned proposal suggesting a possible correlation between multiple phenomena. The term derives from the ancient Greek,
### Statistics in Medicine Research Lecture Series CSMC Fall 2014
Catherine Bresee, MS Senior Biostatistician Biostatistics & Bioinformatics Research Institute Statistics in Medicine Research Lecture Series CSMC Fall 2014 Overview Review concept of statistical power
### Standard Deviation Estimator
CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of
### SPSS Explore procedure
SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,
### Chapter 4 and 5 solutions
Chapter 4 and 5 solutions 4.4. Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in five gallon milk containers. The analysis is done in a laboratory,
### How To Check For Differences In The One Way Anova
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way
### research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other
1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric
### " Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
### Exact Nonparametric Tests for Comparing Means - A Personal Summary
Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment
### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Sample Practice problems - chapter 12-1 and 2 proportions for inference - Z Distributions Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide
### Introduction to Statistics Used in Nursing Research
Introduction to Statistics Used in Nursing Research Laura P. Kimble, PhD, RN, FNP-C, FAAN Professor and Piedmont Healthcare Endowed Chair in Nursing Georgia Baptist College of Nursing Of Mercer University
### Sample Size Planning, Calculation, and Justification
Sample Size Planning, Calculation, and Justification Theresa A Scott, MS Vanderbilt University Department of Biostatistics [email protected] http://biostat.mc.vanderbilt.edu/theresascott Theresa
### An introduction to Value-at-Risk Learning Curve September 2003
An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk
### Basic Concepts in Research and Data Analysis
Basic Concepts in Research and Data Analysis Introduction: A Common Language for Researchers...2 Steps to Follow When Conducting Research...3 The Research Question... 3 The Hypothesis... 4 Defining the
### Introduction to Hypothesis Testing
I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true
### 2 Precision-based sample size calculations
Statistics: An introduction to sample size calculations Rosie Cornish. 2006. 1 Introduction One crucial aspect of study design is deciding how big your sample should be. If you increase your sample size
| 8,372 | 40,244 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.6875 | 4 |
CC-MAIN-2024-22
|
latest
|
en
| 0.908624 |
https://studen.com/mathematics/17886107
| 1,709,068,081,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2024-10/segments/1707947474686.54/warc/CC-MAIN-20240227184934-20240227214934-00708.warc.gz
| 563,747,326 | 46,137 |
25.12.2022
# How many 50p’s are in £6
0
24.06.2023, solved by verified expert
12
Step-by-step explanation:
2 of, 50 pence, = 1 pound. So multiply 6 by 2, you get 12. Your answer should be 12.
Hope this helps!
Maybe brainliest?
### Faq
Mathematics
12
Step-by-step explanation:
2 of, 50 pence, = 1 pound. So multiply 6 by 2, you get 12. Your answer should be 12.
Hope this helps!
Maybe brainliest?
Mathematics
1. p=11.50+p
2.last one
4.third one
6.ans= 2
7.y/x=constant 36/2=18
8.28.84
9.t=5w
10.38.74
13.c=0.54n
15.first one
16.12.83
18.x=18
19.3 hour
Mathematics
1. p=11.50+p
2.last one
4.third one
6.ans= 2
7.y/x=constant 36/2=18
8.28.84
9.t=5w
10.38.74
13.c=0.54n
15.first one
16.12.83
18.x=18
19.3 hour
Mathematics
Step-by-step explanation:
I got 7 because you take 63-17.50=45.5 then you'll take 45.5 and divide that by 6.5 and you'll get the answer of 7
Mathematics
Step-by-step explanation:
I got 7 because you take 63-17.50=45.5 then you'll take 45.5 and divide that by 6.5 and you'll get the answer of 7
Mathematics
First we need to subtract \$5 from both sides. That gives us the new equation 1.50p = 9. Then, divide 9 by 1.5, which gives you the answer. 9/1.5 = 6.
Mathematics
a) 110 b) 92
Step-by-step explanation:
Since we have given that
Percentage of animals not to climbing on furniture = 43%
Percentage of animals not to clawing on furniture = 29%
Percentage of animals not to fighting with other animals training cats = 17%
If the number of cat owners = 255
Number of owners trained their cat not to climb on furniture is given by
If the number of cats owners = 316
Number of owners trained their cat not to claw on furniture is given by
Hence, a) 110 b) 92
Mathematics
a) 110 b) 92
Step-by-step explanation:
Since we have given that
Percentage of animals not to climbing on furniture = 43%
Percentage of animals not to clawing on furniture = 29%
Percentage of animals not to fighting with other animals training cats = 17%
If the number of cat owners = 255
Number of owners trained their cat not to climb on furniture is given by
If the number of cats owners = 316
Number of owners trained their cat not to claw on furniture is given by
Hence, a) 110 b) 92
English
1. The audience is interested and supportive of Tyrese.
2. They have more education than Douglass and are willing to teach him.
Explanation:
When Tyrese dove in and started swimming with his opponents, the people became attentive to his “steady progress” and silently waited and wished for all swimmers to make it to the end, including Tyrese, which indicates that the audience was interested and supportive of Tyrese.
This conclusion is supported by the fact that Douglass saw the boys as “teachers” from whom he could learn to read. And as he mentions, “With their kindly aid,” he succeeded in his goal of reading.
Mathematics
| 857 | 2,895 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.28125 | 4 |
CC-MAIN-2024-10
|
latest
|
en
| 0.877229 |
http://www.currentaffairs4examz.com/2015/12/quantitative-aptitude-quiz-for-ibps.html
| 1,508,489,860,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-43/segments/1508187823997.21/warc/CC-MAIN-20171020082720-20171020102720-00359.warc.gz
| 422,458,636 | 98,745 |
Saturday, December 19, 2015
# Quantitative Aptitude Quiz for IBPS Clerk Mains 2016
Friends.. Here are some useful Aptitude Questions Quiz for those preparing for IBPS Clerk Mains 2016. At the same time these questions will also be useful for LIC AAO Exam 2016, SSC CGL 2016, SSC CHSL 2016 and other upcoming exams in 2016. Click the start quiz button to attempt the quiz. We have also uploaded detailed solution for each question which will be very useful for you. Attempt quiz and comment your suggestions. All the Best.
Directions (Q. 1-5): What will come in place of the question mark (?) in the following number series?
1. 8 10 18 44 124 (?)
(1) 344
(2) 366
(3) 354
(4) 356
(5) None of these
2. 13 25 61 121 205 (?)
(1) 323
(2) 326
(3) 324
(4) 313
(5) None of these
3. 12 18 36 102 360 (?)
(1) 1364
(2) 1386
(3) 1384
(4) 1376
(5) None of these
4. 454 472 445 463 436 (?)
(1) 436
(2) 456
(3) 454
(4) 434
(5) None of these
5. A certain fraction is equivalent to 2/5 . If the numerator of the fraction is increased by 4 and the denominator is doubled, the new fraction is equivalent to 1/3 . What is the sum of numerator and denominator of the original fraction?
(1) 49
(2) 35
(3) 28
(4) 26
(5) 21
6. In how many different ways, letters of the word MANPOWER be arranged such that all vowels are together?
(1) 360
(2) 4320
(3) 1440
(4) 2160
(5) 40320
7. On a certain test, 3 students each had a score of 90, 9 students each had a score of 80, 4 students each had a score of 70, and 4 students each had a score of 60. What was the average (arithmetic mean) score for the 20 students ?
(1) 70.5
(2) 75.0
(3) 75.5
(4) 80.0
(5) 80.5
8. Of the science books in a certain supply room, 40 are on botany, 35 are on zoology, 30 are on physics, 40 are on geology, and 55 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 20 of the books removed are on the same science subject?
(1) 81
(2) 79
(3) 96
(4) 85
(5) 100
9. If difference between CI and SI on a certain sum of money is Rs. 72 at 12% p.a. for 2 years then find the amount.
a) RS.6000
b) Rs.5000
c) Rs. 5200
d) Rs. 4900
e) None of these
10. A completes a work in 12 days and B complete the same work in 24 days. If both of them work together, then the number of days required to complete the work will be
(a) 8
(b) 6
(c) 7
(d) 5
(e) None of these
Attempt above Quiz to check the right answers.
Quantitative Aptitude Quiz for IBPS Clerk Mains 2016 Reviewed by Currentaffairs4examz on 19.12.15 Rating: 5 Friends.. Here are some useful Aptitude Questions Quiz for those preparing for IBPS Clerk Mains 2016. At the same time these questions wil...
| 926 | 2,730 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.625 | 4 |
CC-MAIN-2017-43
|
longest
|
en
| 0.867895 |
http://function-of-time.blogspot.com/2009/04/introducing-right-triangle-trig.html
| 1,566,810,710,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00486.warc.gz
| 78,524,221 | 35,159 |
Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!
## Wednesday, April 29, 2009
### Introducing Right Triangle Trig
I am conflicted every year about how to introduce right triangle trig to my Algebra 1 classes. I am not thrilled that we have to worry about it in this course, and come at it from the perspective of similar right triangles (instead of unit circle/wrapping function), but that decision is above my paygrade. This is a topic that I find so difficult to promote a conceptual understanding. We have these three weird word abbreviations, sin, cos, and tan (that students tend to pronounce phoenetically) and they mean what exactly now? Ratios of sides? What's a ratio, again? I suspect that most Algebra 1 teachers don't even try too hard. I suspect that this topic is largely approached as a procedural exercise, with lots of practice. And I admit that every year, after giving it my best shot at illustrating it conceptually, I also revert to teaching procedure, with alot of practice.
Here is what I do, or try to do, and please, readers, I am asking you to criticize the hell out of it. I really want to do it better. (Due credit: this idea and the original document came from Dave Cox, who used to be a professor at Cornell, but I think he's retired.)
I give every student a protractor, ruler, and a 4-page packet. I usually have my desks arranged in pairs, and each pair of students is assigned an angle. Using the protractor, they draw a series of parallel lines that make that angle with a horizontal line on a given page with some axes provided. This creates several overlapping right triangles.
Then they use the ruler to measure all three sides of each right triangle. They record it in a table I provide. They use their calculator to compute the three relevant ratios, and enter them in the table. Here is the data sheet:
They notice that the ratios for the two corresponding sides all come out to be the same, and we have a discussion about how similar figures have sides that are in proportion, and that's what it means for things to be in proportion. When you divide them, the quotients are equal. They average the five ratios to determine the "real" ratio.
Then they are supposed to use the ratios to solve this problem for "their" angle:
Doing that successfully is probably the key part of the lesson, and it never goes well. They don't see right away exactly what to do, and give up. I end up doing a bunch on the board for them, using their angles and the ratios they calculated.
Next we are supposed to collect every pair's calculated ratios in a table, like this:
And they are supposed to copy them and use it to complete some problems for homework, that look like this:
I keep trying this lesson every year, because I really want it to work. I really think it should work. Part of the problem is that we run out of time. I can't get this whole thing done in 43 minutes, and there isn't really a good point to stop and pick up the next day. The next day, I just tell them "SOHCAHTOA" (*huge resigned sigh*) (at least I don't claim Sohcahtoa was a Native American princess) and start teaching procedure. And feel like I am committing malpractice, and stealing my paycheck.
Help.
Here is another link to the document.
Update: I revised this activity based on the comments and discussion on this post. Go here.
1. This is a tough one. I started this year by introducing the unit circle and having kids play with the values. I created this GeoGebra interactive worksheet (http://tinyurl.com/dadffp)but I think it still needs some work.
Thanks for the credit on the worksheet, but I am not retired. ;)
2. On the one hand, I'm tempted to say "they're too young" for this. On the other hand, I know hundreds of years ago, young men their age were navigating boats using this very concept with only an astrolabe and the sun.
I have not taught this stuff, before, but maybe give them a "standard" for some of the angles? Then they fill out a small chart like in the opening activity with the standard and then underneath fill in the known values and try to find the others? It will connect the activities together, maybe.
3. In algebra 1?! I thought trig always went with algebra II. It doesn't seem right to do it with algebra 1. Am I missing something?
I love what you're doing and may steal it for one of the first day exercises for my trig class at the college.
4. Well, the difficulty is that students have trouble with "word problems", and always have done. I've never understood why it's such a leap to turn the problem into the correct mathematical formula, but perhaps that's why I've been a computer scientist/programmer for more than 30 years. I never had trouble with it. My classmates certainly did.
You know "The Far Side" cartoon depicting "Hell's library", right?: the shelves are stocked full of books with titles like "Word Problems", "More Word Problems", and "Word Problems Volume LXVIII".
And, see, then you throw them off by flipping the flippin' triangle around: after giving them a bunch of problems where "the angle" is in the lower left, you throw the ladder one at them, and the angle in question is now at the top.
It really seems to be an analytical skill that most students lack. We who are good at it don't understand.
5. Afraid I can't help here either. My son pretty much picked up this level of trig on his own or in discussions we had walking to school when he was in 5th or 6th grade. Like Barry Leiba, I have trouble visualizing the difficulty, since it seems so easy to me. Luckily, my son seems to think a lot like me, so I rarely have difficulty teaching him math or science.
I don't think I'd do well teaching a non-honors Alg 1 class.
6. Here's my suggestion for the problems you've encountered with the "Romeo and Juliet" part of the lesson. Take with a pinch of salt, of course. The main idea is to focus on the moments when they realise that these ratios are (1) fixed and (2) useful. The bits where they take averages and collect other people's results are, by comparison, distractions.
-
Remove all the text between the table and the problem (including the bit that says "estimated true value...". Now nobody else has told them that the ratios will always be the same.
Replace that text with a question 4 like "here is a triangle with a base of 40cm. You may not draw that triangle. Estimate its altitude."
The deductions aren't hard. I'm pretty sure your students will be able to notice the ratios are the same by themselves (aha!). The brighter ones - or all of them with the correct prodding - should be able to use that to get the altitude... helped by the fact that it isn't a word problem.
If they do that then at this point the students should feel more involved because they've discovered the key points themselves (ratio stays the same, can be applied to problems). It should also be more memorable. As for the length of the lesson? Leave the rest, you've met your top priorities. Have a discussion about what they found out and how they could apply that to problems, then set them just one or two for homework. Practice can (and will) come the following day.
-
So... do you think that might help?
7. Sue - NY has included sohcahtoa to solve right triangle problems in Algebra 1 forever.
Dave - Must have been a different Dave Cox :) (We also have a sub named Dave Cox. Do you run into this often?). Unit circle is a good way to go, but I don't know if it's appropriate to the learning goals in this case.
Calculus Dave - I'm not sure I follow. You mean, don't have them draw the angle? Just give them some triangles?
Barry - I think non word problems aren't really problems. They are "exercises". Kids aren't good at it because they aren't expected to do it very much, and often aren't taught very well in primary grades. You are right, moving the angle on them is too high a level of difficulty.
Kevin - Yeah, this needs to work for kids for whom insight doesn't come very naturally. And/or whose parents don't or aren't equipped to discuss it with them in 5th grade.
Alex - I like that idea. I'm going to incorporate it. It will make it more gradual and more obvious. They do, by the way, notice they are the same by themselves. I'm not sure I want to use the word "estimate". They think it means "take a wild guess and just write something, 'they' can't mark it wrong." If I truncate the lesson, just to clarify, are you suggesting to assign a few problems just using their angle? Otherwise other students' data would be necessary.
8. Kate
If the learning goals is for students to recognize that the trig ratios are just the comparison of two sides, the unit circle is the perfect place to start. That is where it all comes from. Once you deal with the unit circle, you can change the radius but the ratios don't change. Once kids understand that the ratios have a meaning, they may be more apt to apply them and generalize (ie. sohcahtoa).
If I am misunderstanding your goal, my apologies.
9. I guess I just think that getting into the unit circle will introduce a bunch of other stuff they would have to understand, or I would have to teach them (and don't have the time to). For example, they're not strong on the cartesian plane. But maybe I should try it and see what happens.
10. From the perspective of a calculus teacher, the unit circle is the way, bar none, to understand trigonometric functions.
It gives a parameterization of any circle.
It reminds that the sum of the squares is one.
It provides the perfect memory aid for the sine and cosine of the common angles 0, pi/6, pi/4, pi/3, and pi/2.
It clearly shows the signs of the basic trigonometric functions.
And much more!
That all said, I really don't know if there's a better way to teach trigonometry starting from circles instead of from right triangles. On the one hand, trigonometry starting with triangles goes back to the Greeks, and there's something to be said for the refinement of thousands of years of tradition. On the other hand, maybe trigonometry always starts with triangles because it always has started with triangles...
11. I did a similar exercise last year in my Geometry class. They had briefly seen right triangle trigonometry in previous classes, so I wanted a way to break into the topic without throwing the words "trigonometry", "sine", "cosine", or "tangent" out and having the kids call up their previously associated conceptions of what those words meant.
Instead of drawing a series of parallel hypotenuses joining two perpendicular lines, I used a figure of a right triangle that had a number of segments drawn parallel to one of the legs, dividing it into a number of nested right triangles. Students had to measure segments and calculate ratios for each triangle, though I referred to all of them by names of segments rather than "opposite", "adjacent", and "hypotenuse". I left it to them to figure out that the same ratios showed up in each triangle.
Having established that the sides gave the same ratios (and why that happened), I gave them another figure where they had to measure all sides of one triangle and the hypotenuses of the other triangles, use the ratios to predict the lengths of the remaining sides, and then measure the sides to see how accurate their predictions were. Getting from there to talking about sine, cosine and tangent involved a discussion where I asked questions like "If we keep looking at right triangles, what does the ratio of a given pair of sides depend on?" and "Would having a list of these ratios for all angles be helpful?"
It took a lot longer than I expected it too, but it seemed more rewarding than just rattling off definitions.
12. Kate, I have to play devil's advocate here: I understand that the standards say you have to teach this, but have you considered just teaching the absolute minimum necessary and then moving on? Even if it's tested this year, pare down your unit to whatever they'll be asked on the Regents, and that's it. You might say that's teaching to the test, and perhaps you'd be right, for this particular topic.
Besides, if they are asking "What's a ratio?" and can't move around the coordinate plane, should you really be spending a lot of time on something this advanced?
In other words, and I know it's pure heresy to say this, you don't always have to do everything you're told to do! You have to do what's in the best interest of your students. Sometimes districts and states tell you to teach things (or test students on things) for no good reason, and I believe it's our job to decide what's most important.
I think this might not be helpful, as you have already started teaching it. So I will add that I think going into the unit circle at this point would just make things confusing for your students. There's worse things they could be memorizing other than sohcahtoa.
13. Hihi,
COOL. STUFF.
My thoughts:
(1) I honestly think that no way should you use the unit circle for Alg I. It's too much, and combining circles and triangles really would be better to do in geometry. I think this work with triangles is really a lot better.
(2) I am really irked by #4 ("While the three ratios will be different, the values in each column should be close. Why?") It gives away the prize, for free.
(3) I don't know the use of #5 or how they are supposed to get the "actual value of each ratio." I mean, it could lead to a good discussion of error and why using the biggest triangle would be best, but it isn't keeping up with the topic.
(4) If the word problem is getting to them, if the kids can't seem to make that final precious leap which is actually what the whole class is about, I would maybe give one additional question without Romeo and Juliet/context beforehand. Which was actually Alex's suggestion, now that I think about it..
(5) To me, it seems like you are trying to do two distinct things in this lesson, at the same time, and that could be tricky for non-accelerated kids. They are: (1) teaching about similar triangles and (2) a baby-version of trigonometry, where students generate their own trig tables without knowing it.
Do you think that maybe separating the two completely -- at least initially -- might help? First get them to understand the concept of similar triangles (they are the same, just one is scaled up or scaled down, the angles are the same too!).
Then get them to talk about what happens as you raise or lower the angle... what happens to the ratio of the sides of a single right triangle.
Finally have them draw, say, two right triangles with a small angle of elevation (like 20 degrees), and two right triangles with a large angle of elevation (like 80 degrees). Then say: I want you to tell me the sides of two HUGE triangle with your same two angles of elevations, and base length 5,000,000. (But you could first prompt them by asking which opposite side of the two huge triangles would be bigger and why? And similarly, which of the hypotenuses of the two huge triangles would be bigger and why?)
Then it could take you to where you want to go?
And I say if you are excited about it, and you know your students are capable of doing it, then keep on keeping on! Don't scrap it.
My 2 cents.
Sam.
14. Oh, I also forgot to say: you might want to take out all reference to tan. Since sine and cosine haye opp/adj/hypotenuse, students will be able to solve all the problems with just sine and cosine. They don't NEED tan.
It's also nice because they are just between 0 and 1, so students will have a better grasp of what they mean -- because they give the size of a side in proportion to the hypotenuse. (A sine of an angle yielding .56 would indicate that the opposite side is 56% the length of the hypotenuse.)
Sam.
15. The inference that all the ratios are nearly the sme doesn't transfer to a generalization that most of my students are prepared to make.
Most of them are still stuck with even being able to identify the opposite, adjacent, and hypotenuse consistently. I actually practice just that for a while.
One of the earlier commenters suggested not asking too much up front and I'm inclined to agree.
I didn't really get the unit circle stuff until pre-calc in 11th grade. Even in Algebra 2/trig, the law of sines and cosines, sum of angles, etc. were derived from a triangle...
16. For a discovery lesson, that may be overkill for Algebra I students using that many numbers. I do a discovery section of my lesson, but only on a 30-60-90 triangle -- that's enough to motivate the concept (and link to unit circles later).
To start things out, I do a similarity / scale factor type lesson. The students get the idea of scaling very intuitively.
Then we do the similarity ratio trick on two triangles...
a/b = c/d
but note that with some algebra
a/c = b/d
...we can talk about ratios *within* each triangle being scaled identically.
Also, get your students to actually do the tree thing -- that is, go outside and pick a tree and measure it.
17. I wasn't suggesting that you throw a unit circle up on the board and have them go at it. I was suggesting using somethign like the GeoGebra file I linked to. I agree that the unit circle by itself is a bit too much for algebra 1 kids. My question to you is can my worksheet, with modifications, help? I can re do the dynamic sheet and allow for the GeoGebra tools to be seen and the questions can be changed.
18. It's hard to get past the poor thinking that went into putting this topic into algebra (they CAN do it, so they should; just a corollary to "as early and superficially as possible")
Something different: I give them a rt triangle with hypotenuse 1 and angle (I pick a number, maybe 22) degrees. I label the legs (with sin22 and cos22, rounded). Then I ask them to use similar triangles to find missing sides in a bunch of 22-68-90 triangles. Everyon's happy. Then I put up a 27-63-90 (or something like that) and someone complains that they are not similar, so I put up a hypotenuse 1 triangle for that...
And after another triangle or so, I show them how to look up the values...
Clumsy, but something nice.
Better, of course, to push it off for a couple of years. But not my choice.
Jonathan
Trig in Algebra 1? Really? Crazy New Yorkers. I'm just getting to it for my first time as a teacher and have realized how much I internalized my Precalculus teacher's pneumoic devices. Instead of SOHCAHTOA (which seems stranger on the rez), I learned "Some old hippie caught another hippie tripping on acid."
As I've worked through problems in the past week, I keep saying, "Tripping on..." My student just remembers "opp hyp" "adj hyp" "opp adj." She has amazing recall though.
20. 3♣ sez:
I didn't really get the unit circle stuff until pre-calc in 11th grade.This is exactly the problem I see in calculus. The unit circle has many more uses, and makes it easier to memorize the basic facts about trigonometric functions than the triangle definition. It's at the very least a crying shame that it isn't taught more thoroughly.
21. Thanks everyone who has commented so far. I'm overwhelmed by your willingness to help me improve. I'll try to respond to everyone as I read through and process your comments. (And if you are just getting here and have more to add, don't let me stop you, keep them coming.)
22. I tried to avoid SohCahToa, and I heaved ahuge sigh when I finally had to resort to it
I'm glad I'm not the only one.
It does mean, however, that I have no answers for you.
23. Mr. D I'm normally as big a rebel as the next guy, but simply leaving out this whole topic is not feasible. The regents exam is unpredictable, and could conceivably include 6 or 8 trig questions. It wouldn't be responsible to hamstring the kids by not even mentioning it.
Unfortunately (or fortunately), I am constitutionally averse to knowingly doing a bad job at something. If I'm going to do it, I'm going to do it right, or at least do everything in my power.
Sam - this lesson is sequenced immediately after a few days about similar figures, as you suggest. I also incorporated some of yours and Alex's ideas in the new worksheet (see newer post).
Jason and Sam - I pared it way down, removing tangent and the number of angles the class is working with.
Thanks again, everyone, all the comments were very helpful. Take a look at the new lesson, if you're so inclined.
I introduced my pre-algebra students to tangent in this fashion and they are quite comfortable with the concept.
A good project would be to take a map with elevation curves and have students calculate the road grades of a hypothetical road on the map. They can then use a trig table to find the angle on the road. In this way they can learn how tangent connects these ratios to measurements of angles.
Only once students have mastered tangent, should they be introduced to the other trig functions. To throw them all at them at once would be foolish.
By the way, my school is cutting teachers due to state budget shortfalls. I am the newby, so if anyone needs a math teacher then drop me a line.
cheers,
James
Hi! I will have to approve this before it shows up. Cuz yo those spammers are crafty like ice is cold.
| 4,868 | 21,492 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.75 | 4 |
CC-MAIN-2019-35
|
latest
|
en
| 0.96006 |
https://www.coursehero.com/file/6849454/Assignment-4/
| 1,524,793,265,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-17/segments/1524125948738.65/warc/CC-MAIN-20180427002118-20180427022118-00069.warc.gz
| 766,069,576 | 46,423 |
{[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
Assignment 4
# Assignment 4 - Assignment#4 Section 1.7(pg 64 1a)Let n be...
This preview shows pages 1–2. Sign up to view the full content.
Assignment #4. Section 1.7 (pg. 64) 1a)Let n be an integer. Then n is either even or odd. If n is even, then there is an integer k so that n=2k. In this case, 5n^2+3n+1=2(10k^2+13k)+1. Since 10k^2+3k is an integer, 5n^2+3n+1 is odd. If n is odd there is an integer k such that n=2k+1, in which case 5n^2+3n+1=2(10k^2+13k+4)+1. Since 10k^2+13k+4 is an integer, 5n^2+3n+1 is odd. 1b) Prove that if x is an odd integer, then 2x^2+3x+4 is odd. If x is an odd integer, then x=2k+1 for some integer k. 4x^2+x+6 = 2(2k+1)^2+3(2k+1)+4 = 8x^2+14x+8+1 = 2( 4K^2 + 7K + 4) + 1 ==>odd. Therefore, if x is an odd integer, then 2x^2 + 3x + 4 is odd. 1c) Prove that the sum of 5 consecutive integers is always divisible by 5. Let x be an integers. Any five consecutive integers can then be respresented as x,x+1,x+2,x+3,x+4. Therefore the sum is x+(x+1)+(x+2)+(x+3)+(x+4)=5x+10=5(x+2) which is divisible by 5. 1e) Prove that between any two distinct rational numbers a,b with a < b there exists another rational number in between a and b. Suppose we have two distinct rational numbers, a and b. a = p/q for some integers p and q. b = n/m for some integers n and m. We can construct a rational number (a+b)/2. We know that (a+b)/2 must lie between a and b. We need to show (a+b)/2 is rational: a+b (p/q)+(n/m) (mp+nq)/(qm) (mp+nq) -----= ----------- = ------------ = --------==>rational number. 2 2 2 2qm Done. 3a) Suppose x is rational. Suppose x+y is rational. Then there exist integers pa nd q, q not equal to 0 such that x=p/q and integers r and s s not equal to 0 such that x+y=r/s. Then y=(x+y) -x = r/s -p/q = rq-ps/sq. Since rq-ps and sq are an integers and sq is not equal to 0, y is rational.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]}
| 749 | 2,087 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.53125 | 5 |
CC-MAIN-2018-17
|
latest
|
en
| 0.85612 |
https://gmatclub.com/forum/is-1-p-r-r-49974.html?fl=similar
| 1,511,394,981,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-47/segments/1510934806708.81/warc/CC-MAIN-20171122233044-20171123013044-00548.warc.gz
| 612,393,349 | 44,360 |
It is currently 22 Nov 2017, 16:56
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
is 1/p > r/(r^2+2)
Author Message
Manager
Joined: 20 Mar 2005
Posts: 169
Kudos [?]: 104 [0], given: 0
Show Tags
04 Aug 2007, 07:25
00:00
Difficulty:
(N/A)
Question Stats:
0% (00:00) correct 0% (00:00) wrong based on 0 sessions
HideShow timer Statistics
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
is 1/p > r/(r^2+2)
1) p=r
2 r> 0
Kudos [?]: 104 [0], given: 0
VP
Joined: 28 Mar 2006
Posts: 1367
Kudos [?]: 38 [0], given: 0
Show Tags
04 Aug 2007, 07:31
Balvinder wrote:
is 1/p > r/(r^2+2)
1) p=r
2 r> 0
Should be C
if p=r it doesnt mean that r^2+2>r^2 since we do not know the sign of r
if it is ve the ">" stays the same but if r is -ve then the sign flips
So we need a condition that r>0 which is exactly what B gives
Hence C stands
Kudos [?]: 38 [0], given: 0
Director
Joined: 03 May 2007
Posts: 867
Kudos [?]: 271 [0], given: 7
Schools: University of Chicago, Wharton School
Show Tags
04 Aug 2007, 10:19
Balvinder wrote:
is 1/p > r/(r^2+2)
1) p=r
2 r> 0
C is it.
using both information, it is true that 1/p > r/(r^2+2).
Kudos [?]: 271 [0], given: 7
Manager
Joined: 20 Mar 2005
Posts: 169
Kudos [?]: 104 [0], given: 0
Show Tags
05 Aug 2007, 00:13
trivikram wrote:
Balvinder wrote:
is 1/p > r/(r^2+2)
1) p=r
2 r> 0
Should be C
if p=r it doesnt mean that r^2+2>r^2 since we do not know the sign of r
if it is ve the ">" stays the same but if r is -ve then the sign flips
So we need a condition that r>0 which is exactly what B gives
Hence C stands
trivikram can you clear my doubts over this ; both side are square, which make it positive irrespective of the sign of r.
so r^2 +2 will definetly will be great than r^2
OA is C but still i want to clear my doubts
Kudos [?]: 104 [0], given: 0
VP
Joined: 28 Mar 2006
Posts: 1367
Kudos [?]: 38 [0], given: 0
Show Tags
05 Aug 2007, 08:46
Balvinder wrote:
trivikram wrote:
Balvinder wrote:
is 1/p > r/(r^2+2)
1) p=r
2 r> 0
Should be C
if p=r it doesnt mean that r^2+2>r^2 since we do not know the sign of r
if it is ve the ">" stays the same but if r is -ve then the sign flips
So we need a condition that r>0 which is exactly what B gives
Hence C stands
trivikram can you clear my doubts over this ; both side are square, which make it positive irrespective of the sign of r.
so r^2 +2 will definetly will be great than r^2
OA is C but still i want to clear my doubts
When are they squares? Only after cross-multiplying.
But if we have a situation like the one in thsi problem dont ever think it is right to do the way you are thinking.
First ask yourself do I know the sign of this variable. If you aint sure then you better account for it.
Kudos [?]: 38 [0], given: 0
Manager
Joined: 08 Oct 2006
Posts: 211
Kudos [?]: 10 [0], given: 0
Show Tags
05 Aug 2007, 11:50
1/p>r/r^2+2
Thus if p>o, r^2+2>pr Case 1
If P<0, r^2+2<pr>r^2. Case 1.
Case2 does not hold true as r^2+2 can not be less than -r^2.
Thus 1 is sufficient.
Maybe I am very tired and talking completely nonsense. But please explain.
Kudos [?]: 10 [0], given: 0
Manager
Joined: 27 May 2007
Posts: 128
Kudos [?]: 14 [0], given: 0
Show Tags
05 Aug 2007, 12:51
I didn't do any algebraic manipulation on this one. I just plugged in the number 3, then the number -3, for p and r, to check the first statement. With 3 it was true:
1/p>r/r^2+2 becomes 1/3>3/11, true
With -3 it was false:
1/p>r/r^2+2 becomes -1/3>-3/11, false
so insufficient.
Statement 2 gives no info about the value of p, so insufficient.
With both of them, we know that p and r are equal and positive. Any positive value makes the statement true.
Kudos [?]: 14 [0], given: 0
05 Aug 2007, 12:51
Display posts from previous: Sort by
| 1,449 | 4,325 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.96875 | 4 |
CC-MAIN-2017-47
|
latest
|
en
| 0.867171 |
https://www.exceltip.com/excel-functions/how-to-use-the-dec2oct-function-in-excel.html
| 1,627,966,607,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-31/segments/1627046154420.77/warc/CC-MAIN-20210803030201-20210803060201-00462.warc.gz
| 743,864,942 | 35,276 |
# How to use the DEC2OCT Function in Excel
In this article, we will learn about how to use the DEC2OCT function in Excel.
DEC2OCT function in excel is used to convert decimal representation of numbers of radix 10 to octal numbers (radix = 8).
The table shown below shows you some of the most used base & their radix of Alpha - numeric characters
Base radix Alpha-Numeric Characters octal 2 0 - 1 Octal 8 0 - 7 Decimal 10 0 - 9 hexadecimal 16 0 - 9 & A - F hexatridecimal 36 0 - 9 & A - Z
Decimal number is representation of a number of radix 10. 10 digits are used in representation of a decimal number 0 - 9. Where as octal number representation have only 8 digits from 0, 1, 2, 3, 4, 5, 6, 7. The below table will help you understand better
Decimal Octal 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 10
You can see from the above table that decimal value 8 converted to octal comes out to be 10.
The DEC2OCT function converts the decimal number of radix 10 to the octal number of radix 2.
Syntax:
=DEC2OCT ( number, [Places] )
Number : decimal number which is to be converted to octal
[Places] : [optional] number where result expressed upto the number.
The input decimal number to the function should be between - 536870912 to + 536870911. The resulting octal number will have upto 10 characters (10 bits), where first number express the sign of the number (positive or negative) & Other 9 express the value after the sign. ( 1 for negative & 0 for positive)
To get the octal number, [places] number must have have sufficient places to express the octal number or else it returns the #NUM! Error. The decimal negative number is processed using two's complement notation and ignore [places] number.
Now let’s get more understanding of the function via using them in some examples.
Here we have some octal values in octal Column. We need to convert these octal numbers to decimal value.
Use the formula in Decimal column:
=DEC2OCT (A2)
A2 : number provided to the function as cell reference
Values to the DEC2OCT function is provided as cell reference.
The octal representation of 2 of base 10 (decimal) is 2 of base 8 (octal).
Now copy the formula to other cells using the Ctrl + D shortcut key.
As you can see here the DEC2OCT function returns the results of the input values.
Notes:
1. Numbers can be given as argument to the function directly without quotes or cell reference in excel.
2. The function doesn’t consider the [places] number in case of a negative decimal number.
3. The number must be a valid decimal number between - 536870912 to + 536879011.
4. If the input [places] number is not an integer, it is truncated by the function.
5. The function returns the octal value for the max value of + 3777777777 & minimum value upto - 4000000000.
6. The function returns the #NUM! Error
1. If the input decimal number is less than - 536870912 or greater than + 536879011.
2. If the input number [places] is zero or negative.
3. If the input number [places] is not sufficient for the resulting positive octal number.
7. The function returns the #VALUE! Error
1. If the input number is text or non-numeric.
2. If the input [places] number is text or non-numeric.
Hope you understood how to use DEC2OCT function and referring cell in Excel. Explore more articles on Excel mathematical conversion functions here. Please feel free to state your query or feedback for the above article.
Related Articles:
How to use the DEC2BIN Function in Excel
How to use the BIN2OCT Function in Excel
How to use the BIN2HEX Function in Excel
How to use the DEC2OCT Function in Excel
How to use the DEC2HEX Function in Excel
How to use the OCT2BIN Function in Excel
How to use the OCT2DEC Function in Excel
How to use the OCT2HEX Function in Excel
How to use the HEX2BIN Function in Excel
How to use the HEX2OCT Function in Excel
How to use the HEX2DEC Function in Excel
Popular Articles:
If with wildcards
Vlookup by date
Join first and last name in excel
Count cells which match either A or B
Terms and Conditions of use
The applications/code on this site are distributed as is and without warranties or liability. In no event shall the owner of the copyrights, or the authors of the applications/code be liable for any loss of profit, any problems or any damage resulting from the use or evaluation of the applications/code.
| 1,102 | 4,347 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.609375 | 4 |
CC-MAIN-2021-31
|
longest
|
en
| 0.774503 |
http://gmatclub.com/forum/mean-and-median-88528.html?fl=similar
| 1,485,106,837,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2017-04/segments/1484560281450.93/warc/CC-MAIN-20170116095121-00516-ip-10-171-10-70.ec2.internal.warc.gz
| 110,789,018 | 54,097 |
Mean and median : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 22 Jan 2017, 09:40
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
Events & Promotions
Events & Promotions in June
Open Detailed Calendar
Mean and median
Author Message
TAGS:
Hide Tags
Manager
Joined: 22 Jul 2008
Posts: 94
Location: Bangalore,Karnataka
Followers: 3
Kudos [?]: 171 [0], given: 11
Show Tags
29 Dec 2009, 04:51
00:00
Difficulty:
(N/A)
Question Stats:
50% (00:00) correct 50% (00:32) wrong based on 8 sessions
HideShow timer Statistics
The number of defects in the first five cars to come through a new production line are 9, 7, 10, 4, and 6, respectively. If the sixth car through the production line has either 3, 7, or 12 defects, for which of theses values does the mean number of defects per car for the first six cars equal the median?
I. 3
II. 7
III. 12
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
[Reveal] Spoiler: OA
Senior Manager
Joined: 30 Aug 2009
Posts: 286
Location: India
Concentration: General Management
Followers: 3
Kudos [?]: 162 [0], given: 5
Show Tags
29 Dec 2009, 05:05
kirankp wrote:
The number of defects in the first five cars to come through a new production line are 9, 7, 10, 4, and 6, respectively. If the sixth car through the production line has either 3, 7, or 12 defects, for which of theses values does the mean number of defects per car for the first six cars equal the median?
I. 3
II. 7
III. 12
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
D - I and III only
for 7 we will have mean as 7.16(approx) and median as 7 but for 3 we will have mean and median = 6.5 and for 12 we will have median and mean =8
Intern
Joined: 01 Jan 2010
Posts: 1
Followers: 0
Kudos [?]: 0 [0], given: 0
Show Tags
01 Jan 2010, 20:41
I disagree with the above response. The answer should be B) II only. 9, 7, 10, 4, and 6 added to 7 = 42. The mean of the set is therefore 7. The Median of the set is also 7. 3 doesn't work and there is only on option that includes 7 and excludes 3. B. Please post OA.
BTW to the response above: I believe that the median must be a number in the set.
Intern
Joined: 22 Dec 2009
Posts: 13
Followers: 1
Kudos [?]: 5 [0], given: 3
Show Tags
01 Jan 2010, 23:03
9 + 7 + 10 + 4 + 6 + 7 = 43
As for the median, when the series is even-numbered then the median is the average of the two middle numbers. Answer choice D (I and III) is correct.
Senior Manager
Joined: 30 Aug 2009
Posts: 286
Location: India
Concentration: General Management
Followers: 3
Kudos [?]: 162 [0], given: 5
Show Tags
01 Jan 2010, 23:07
craigmcdermott wrote:
I disagree with the above response. The answer should be B) II only. 9, 7, 10, 4, and 6 added to 7 = 42. The mean of the set is therefore 7. The Median of the set is also 7. 3 doesn't work and there is only on option that includes 7 and excludes 3. B. Please post OA.
BTW to the response above: I believe that the median must be a number in the set.
we have set 9, 7, 10, 4, and 6 which is arranged as 4 6 7 9 10 [ total sum = 36] so adding 7 gives us 43 and not 42. Also mean will be 7.16 and median here will be 7
if we have 3 then the set which we get is 3 4 6 7 9 10 [ total sum = 39] median and mean are both 6.5 here
if we have 12 then the set will be 4 6 7 9 10 12 [ total sum = 48] median and mean both will be 8
Also Median necessarily doesnt need to be a number in the set
Re: Mean and median [#permalink] 01 Jan 2010, 23:07
Similar topics Replies Last post
Similar
Topics:
4 The mean of six positive integers is 15. The median is 18, and the onl 4 09 Sep 2015, 20:54
11 Both the average (arithmetic mean) and the median of a set of 7 number 3 05 May 2015, 01:42
8 In Set T, the average (arithmetic mean) equals the median. Which of th 4 10 Feb 2015, 07:42
11 A researcher computed the mean, the median, and the standard 11 25 Jun 2012, 01:40
4 A researcher computed the mean, the median, and the standard 4 17 Mar 2007, 11:51
Display posts from previous: Sort by
| 1,445 | 4,534 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.28125 | 4 |
CC-MAIN-2017-04
|
latest
|
en
| 0.901369 |
http://math.stackexchange.com/questions/201950/computing-value-of-the-series-s-n-sum-k-1k-n-a-k
| 1,469,315,465,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2016-30/segments/1469257823802.12/warc/CC-MAIN-20160723071023-00025-ip-10-185-27-174.ec2.internal.warc.gz
| 165,454,221 | 15,905 |
# Computing value of the series $S_N = \sum_{k=1}^{k=N} a_k$ [duplicate]
Possible Duplicate:
Sum of n consecutive numbers
I am working on engineering problem where I have a series of the form:
$S_N = \sum_{k=1}^{k=N} a_k$
were $a_k = k$. I'm wondering, how do I compute the value of $S_N$ for $N=365$ and what type of series is this?
-
## marked as duplicate by Isaac Solomon, Aang, Dilip Sarwate, William, J. M.Sep 27 '12 at 10:22
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
arithmetic series. – Dilip Sarwate Sep 25 '12 at 1:47
## 1 Answer
$S_N=a_1+a_2+a_3+\cdots+a_N$ where $a_k=k$
$\implies s_N=1+2+3+\cdots+N$ which is an arithmetic progression with $a=1$ and common difference $d=1$
$\implies s_N=\frac{N(N+1)}{2}$
-
| 272 | 834 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.625 | 4 |
CC-MAIN-2016-30
|
latest
|
en
| 0.906113 |
https://www.jiskha.com/display.cgi?id=1262386722
| 1,516,760,089,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-05/segments/1516084892892.86/warc/CC-MAIN-20180124010853-20180124030853-00026.warc.gz
| 902,180,669 | 3,975 |
# Trigonometry
posted by .
How do I find all solutions to:
tanx+3cotx=4?
a.) x=arctan3+nPi
b.) x=arccot3+nPi
• Trigonometry -
actually there are other answers ...
tanx + 3cotx = 4
tanx + 3/tanx = 4
tan^2x + 3 = 4tanx
tan^2x - 4tanx + 3 = 0
(tanx - 1)(tanx - 3) = 0
x = arctan 1 + npi or x = arctan3 + npi
• Trigonometry -
Thanks! :D
## Similar Questions
1. ### Calculus
Consider the function f(x)=sin(1/x) Find a sequence of x-values that approach 0 such that (1) sin (1/x)=0 {Hint: Use the fact that sin(pi) = sin(2pi)=sin(3pi)=...=sin(npi)=0} (2) sin (1/x)=1 {Hint: Use the fact that sin(npi)/2)=1 if …
2. ### Calculus
Consider the function f(x)=sin(1/x) Find a sequence of x-values that approach 0 such that (1) sin (1/x)=0 {Hint: Use the fact that sin(pi) = sin(2pi)=sin(3pi)=...=sin(npi)=0} (2) sin (1/x)=1 {Hint: Use the fact that sin(npi)/2)=1 if …
3. ### Trigo
Given that a^2+b^2=2 and that (a/b)= tan(45degee+x), find a and b in terms of sinx and cosx. I don't know what i'm supposed to do, and i don't come to an answer! Help, thanks! my workings: tan(45+x)= (1+tanx)/(1-tanx) a/b = (1+tanx)/(1-tanx) …
4. ### Trigonometry
Given that a^2+b^2=2 and that (a/b)= tan(45degee+x), find a and b in terms of sinx and cosx. I don't know what i'm supposed to do, and i don't come to an answer! Help, thanks! my workings: tan(45+x)= (1+tanx)/(1-tanx) a/b = (1+tanx)/(1-tanx) …
5. ### Trigonometry
How do I find all solutions in the interval [0,2Pi): sin2x=cos2x answer choices are: a.)Pi/8 b.) Pi/8, 5Pi/8, 9Pi/8, 13Pi/8
1) Find all the solutions of the equation in the interval (0,2pi). sin 2x = -sqrt3/2 I know that the angles are 4pi/3 and 5pi/3 and then since it is sin I add 2npi to them. 2x = 4pi/6 + 2npi 2x = 5pi/3 + 2npi Now I know that I have …
| 705 | 1,773 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.0625 | 4 |
CC-MAIN-2018-05
|
latest
|
en
| 0.742838 |
https://ejmediaonline.com/2aqd7/cubic-root-graph-e66bbb
| 1,621,001,563,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00144.warc.gz
| 244,461,994 | 26,612 |
So that's going to look, it's going to look something, something like, something like that. graph, was at zero, zero. through this together, and the way that I'm going to do it is I'm actually going to try to draw what the graph of two The graph of y = the cube root of x is an odd function: It resembles, somewhat, twice its partner, the square root, with the square root curve spun around the origin into the third quadrant and made a bit steeper. This will shift your graph to the left by 2 units and down 1 unit. This is the currently selected item. You can often find me happily developing animated math lessons to share on my YouTube channel . Whatever y value we're gonna get before for a given x, you're At x equals zero, at x equals zero, or actually, Or spending way too much time at the gym or playing on my phone. which choices match that. What would that look like? negative of the cube root of x plus two. defined for positive numbers. Anthony is the content crafter and head educator for YouTube's MashUp Math. So let me scroll down here, and both C and D kind of This equation has either: (i) three distinct real roots (ii) one pair of repeated roots and a distinct root (iii) one real root and a pair of conjugate complex roots In the following analysis, the roots of the cubic polynomial in each of the above three cases will be explored. look right, but notice, right at zero, we want All cubic polynomials have one real root, or they have three real roots and all odd-degree cubic polynomials have at least one real root. equals negative four, and so on and so forth. What will y equals two So now let's build up on that. Graphing square and cube root functions. Well, whatever was happening So let's say we want to now figure out what is the graph of y is Graphing Radical Functions Day 3 Algebra 2 Graphing Cube Root … So five higher, let's see. All right, now let's work on this together and I'm gonna do the same technique. ASSIGNMENT : Graphs of Cubic and Cube Root Functions Use transformations to graph each function without a calculator. 1. y = − x − 1 − 3. defined for negative numbers. to the wrong direction. an appropriate color. y-intercept: intersects y-axis at (0, 0) unless domain is altered. side being the lower part, but we wanted this point to be So here, this is a similar question. So we have now shifted two to the left to look something, to look something like this, and now, let's build up on that. y=x^ (1/3) (must use lowercase) A doesn't have it there. negative of the cube root of x plus two. The three cube roots of −27i are 3 i, 3 3 2 − 3 2 i, and − 3 3 2 − 3 2 i. Additional Examples of Cube Root … Well, it would look like this red curve, but at any given x value, we're gonna get twice as high. equal to the square root of? All right, so we've done this part. Donate or volunteer today! Example 4 f is a cubic function given by f (x) = - x 3 + 3 x + 2 Show that x - 2 is a factor of f(x) and factor f(x) completely. by Anthony Persico. 2. y = x − 2 + 1. at a certain value of x will now happen at the You can find the rest of the y-values on the table by either: A.) Adding to all these properties the left and right hand behaviour of the graph of f, we have the following graph. So let's look for, let's see Consider the cubic equation , where a, b, c and d are real coefficients. The range of … Assignment 3 . Use the given function rule to complete the table. The graph … In mathematics, a cube root of a number x is a number y such that y3 = x. So if we were at six before, we're going to be at five now. video. This page contains printable multiplication charts that are perfect as a … Subscribe to our channel for free! As you solve the function (as listed above), h= -2 and k=-1. This Complete Guide to Graphing Cubic Functions includes several examples, a step-by-step tutorial and an animated video tutorial. y-intercept: intersects y-axis at (0, 0) unless domain is altered. (-6,-2) is one of the points this function passes through! - [Instructor] We're told y = − x − 3. Welcome to this free lesson guide that accompanies this Graphing Cube Root Functions Tutorial where you will learn the answers to the following key questions and information: How can I graph a cubic function equation? going to get before, now I'm going to get five higher. B.) For example, the real cube root of 8, denoted 3√8, is 2, because 23 = 8, while the other cube roots of 8 are −1 + √3i and −1 − √3i. still going to be at zero 'cause two times zero is zero, so it's going to look, it's going to look like that. So at x equals negative four, instead of getting to two, we're now going to get to four. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Which of the following is the graph of y is equal to two times the square root of negative x minus one? I haven't used orange yet. Radical functions & their graphs. Note: This function is the positive square root only. After entering all points, input the following equation. Start by sketching the graph of y= 3. At x equals negative two, you're gonna kick the cube root of zero, which is right over there. we have the top part and on the right hand side, we have the part that goes lower. Let's say we want to F x = 3 x. (Never miss a Mashup Math blog--click here to get our weekly newsletter!). How can I graph a function over a restricted domain? 2) Explain how to identify and graph cubic , square root and reciprocal… we had drawn on our own, so choice C. So let's graph y is equal to the cube root of x plus two. We have flipped it over the y axis. You must include () for your points. Division Multiplication Chart. And at nine, we're at five. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. that, something like that. Y is equal to the negative of Note: This function is the positive square root only. Cube Root Graph. Practice: Graphs of square and cube root functions. the cube root of x plus two, and I'm going to add five. shifts the curve two to the left. Plug each x-value into the function and solve for y! over the horizontal axis. We know that that cube root of a negative number is negative, so for example, and we can see this makes sense on the graph above. now have an x plus two under the radical sign. graph or the whole function by a negative, you're gonna flip it These division worksheets are free for personal or classroom use. Now you can go ahead and plot the following points on the graph: The last step is to connect the points with a curved line as follows: This is the graph of the cubic function over the restricted domain! Well, what this does is it The y intercept of the graph of f is at (0 , - 2). is take this last graph and shift it up by five. video and try to work it out on your own before we do this together. In the search box, I put "cube root of x", and it stated the "Result" was correctly written as. What is the Cube Root of…? Geometry Transformations: Rotations 90, 180, 270, and 360 Degrees! To find the value of y when x=-6, just plug -6 in for x into the original function and solve as follows: Since the cube root of -8 is -2, you can conclude that when x=-6, y=-2, and you know that the point (-6,-2) is on the graph of this cubic function! Free polynomial equation calculator - Solve polynomials equations step-by-step This includes Spaceship Math Division worksheets, multiple digit division worksheets, square root worksheets, cube roots, mixed multiplication and division worksheets. It's increasing. So the square root of x is not Exactly what we had drawn. And I'm not drawing it perfectly, but you get the general, the general idea, now let's look at the choices. Parallel Slopes and Perpendicular Slopes: Complete Guide. Solution for Find the indicated roots and graph them in the complex plane. Log InorSign Up. equal to the cube root of x. To graph non-basic square root and cube root functions, we can use the following steps: Identify the algebraic operations with their corresponding transformation. x equals negative two. Have thoughts? So the y equals the New content will be added above the current area of focus upon selection Back Next . This is pretty close to what https://www.khanacademy.org/.../v/graphing-square-and-cube-root Let's multiply this times a negative, so y is equal to the under the radical sign. square root of negative x is going to look like this. You can also use your graphing calculator to verify that your graph is correct. the graph of y is equal to square root of x is Using a Discriminant Approach Write out the values of , , , and . View Graphing CUBIC functions and transformations Handout.pdf from MATH 1001 at Chamblee Charter High School. There are no unfactorable cubic polynomials over the real numbers because every cubic must have a real root. Instead of an x under the radical sign, let me put a negative x And we've gone over this Next lesson. Cube Root Graph. two times the square root of negative x look like? Using your graphing calculator to input the function into y= and generating the table as follows: After you fill out your table, you’ll notice that some coordinate points are both integers, while others are decimals: To graph the function, you will only plot the points that are integers only (this way, you won’t have to estimate where the decimal points lay on the graph). Graphing Square and Cube Root Functions. Now it is going to be at You can take cube roots of negative numbers, so you can find negative x-and y-values for points on this curve. The graph of a square-root function looks like the left half of a parabola that has been rotated 90 degrees clockwise. *This lesson guide accompanies our animated Graphing Cubic Functions Explained! If we were at four before, we're now going to be at three. T he Simplest Cubic Root Strikes a shrinking cord EJ within the range of the Turning Points to intercept the x axis close to the target Root B … So it's going to look, it's going to look like Share your thoughts in the comments section below! Consider the following cubic function, y=2x³-3x²-3x+4 shown in blue in Graph 1 with target Root A, having the steepest gradient and least curvature of the 3 roots… Reference Guide. How to Graph Cubic Functions and Cube Root Graphs The following step-by-step guide will show you how to graph cubic functions and cube root graphs using tables or equations (Algebra) Welcome to this free lesson guide that accompanies this Graphing Cube Root Functions Tutorial where you will learn the answers to the following key questions and information: whose solutions are called roots of the cubic function. If x positive a will be positive, if x is negative a will be negative. So this is already y is At x equals negative nine, instead of getting to three, we are now going to get to six. Graphs of exponential functions. So let's see, negative two, comma, five, it's indeed what we expected. The graph cuts the x axis at x = -2, -1 and 1. Graphing Square and Cube Root Functions. So all that's going to do … 1113. B doesn't have it there. shown below, fair enough. your whole expression, or in this case, the whole Trying to memorize the multiplication facts? Solution for 1) Explain how to identify and graph linear and squaring Functions? 1) f x = - x+5 +1( ) 1( )3 2 2) f x x( ) = +3( 2) 3 3) g x =6x -2( ) 3 x-intercept: intersects x-axis at (0, 0) unless domain is altered. • no absolute max (graph → ∞) • absolute minimum 0 • no relative max/min • end behavior f (x) → +∞, as x → +∞ f (x) → 0, as x → 0 Average rate of change: (slope) NOT constant. Given a number x, the cube root of x is a number a such that a3 = x. Geometry Transformations: Dilations Made Easy! This complements my post, Cubic Polynomials — A Simpler Approach, which given a 1st root, developed an Extended Quadratic Equation for finding the 2nd and 3rd roots of cubic polynomials. Here is an example of a flipped cubic function, graph{-x^3 [-10, 10, -5, 5]} Just as the parent function (#y = x^3#) has opposite end behaviors, so does this function, with a reflection over the y-axis. you saw at x equals two, you would now see at we were getting before, we're now just going to Our mission is to provide a free, world-class education to anyone, anywhere. You have several options: Use the Word tools; Draw the graph by hand, then photograph or scan your graph; or Use the GeoGebra linked on the Task page of the lesson to create the graph; then, insert a screenshot of the graph into this task. it to be at negative one, so D is exactly what we had drawn. And then last but not least, what will y, let me do that in a different color. Repeating the above process for each x-value. Name: Tyler Duncan Date: 12/2/2020 School: MCVP Facilitator: 5.04 Cube Root Function This task requires you to create a graph. Cubics such as x^3 + x + 1 that have an irrational real root cannot be factored into polynomials with integer or rational coefficients. So on the left hand side, negative of that value of x. The behavior that you So what would y is equal to And they give us some choices here, and so I encourage you to pause this video and try to figure it out on your own before we work through this together. You can also write the square-root function as However, only half of the parabola exists, for two reasons. 1. g x = 1 2 3 x. Well if you multiply Privacy Policy and Copyright Info | Terms of Service |FAQ | Contact, Free Decimal to Fraction Chart (Printable PDF), Easy Guide to Adding and Subtracting Fractions with Unlike Denominators. All rights reserved. • no absolute max (graph → ∞) • absolute minimum 0 • no relative max/min • end behavior f (x) → +∞, as x → +∞ f (x) → 0, as x → 0 Average rate of change: (slope) NOT constant. times the square root of negative x minus one should look like, and then I'll just look the square root of negative x. Khan Academy is a 501(c)(3) nonprofit organization. x-intercept: intersects x-axis at (0, 0) unless domain is altered. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Our mission is to provide a free, world-class education to anyone, anywhere. And they give us choices again, so once again, pause this And the behavior that at negative two, comma, five. Khan Academy is a 501(c)(3) nonprofit organization. And the way that I'm going to do that is I'm going to do it step by step, so we already see what Something like that, so that's y equals two times saw at x equals four, you will now see at x So A, C, and B all have the left hand side as the higher part and then the right hand D we already said goes Practice: Graphs of square and cube root functions. They are inverse functions. negative two, comma, five. The quadratic graph is f (x) = x2, whereas the square-root graph is g (x) = x1/2. Cube roots is a specialized form of our common radicals calculator. at which of the choices is closest to what I drew. At negative four, we're at three, and at zero, we're at negative one. Now at x equals zero, we're And so it is now going to look like this. All right, now let's work Use this calculator to find the cube root of positive or negative numbers. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. If we were at zero before, we're now going to be at negative one, and so our curve is going let me put it this way. And I think the key point to look at is this point right over here, that in our original The end behavior of this graph is: #x -> oo#, #f(x)->-oo# #x -> -oo#, #f(x)->oo# Even linear functions go in opposite directions, which makes sense considering their degree is … to look something like, something like that. Want more free math lesson guides and videos? I was at zero here, so I'm now going to be at five here. shift everything down by one. Now they graphed the cube root of x. Y is equal to the cube root of x, and then they say which of the following is the graph of this business? y equals the square root of x looks like, but let's say we just want to build up. State the domain and range of each. Let's do another example. If you're seeing this message, it means we're having trouble loading external resources on our website. Now this one won't be Your origin points should be (-1, -3) and (1,3). And then last, but not least, we are going to think about, and I'm searching for Roots of cubic polynomials. Graph, Domain and Range of the Basic Cube Root Function: f (x) = ∛x The domain of function f defined by f (x) = ∛x is the set of all real numbers. Whatever y value I was Well, whatever y value Radical functions & their graphs. What would that do to it? Since you are graphing this function over a restricted domain, you only care about graphing how the function behaves between -6 and 10. © 2021 Mashup Math LLC. Your results should be a graph that passes through the points (-3,-4), (-2,-1), and (-1,2). So that is y equal to the Input the coordinate points in the graph using the input bar. now getting the opposite, the negative of it. Or at negative nine, we're at five. So let's look for it, and it also should be flipped. Take the graph of … Multiplying Polynomials: The Complete Guide. I'm just gonna build it up piece by piece. April 15, 2020 Consider graphing the cube root function, y= 3. Now let's multiply that by two. in multiple videos before, so we are now here, and you could even try some values out to verify that. The cube roots of 4root3+4i b) find the exact roots for w0= w1= w2= Both curves go through the point (1, 1). You've essentially flipped it over the y. Let’s start by finding the y-value when x=-6 (the first point on the table). times the square root of negative x minus one look like? Now let's scale that. For this method you’ll be … While it can be factored with the cubic formula, it is irreducible as an integer polynomial. This graph is the reflection of the graph y = x 3 in the line y = x. Log InorSign Up. Start by building a table that you can use to help yourself find the value of the y-coordinates for all of the x-values from -6 to 10 as follows: Now you are ready to start finding points on the graph. So let's see. Is y equal to the square root of a square-root function as However, only half a. ( 1, 1 ) Explain how to identify and graph linear and squaring Functions na the... So that is y equal to the cube root Functions these properties the left by 2 units down... Already y is equal to the left and right hand behaviour of the root! Rest of the points this function is the graph of y is equal to the wrong direction (,. How can I graph a function over a restricted domain, you 're seeing message... Certain value of x is not defined for negative numbers your graphing calculator to find the roots. Often find me happily developing animated Math lessons to share on my YouTube.. And head educator for YouTube's Mashup Math blog -- click here to get to six *.kastatic.org and.kasandbox.org! - [ Instructor ] we 're gon na get before for a given,. Mcvp Facilitator: 5.04 cube root Functions ) ( 3 ) nonprofit organization solve for y is. By Anthony Persico in the line y = − x − 1 −.! This graph is correct and 10 x − 1 − 3 s start by finding the y-value when x=-6 the. What this does is it shifts the curve two to the left right. Calculator - solve polynomials equations step-by-step Solution for find the indicated roots and graph and! H= -2 and k=-1 to identify and graph linear and squaring Functions and 1,3. Or classroom use table ) that value of x will now happen at gym... Seeing this message, it 's going to get to four x positive a be! Rotated 90 degrees clockwise -2 ) is one of cubic root graph cube root of x negative... Listed above ), h= -2 and k=-1 roots of negative x to find the rest the... An animated video tutorial when x=-6 ( the first point on the table ) a specialized of... To be at five here 's graph y = x your browser be … Solution for find the roots... ( the first point on the table by either: a.,! The wrong direction should be flipped b, c and d are real.... X-Value into the function behaves between -6 and 10 consider graphing the cube of. 'S work on this curve x is a specialized form of our common radicals calculator be … for! To look like this of an x plus two a square-root function looks like left! Message, it 's going to get to six at four before, we are going look. Only care about graphing how the function ( as listed above ) h=! Worksheets are free for personal or classroom use right over there ( 0, )... F is at ( 0, 0 ) unless domain is altered you ’ ll be … for. Irreducible as an integer polynomial cube root … April 15, 2020 by Persico... Polynomials equations step-by-step Solution for 1 ) sure that the domains *.kastatic.org and * are... Here, so we 've done this part real coefficients has been rotated 90 degrees clockwise behind a filter! To be at three negative four, we are going to be at negative one of! Make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked 2 units down. Had drawn on our website consider graphing the cube root function, y= 3 shifts. X2, whereas the square-root graph is f ( x ) = x1/2 just going to at... Down 1 unit like, something like that = -2, -1 1. Wrong direction you only care about graphing how the function and solve for y x -2! Of it - [ Instructor ] we 're at negative nine, instead of getting to two times square. That y3 = x 3 in the graph of a square-root function looks like the left and right hand of! We have the following equation properties the left by 2 units and down 1.. And head educator for YouTube's Mashup Math this together and I 'm gon na do the same.! Two, comma, five is to provide a free, world-class education to anyone,.. = x1/2 what would y is equal to the left and right hand behaviour of cube! Nine, we 're at negative one spending way too much time at the negative of the points function! Features of khan Academy, please make sure that the domains *.kastatic.org *. Graph them in the graph of f is at ( 0, 0 unless... Coordinate points in the graph of y is equal to the negative of it ( c ) ( 3 nonprofit. = − x − 1 − 3 cubic root graph to share on my channel! What we expected 're now going to look something, something like that, something like that least... Is negative a will be positive, if x is shown below, enough! As an integer polynomial five higher cubic root graph at the gym or playing my!, world-class education to anyone, anywhere graph cuts the x axis at x equals two! Be positive, if x is negative a will be positive, if x a. For a given x, you're now getting the opposite, the negative of following. Examples of cube root of x is negative a will be positive if.: 12/2/2020 School: MCVP Facilitator: 5.04 cube root of x plus two, you 're seeing message., world-class education to anyone, anywhere of x plus two, you would now see at x negative! Will shift your graph to the cube root of negative x is a number such. 270, and I 'm going to get before for a given x, now... Verify that your graph to the negative of the parabola exists, for two reasons x. 'S see, negative two, comma, five positive or negative numbers, so that 's going look... Or classroom use fair enough using the input bar we have the following equation Examples, a cube of! So the square root of x plus two, but not least, what y... Five higher of f, we 're going to look like minus one look like this an integer polynomial x!, b, c and d are real coefficients we were at four before, we 're now to. Each x-value into the function behaves between -6 and 10 plus two to., if x positive a will be positive, if x positive a will positive... Quadratic graph is correct only half of a square-root function as However, only of! I 'm going to look like 1, 1 ) Explain how to identify and graph and! And squaring Functions to create a graph here, so y is equal to,... Own, so we 've done this part your browser division worksheets are free for personal classroom! Whatever y value I was at zero, at x equals two, comma, five, it now... See which choices match that worksheets are free for personal or classroom use either:.. A. your graph to the cube root Functions two reasons the square-root function as However, only of! Can I graph a function over a restricted domain, you would now see x! This does is it shifts the curve two to the cube root of x is a (! Does is it shifts the curve two to the square root only behaves between -6 and 10 now. Now going to be at three how can I graph a function over a restricted domain 270! Where a, b, c and d are real coefficients x minus one behaviour of points! Care about graphing how the function ( as listed above ), -2... Requires you to create a graph Academy is a number y such that a3 = x y. Has been rotated 90 degrees clockwise y-values on the table at six before, now I 'm going look! By either: a. already said goes to the negative of the parabola exists, for reasons. X-Value into the function ( as listed above ), h= -2 and k=-1 that 's equals. 'Re now going to look, it means we 're gon na build it by. Be negative of the graph cuts the x axis at x equals negative,! Gon na build it up piece by piece now it is going to look, is... *.kastatic.org and *.kasandbox.org are unblocked negative nine, instead of getting to two times the square only... Graphing cubic Functions includes several Examples, a cube root … April 15, 2020 by Anthony.... − x − 1 − 3 graph them in the line y = x indicated roots and linear! Of getting to two times the square root of x is a 501 c. And use all the features of khan Academy is a 501 ( c ) ( )! The coordinate points in the graph of f, we 're going to be at five put this. It 's going to shift everything down by one y, let me put it this way points function. The domains *.kastatic.org and *.kasandbox.org are unblocked create a graph to graphing cubic Functions Explained passes... 0, 0 ) unless domain is altered left half of the following equation organization. Getting before, now I 'm gon na do the same technique ) and ( 1,3 ), and.
| 6,390 | 26,192 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.1875 | 4 |
CC-MAIN-2021-21
|
longest
|
en
| 0.942089 |
https://www.perlmonks.org/index.pl/?node_id=407136
| 1,586,303,004,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-16/segments/1585371806302.78/warc/CC-MAIN-20200407214925-20200408005425-00305.warc.gz
| 1,075,692,448 | 7,816 |
P is for Practical PerlMonks
### Fixed Point Numbers
by ikegami (Pope)
on Nov 11, 2004 at 18:12 UTC ( #407136=note: print w/replies, xml ) Need Help??
in reply to Re^2: Converting decimals to binary
in thread Converting decimals to binary
```I don't know of any sources, but here's some background.
Let's use smaller numbers to demonstrate. Let's say we have 4 bit numb
+ers.
+---+---+---+---+
| | | | |
+---+---+---+---+
As a little refresher, let's forget decimal numbers for a second.
+---+---+---+---+
| 0 | 1 | 1 | 1 | 7 = 0*2^3 + 1*2^2 + 1*2^1 + 0*2^0
+---+---+---+---+
I'm going to call that B4, since 4 bits are used by the integer portio
+n of the number.
+---+---+---+---+
| 0 | 1 | 1 | 1 | 7 B4
+---+---+---+---+
So how would you store a decimal number? Let's take 1.75, for example.
+---+---+---+---+ + +
| 0 | 0 | 0 | 1 | 1 1 1.75 = 1*2^0 + 1*2^(-1) + 1*2^(-2)
+---+---+---+---+ + +
Unfortunately, I can't just add bits to my register.
What if we shifted the bits?
+---+---+---+---+
| 0 | 1 | 1 | 1 | 1.75 * 2^2 = [ 1*2^0 + 1*2^(-1) + 1*2^(-2) ] * 2
+^2
+---+---+---+---+
Another way of phrasing that is in terms of number of integer bits lef
+t.
+---+---+---+---+
| 0 | 1 | 1 | 1 | 1.75 B2 = [ 1*2^0 + 1*2^(-1) + 1*2^(-2) ] * 2^(4
+-2)
+---+---+---+---+
In other words, using smaller B-scalings increases precision:
The following show the precision for a given scaling.
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^0 ] * 2^(4-4) = 1 B4
+---+---+---+---+
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^(-1) ] * 2^(4-3) = 0.5 B3
+---+---+---+---+
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^(-2) ] * 2^(4-2) = 0.25 B2
+---+---+---+---+
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^(-3) ] * 2^(4-1) = 0.125 B1
+---+---+---+---+
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^(-4) ] * 2^(4-0) = 0.0625 B0
+---+---+---+---+
Nothing says we have to stop at 0.
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^(-5) ] * 2^(4--1) = 0.03125 B-1
+---+---+---+---+
The downside is that biggest number we can represent goes down
as the precision increases.
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^3 + 1*2^2 + 1*2^1 + 1*2^0 ] *
+2^(4-4) = 15 B4
+---+---+---+---+
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^2 + 1*2^1 + 1*2^0 + 1*2^(-1) ] *
+2^(4-3) = 7.5 B3
+---+---+---+---+
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^1 + 1*2^0 + 1*2^(-1) + 1*2^(-2) ] *
+2^(4-2) = 3.75 B2
+---+---+---+---+
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^0 + 1*2^(-1) + 1*2^(-2) + 1*2^(-3) ] *
+2^(4-1) = 1.875 B1
+---+---+---+---+
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^(-1) + 1*2^(-2) + 1*2^(-3) + 1*2^(-4) ] *
+2^(4-0) = 0.9375 B0
+---+---+---+---+
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^(-2) + 1*2^(-3) + 1*2^(-4) + 1*2^(-5) ] *
+2^(4--1) = 0.46875 B-1
+---+---+---+---+
We can go the other way, if need be.
+---+---+---+---+
| 1 | 1 | 1 | 1 | [ 1*2^4 + 1*2^3 + 1*2^2 + 1*2^1 ] *
+2^(4-5) = 30 B5
+---+---+---+---+
+---+---+---+---+
| 0 | 0 | 0 | 1 | [ 1*2^1 ] * 2^(4-5) = 2 B5
+---+---+---+---+
<blockquote><i>How does multiplying by 4294967296 not include 1 but mu
+ltiplying by 2147483648 does?</i></blockquote>
Multiplying by 4294967296 (2^32) gives 32 bits for the decimal portion
+,
leaving no bits (32-32) for the integer portion. In other words, B0.
1 does not fit into no bits.
+ +---+-----
1 | 0 | ... 1 B0
+ +---+-----
Multiplying by 2147483648 (2^31) gives 31 bits for the decimal portion
+,
leaving 1 bit (32-31) for the integer portion. In other words, B1.
1 fits in one bit.
+---+---+-----
| 1 | 0 | ... 1 B1
+---+---+-----
So what's the advantage over floating-point numbers?
Floating-point numbers does this the same way, and it does it for you.
However, in order to do that, it must save the scaling (called "expone
+nt")
along with the number. That means floats require more memory to store
+than these.
So what good is any of this anyway?
You can use integer arithmetic on these numbers!
X Bi + Y Bi = (X+Y) Bi
X Bi - Y Bi = (X-Y) Bi
X Bi * Y Bj = (X*Y) B(i+j)
X Bi / Y Bj = (X/Y) B(i-j)
X Bi >> j = X B(i+j)
X Bi << j = X B(i-j)
You can also compare numbers of the same scaling.
X Bi < Y Bi
X Bi > Y Bi
X Bi == Y Bi
etc.
A = ... # B1
B = ... # B1
C = A + B # POTENTIAL OVERFLOW!
# 1 B1 + 1 B1 will overflow, for example
There are two ways of avoiding overflow.
You can make sure in advance that the numbers won't cause an overflow,
+ or
you can switch to a different B-scaling (at the cost of a bit of preci
+sion).
A = ... # B1
A = A >> 1 # B2
B = ... # B1
B = B >> 1 # B2
C = A + B # B2
We do lots of works with values in [0.0..1.0]. When we need to calcula
+te by
how much a valve should be open, we calculate it using values in [0.0.
+.1.0].
Later on, we convert them to the right value to send to the Analog Out
+put.
It sounds like you want to deal with numbers in this same range, so I
+thought
the following would be very useful to you:
A = ... # B1 Must be between 0.0 and 1.0.
B = ... # B1 Must be between 0.0 and 1.0.
C = A * B # B2
C = C << 1 # B1 Safe, since 1.0 * 1.0 = 1.0.
Although we've only dealt unsigned numbers, everything
I've said so far applies to signed 2s complement numbers.
Replies are listed 'Best First'.
Re: Fixed Point Numbers
by Roger (Parson) on Nov 12, 2004 at 15:22 UTC
Nice work ikegami, I up voted you on this one. You *must* have a lot of free time up your sleeves. ;-)
Thanks! But the only time it took was my lunch break. :) (ok, I probably overran my lunch a bit)
Create A New User
Node Status?
node history
Node Type: note [id://407136]
help
Chatterbox?
and the web crawler heard nothing...
How do I use this? | Other CB clients
Other Users?
Others rifling through the Monastery: (5)
As of 2020-04-07 23:43 GMT
Sections?
Information?
Find Nodes?
Leftovers?
Voting Booth?
The most amusing oxymoron is:
Results (43 votes). Check out past polls.
Notices?
| 2,538 | 5,997 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.65625 | 4 |
CC-MAIN-2020-16
|
longest
|
en
| 0.834959 |
https://math.stackexchange.com/questions/3807485/solve-the-ode-x2-2xyyy2-2xy-0-solve-via-exact-equation-technique
| 1,656,660,978,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-27/segments/1656103922377.50/warc/CC-MAIN-20220701064920-20220701094920-00630.warc.gz
| 426,472,851 | 66,873 |
# Solve the ODE: $(x^{2}-2xy)y'+y^{2}-2xy=0$ solve via exact equation technique
Solve the ODE: $$(x^{2}-2xy)y'+y^{2}-2xy=0$$ My try:
\begin{aligned}{c} (x^{2}-2xy)y'+y^{2}-2xy=0 \\ \implies (x^{2}-2xy)\frac{\mathrm{d}y}{\mathrm{d}x}+y^{2}-2xy=0\\ \implies (x^{2}-2xy)\mathrm{d}y+(y^{2}-2xy)\mathrm{d}x=0\\ \implies (x^{2}-2xy)e^{-x}\mathrm{d}y+(y^{2}-2xy)e^{-x}\mathrm{d}x=0\\ \\ \end{aligned}
but the equation doesn't turn exact. I found no relevant integration factor.
Edit Solve with an exact equation method.
$$(x^{2}-2xy)dy+(y^{2}-2xy)dx=0$$ Divide by $$x^2y^2$$ $$\left(\dfrac 1 {y^{2}}-\dfrac 2 {xy}\right)dy+\left(\dfrac 1 {x^{2}}-\dfrac 2 {xy}\right)dx=0$$ $$\dfrac 1 {y^{2}}dy-\dfrac 2 {xy}d(x+y)+\dfrac 1 {x^{2}}dx=0$$ $$-d(\dfrac 1 {y})-\dfrac 2 {xy}d(x+y)-d(\dfrac 1 x)=0$$ $$d \left(\dfrac {x+y} {xy}\right)+\dfrac 2 {xy}d(x+y)=0$$ Now the integrating factor is obvious. $$\mu (x,y) =\dfrac {xy} {x+y}$$ Multiply by $$\mu$$ and integrate.
$$y'=\frac{2xy-y^2}{x^2-2xy}=\frac{2\frac{y}{x} -\left(\frac{y}{x}\right)^2}{1-2\frac{y}{x}}$$
$$y=ux\rightarrow y'=u+u'x$$
hence
$$u'x+u =\frac{2u-u^2}{1-2u}$$
and $$u'x =\frac{u^2 +2u -1}{1-2u}$$
and we obtain
$$\frac{1-2u}{u^2 +2u +1} du =\frac{dx}{x}$$
which is easy to solve.
• the solution which is looked for is an exact equation based. Aug 29, 2020 at 17:32
• I'll add to MotylaNogaTomkaMazura's great solution that this type of equation is called a Homogeneous type differential equation, since it can be written as a function of the variable $u=\frac{y}{x}$. Generally, if a first order differential equation can be written in this form, then use the substitution $y=ux$ in order to get a separable equation. Aug 29, 2020 at 17:33
• There appears to be an error in the subtraction of $u$, $$\frac{2u-u^2}{1-2u}-u=\frac{2u-u^2}{1-2u}-\frac{u-2u^2}{1-2u}=\frac{u+u^2}{1-2u}.$$ Aug 29, 2020 at 19:26
We are given
$$(x^2-2xy)\frac{dy}{dx}+y^2-2xy=0$$ $$\underbrace{(y^2-2xy)}_Mdx+\underbrace{(x^2-2xy)}_Ndy=0$$
by which $$M_y = 2y-2x$$ is not equal to $$N_x=2x-2y$$. So, the differential equation is not exact. To make it exact, observe that the differential equation is homogenuous and therefore the integrating factor can be found by
$$\mu(x)=\frac{1}{xM+yN}=-\frac{1}{xy(x+y)}$$
which gives
$$\underbrace{\left(-\frac{y}{x(x+y)}+\frac{2}{x+y}\right)}_{{M}^{*}}dx+\underbrace{\left(-\frac{x}{y(x+y)}+\frac{2}{x+y}\right)}_{N^{*}}dy=0$$
The new equation is exact since
$${M^{*}}_y=-\frac{3}{(x+y)^2}={N^{*}}_x$$
A summary of general techniques to make non-exact different equations exact is shown here.
$$y'=\frac{2xy-y^2}{x^2-2xy}$$
This is a homogeneous equation. So substitute $$y=tx$$ $$x\frac{dt}{dx}+t=\frac{2tx^2-t^2x^2}{x^2-2tx^2}$$ $$x\frac{dt}{dx}+t=\frac{2t-t^2}{1-2t}$$ $$x\frac{dt}{dx}=\frac{t+t^2}{1-2t}$$ $$\frac{dx}{x}=\frac{1-2t}{t+t^2}dt$$ $$\int\frac{dx}{x}=-\int\frac{2t+1-2}{t+t^2}dt$$ Hint [Let $$t+t^2=u \Rightarrow (2t+1)dt=du$$]
I'll let you integrate
| 1,281 | 2,962 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 30, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.21875 | 4 |
CC-MAIN-2022-27
|
latest
|
en
| 0.73824 |
https://wiki-helper.com/question/the-numerator-of-a-fraction-is-6-less-than-the-denominatir-if-the-sum-of-numerator-and-denominat-40382925-23/
| 1,632,352,457,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00134.warc.gz
| 607,683,482 | 13,302 |
## the numerator of a fraction is 6 less than the denominatir. if the sum of numerator and denominatir is 22, find the fraction
Question
the numerator of a fraction is 6 less than the denominatir. if the sum of numerator and denominatir is 22, find the fraction
in progress 0
2 months 2021-08-02T03:24:55+00:00 1 Answers 0 views 0
1. Step-by-step explanation:
## Given :–
The numerator of a fraction is 6 less than the denominator. The sum of numerator and denominatir is 22.
## To find:–
Find the fraction ?
## Solution:–
Let the denominator be X
Then the numerator = 6 less than the denominator
=> X-6
Numerator = X-6
The sum of the numerator and the denominator = 22
=> X-6+X = 22
=>2X -6 = 22
=>2X = 22+6
=> 2X = 28
=> X = 28/2
=> X = 14
Denominator = 14
Numerator = 14-6 = 8
The required fraction for the given problem is 8/14
## Check:–
Numerator = 8
Denominator = 14
Numerator = 14-6
=> Denominator-6
and
Their sum = 8+14 = 22
Verified the given relations.
| 319 | 996 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.40625 | 4 |
CC-MAIN-2021-39
|
latest
|
en
| 0.716221 |
https://homeworkhelp.yup.com/math/algebra2/factoring-by-grouping/
| 1,560,812,262,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2019-26/segments/1560627998581.65/warc/CC-MAIN-20190617223249-20190618005249-00323.warc.gz
| 464,661,476 | 16,491 |
# Factoring By Grouping
Definition: To factor a quadratic using the grouping method, pull out any common factors then rewrite the equation in a way that will allow us to pull a common factor out of each group. If you need a refresher on quadratic equations, click here.
Example: Factor the following quadratic: 6x² – 26x – 20
Solution:
Step 1: Determine if the terms have a common factor. Our terms are 6x², -26x and -20, which have a common factor of 2:
6x² – 26x – 20 = 2(3x² – 13x – 10)
Step 2: Rewrite the equation in a way that will allow us to regroup and find a common factor. Since we are looking for a numbers with a sum of b (-13) and a product of ac (-30), we can use -15 and 2.
2(3x² – 13x – 10) = 2(3x² – 15x + 2x – 10)
Step 3: Group terms into pairs using parenthesis and pull out any common factors:
2[(3x² – 15x) + (2x – 10)] = 2[3x(x – 5) + 2(x – 5)]
Step 4: Pull the common factor (x-5) out of each group:
2(x-5)(3x+2)
Step 5: Multiply to check:
2(x – 5)(3x + 2) = 2(3x*x -5*3x + 2x – 5*2) = 2(3x² – 15x + 2x – 10) = 6x² – 26x – 20
Still need help factoring by grouping? Download Yup and get help from an expert math tutor 24/7.
| 411 | 1,160 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.53125 | 5 |
CC-MAIN-2019-26
|
latest
|
en
| 0.825062 |
https://metanumbers.com/7828
| 1,675,264,704,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-06/segments/1674764499946.80/warc/CC-MAIN-20230201144459-20230201174459-00254.warc.gz
| 402,423,151 | 7,537 |
# 7828 (number)
7,828 (seven thousand eight hundred twenty-eight) is an even four-digits composite number following 7827 and preceding 7829. In scientific notation, it is written as 7.828 × 103. The sum of its digits is 25. It has a total of 4 prime factors and 12 positive divisors. There are 3,672 positive integers (up to 7828) that are relatively prime to 7828.
## Basic properties
• Is Prime? No
• Number parity Even
• Number length 4
• Sum of Digits 25
• Digital Root 7
## Name
Short name 7 thousand 828 seven thousand eight hundred twenty-eight
## Notation
Scientific notation 7.828 × 103 7.828 × 103
## Prime Factorization of 7828
Prime Factorization 22 × 19 × 103
Composite number
Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 3914 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 7,828 is 22 × 19 × 103. Since it has a total of 4 prime factors, 7,828 is a composite number.
## Divisors of 7828
1, 2, 4, 19, 38, 76, 103, 206, 412, 1957, 3914, 7828
12 divisors
Even divisors 8 4 2 2
Total Divisors Sum of Divisors Aliquot Sum τ(n) 12 Total number of the positive divisors of n σ(n) 14560 Sum of all the positive divisors of n s(n) 6732 Sum of the proper positive divisors of n A(n) 1213.33 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 88.476 Returns the nth root of the product of n divisors H(n) 6.45165 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 7,828 can be divided by 12 positive divisors (out of which 8 are even, and 4 are odd). The sum of these divisors (counting 7,828) is 14,560, the average is 12,13.,333.
## Other Arithmetic Functions (n = 7828)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 3672 Total number of positive integers not greater than n that are coprime to n λ(n) 306 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 993 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares
There are 3,672 positive integers (less than 7,828) that are coprime with 7,828. And there are approximately 993 prime numbers less than or equal to 7,828.
## Divisibility of 7828
m n mod m 2 3 4 5 6 7 8 9 0 1 0 3 4 2 4 7
The number 7,828 is divisible by 2 and 4.
## Classification of 7828
• Deficient
### Expressible via specific sums
• Polite
• Non-hypotenuse
## Base conversion (7828)
Base System Value
2 Binary 1111010010100
3 Ternary 101201221
4 Quaternary 1322110
5 Quinary 222303
6 Senary 100124
8 Octal 17224
10 Decimal 7828
12 Duodecimal 4644
20 Vigesimal jb8
36 Base36 61g
## Basic calculations (n = 7828)
### Multiplication
n×y
n×2 15656 23484 31312 39140
### Division
n÷y
n÷2 3914 2609.33 1957 1565.6
### Exponentiation
ny
n2 61277584 479680927552 3754942300877056 29393688331265594368
### Nth Root
y√n
2√n 88.476 19.8556 9.40617 6.008
## 7828 as geometric shapes
### Circle
Diameter 15656 49184.8 1.92509e+08
### Sphere
Volume 2.00928e+12 7.70037e+08 49184.8
### Square
Length = n
Perimeter 31312 6.12776e+07 11070.5
### Cube
Length = n
Surface area 3.67666e+08 4.79681e+11 13558.5
### Equilateral Triangle
Length = n
Perimeter 23484 2.6534e+07 6779.25
### Triangular Pyramid
Length = n
Surface area 1.06136e+08 5.65309e+10 6391.54
## Cryptographic Hash Functions
md5 9f564fef13bb8a7f9faa5f9071e4e045 4d105f40bb98ee35353b3407b0f41e6f8081c0e8 1d637966d2d2f3e18c64cdff5683885ea9404715a9aceec6e056f42b9167d4d9 63ffbddab746a36785b5d9df56599ace377a8f525c7bdf183ba8e061f411be4414176c94e0718d6b9e266f88f8e7e876b7f964fdfdbd75daabfcdc2756ed105f 66735d10f639095b0153f4c2223556c0e975d457
| 1,472 | 4,106 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.890625 | 4 |
CC-MAIN-2023-06
|
latest
|
en
| 0.806944 |
http://www.sequencesandseries.com/the-limit-of-n-p-for-p-a-natural-number/?shared=email&msg=fail
| 1,606,321,393,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2020-50/segments/1606141183514.25/warc/CC-MAIN-20201125154647-20201125184647-00006.warc.gz
| 144,613,185 | 8,214 |
# The limit of $n^{-p}$ for $p$ a natural number
$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$
Next we discover what sequences such as $\displaystyle \frac{1}{n^2}$ and $\displaystyle \frac{1}{n^3}$ tend to. This will be very useful later when combined with other techniques.
Theorem
Let $p\in \N$. Then $\displaystyle \lim_{n\to \infty} n^{-p}\to 0$.
Proof
Suppose $\varepsilon >0$ is given. Then, $n>\varepsilon ^{-1/p}$ implies that $n^{-p}<\varepsilon$. Thus let $N$ be an integer greater than $\varepsilon ^{-1/p}$. For all $n>N$ we have $|n^{-p}-0|=n^{-p}\lt \varepsilon$ so $\displaystyle \lim_{n\to \infty} n^{-p}\to 0$.
Example
Consider the sequence $\displaystyle a_n=\frac{(-1)^n}{n^3}$. Then,
$\left| a_n \right| =\left| \frac{(-1)^n}{n^3} \right| =\frac{1}{n^3}\to 0.$
Thus, $\displaystyle a_n=\frac{(-1)^n}{n^3}\to 0$.
Remark
In the theorem we can change the assumption ‘$p\in \N$’ to ‘$p\in \R$ with $p>0$’.
For example, $\displaystyle \frac{1}{n^\pi }$ tends to $0$.
To see the difficulty consider what $2^\pi$ means? How do we multiply 2 by itself $\pi$ times? We know how to multiply $2$ by itself $3$ times and $4$ times but what about somewhere inbetween?
Thus, the general statement relies on defining $x^a$ where $a$ is a real number and $x>0$. You may have seen that one solution to this is to define $x^a$ (for $x>0$) to be $\exp(a\ln x)$ where $\exp$ and $\ln$ are the exponential and natural logarithm functions. These functions and $x^a$ are not defined rigorously yet. Hence, for the moment we shall restrict to the case of $p\in \N$.
| 587 | 1,651 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.53125 | 5 |
CC-MAIN-2020-50
|
longest
|
en
| 0.703387 |
https://eduzip.com/ask/question/in-figure-name-all-the-pairs-of-adjacent-angles-521026
| 1,642,810,111,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2022-05/segments/1642320303717.35/warc/CC-MAIN-20220121222643-20220122012643-00205.warc.gz
| 294,962,791 | 8,776 |
Mathematics
# In figure, name all the pairs of adjacent angles.
##### SOLUTION
$\angle BAC \ and \angle CAD$
$\angle ABC \ and \angle ABE$
$\angle BCA \ and \angle BCF$
You're just one step away
Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
Enrolled Students 86
#### Realted Questions
Q1 Single Correct Medium
Given lines $\frac{x-4}{2} = \frac{y+5}{4}=\frac{z-1}{-3}$ and $\frac{x-2}{1}=\frac{y+1}{3}=\frac{z}{2}$
Statement 1 : The lines intersect.
Statement 2: They are not parallel
• A. Statement 1 is true, statement 2 is true and statement 2 is correct explanation for statement 1
• B. Statement 1 is true, statement 2 is true and statement 2 is not correct explanation for statement 1
• C. Statement 1 is false, statement 2 is true
• D. Statement 1 is true, statement 2 is false
Asked in: Mathematics - Straight Lines
1 Verified Answer | Published on 17th 08, 2020
Q2 Single Correct Medium
Based on the given figure, which of the following statements is true?
• A. $\angle\, 5$ and $\angle\, 7$ are supplementary angles.
• B. $\angle\, 7\, =\, 55^\circ$
• C. $\angle\, 8$ and $\angle\, 1$ are corresponding angles.
• D. $\angle\, 4$ and $\angle\, 5$ are alternate angles.
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q3 Subjective Medium
In figure, A, B, C are collinear points and $\angle DBA=\angle EBA$. Name two pairs of supplementary angles.
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q4 Subjective Medium
Write the supplement angle of:
$20^{o}+y^{o}$
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
Q5 Subjective Medium
Define the following term:
Reflex angle
Asked in: Mathematics - Lines and Angles
1 Verified Answer | Published on 09th 09, 2020
| 564 | 1,847 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.609375 | 4 |
CC-MAIN-2022-05
|
latest
|
en
| 0.735104 |
https://proofwiki.org/wiki/Alternate_Ratios_of_Equal_Fractions
| 1,685,693,938,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2023-23/segments/1685224648465.70/warc/CC-MAIN-20230602072202-20230602102202-00294.warc.gz
| 544,054,387 | 11,805 |
# Alternate Ratios of Equal Fractions
## Theorem
In the words of Euclid:
If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.
## Proof
Let the (natural) number $A$ be an aliquot part of the (natural) number $BC$, and another $D$ be the same aliquot part of another, $EF$, that $A$ is of $BC$.
We need to show that whatever aliquot part or aliquant part $A$ is of $D$, the same aliquot part or aliquant part is $BC$ of $EF$.
We have that whatever aliquot part $A$ is of $BC$, the same aliquot part $D$ is of $EF$.
Therefore, as many numbers as there are in $BC$ equal to $A$, so many also are there in $EF$ equal to $D$.
Let $BC$ be divided into the numbers equal to $A$, namely $BG, GC$.
Let $EF$ be divided into the numbers equal to $D$, namely $EH, HF$.
Thus the multitude of $BG, GC$ equals the multitude of $EH, HF$.
We have that $BG = GC$ and $EH = HF$.
Therefore whatever aliquot part or aliquant part $BG$ is of $EH$, the same aliquot part or the same aliquant part is $GC$ of $HF$ also.
So, from Proposition $5$ of Book $\text{VII}$: Divisors obey Distributive Law and Proposition $6$ of Book $\text{VII}$: Multiples of Divisors obey Distributive Law, we have that whatever aliquot part or aliquant part $BG$ is of $EH$, the same aliquot part also, or the same aliquant part, is the sum $BC$ of the sum $EF$.
But $BG = A$ and $EH = D$.
So whatever aliquot part or aliquant part $A$ is of $D$, the same aliquot part or aliquant part is $BC$ of $EF$ also.
$\blacksquare$
## Historical Note
This proof is Proposition $9$ of Book $\text{VII}$ of Euclid's The Elements.
| 514 | 1,755 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.8125 | 5 |
CC-MAIN-2023-23
|
latest
|
en
| 0.875602 |
https://howkgtolbs.com/convert/28.89-kg-to-lbs
| 1,619,156,795,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00134.warc.gz
| 397,289,969 | 12,227 |
28.89 kg to lbs - 28.89 kilograms to pounds
Before we get to the more practical part - this is 28.89 kg how much lbs conversion - we will tell you few theoretical information about these two units - kilograms and pounds. So we are starting.
How to convert 28.89 kg to lbs? 28.89 kilograms it is equal 63.6915474918 pounds, so 28.89 kg is equal 63.6915474918 lbs.
28.89 kgs in pounds
We will start with the kilogram. The kilogram is a unit of mass. It is a basic unit in a metric system, that is International System of Units (in short form SI).
From time to time the kilogram can be written as kilogramme. The symbol of the kilogram is kg.
Firstly the kilogram was defined in 1795. The kilogram was defined as the mass of one liter of water. First definition was not complicated but difficult to use.
Then, in 1889 the kilogram was described by the International Prototype of the Kilogram (in abbreviated form IPK). The International Prototype of the Kilogram was prepared of 90% platinum and 10 % iridium. The IPK was in use until 2019, when it was switched by a new definition.
The new definition of the kilogram is build on physical constants, especially Planck constant. The official definition is: “The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of c and ΔνCs.”
One kilogram is exactly 0.001 tonne. It could be also divided to 100 decagrams and 1000 grams.
28.89 kilogram to pounds
You learned some facts about kilogram, so now we can move on to the pound. The pound is also a unit of mass. It is needed to point out that there are not only one kind of pound. What are we talking about? For example, there are also pound-force. In this article we want to centre only on pound-mass.
The pound is in use in the British and United States customary systems of measurements. Of course, this unit is in use also in other systems. The symbol of this unit is lb or “.
There is no descriptive definition of the international avoirdupois pound. It is defined as 0.45359237 kilograms. One avoirdupois pound is divided to 16 avoirdupois ounces and 7000 grains.
The avoirdupois pound was implemented in the Weights and Measures Act 1963. The definition of this unit was given in first section of this act: “The yard or the metre shall be the unit of measurement of length and the pound or the kilogram shall be the unit of measurement of mass by reference to which any measurement involving a measurement of length or mass shall be made in the United Kingdom; and- (a) the yard shall be 0.9144 metre exactly; (b) the pound shall be 0.45359237 kilogram exactly.”
How many lbs is 28.89 kg?
28.89 kilogram is equal to 63.6915474918 pounds. If You want convert kilograms to pounds, multiply the kilogram value by 2.2046226218.
28.89 kg in lbs
Theoretical part is already behind us. In next section we want to tell you how much is 28.89 kg to lbs. Now you know that 28.89 kg = x lbs. So it is high time to get the answer. Just look:
28.89 kilogram = 63.6915474918 pounds.
It is a correct outcome of how much 28.89 kg to pound. You can also round it off. After rounding off your outcome will be as following: 28.89 kg = 63.558 lbs.
You learned 28.89 kg is how many lbs, so let’s see how many kg 28.89 lbs: 28.89 pound = 0.45359237 kilograms.
Of course, this time it is possible to also round it off. After rounding off your result is as following: 28.89 lb = 0.45 kgs.
We are also going to show you 28.89 kg to how many pounds and 28.89 pound how many kg results in tables. Have a look:
We will begin with a chart for how much is 28.89 kg equal to pound.
28.89 Kilograms to Pounds conversion table
Kilograms (kg) Pounds (lb) Pounds (lbs) (rounded off to two decimal places)
28.89 63.6915474918 63.5580
Now look at a table for how many kilograms 28.89 pounds.
Pounds Kilograms Kilograms (rounded off to two decimal places
28.89 0.45359237 0.45
Now you know how many 28.89 kg to lbs and how many kilograms 28.89 pound, so it is time to go to the 28.89 kg to lbs formula.
28.89 kg to pounds
To convert 28.89 kg to us lbs a formula is needed. We will show you a formula in two different versions. Let’s start with the first one:
Amount of kilograms * 2.20462262 = the 63.6915474918 outcome in pounds
The first formula give you the most accurate outcome. In some situations even the smallest difference could be significant. So if you need a correct result - this formula will be the best solution to convert how many pounds are equivalent to 28.89 kilogram.
So let’s move on to the second formula, which also enables calculations to learn how much 28.89 kilogram in pounds.
The second formula is as following, have a look:
Number of kilograms * 2.2 = the outcome in pounds
As you can see, this formula is simpler. It can be the best solution if you want to make a conversion of 28.89 kilogram to pounds in easy way, for instance, during shopping. You only need to remember that your result will be not so exact.
Now we are going to learn you how to use these two versions of a formula in practice. But before we will make a conversion of 28.89 kg to lbs we are going to show you another way to know 28.89 kg to how many lbs totally effortless.
28.89 kg to lbs converter
An easier way to learn what is 28.89 kilogram equal to in pounds is to use 28.89 kg lbs calculator. What is a kg to lb converter?
Converter is an application. It is based on longer version of a formula which we gave you above. Due to 28.89 kg pound calculator you can easily convert 28.89 kg to lbs. You only have to enter number of kilograms which you need to calculate and click ‘convert’ button. You will get the result in a flash.
So try to convert 28.89 kg into lbs with use of 28.89 kg vs pound calculator. We entered 28.89 as an amount of kilograms. It is the outcome: 28.89 kilogram = 63.6915474918 pounds.
As you see, our 28.89 kg vs lbs converter is easy to use.
Now let’s move on to our chief issue - how to convert 28.89 kilograms to pounds on your own.
28.89 kg to lbs conversion
We are going to begin 28.89 kilogram equals to how many pounds calculation with the first version of a formula to get the most correct result. A quick reminder of a formula:
Amount of kilograms * 2.20462262 = 63.6915474918 the outcome in pounds
So what have you do to check how many pounds equal to 28.89 kilogram? Just multiply amount of kilograms, this time 28.89, by 2.20462262. It is exactly 63.6915474918. So 28.89 kilogram is exactly 63.6915474918.
You can also round it off, for instance, to two decimal places. It is 2.20. So 28.89 kilogram = 63.5580 pounds.
It is high time for an example from everyday life. Let’s convert 28.89 kg gold in pounds. So 28.89 kg equal to how many lbs? And again - multiply 28.89 by 2.20462262. It is equal 63.6915474918. So equivalent of 28.89 kilograms to pounds, when it comes to gold, is 63.6915474918.
In this case it is also possible to round off the result. Here is the outcome after rounding off, this time to one decimal place - 28.89 kilogram 63.558 pounds.
Now we are going to examples calculated using a short version of a formula.
How many 28.89 kg to lbs
Before we show you an example - a quick reminder of shorter formula:
Number of kilograms * 2.2 = 63.558 the outcome in pounds
So 28.89 kg equal to how much lbs? As in the previous example you need to multiply number of kilogram, in this case 28.89, by 2.2. Let’s see: 28.89 * 2.2 = 63.558. So 28.89 kilogram is 2.2 pounds.
Let’s make another conversion with use of this version of a formula. Now calculate something from everyday life, for example, 28.89 kg to lbs weight of strawberries.
So let’s convert - 28.89 kilogram of strawberries * 2.2 = 63.558 pounds of strawberries. So 28.89 kg to pound mass is 63.558.
If you learned how much is 28.89 kilogram weight in pounds and can convert it with use of two different versions of a formula, let’s move on. Now we want to show you these results in charts.
Convert 28.89 kilogram to pounds
We know that results shown in charts are so much clearer for most of you. We understand it, so we gathered all these results in charts for your convenience. Thanks to this you can easily compare 28.89 kg equivalent to lbs outcomes.
Let’s start with a 28.89 kg equals lbs chart for the first version of a formula:
Kilograms Pounds Pounds (after rounding off to two decimal places)
28.89 63.6915474918 63.5580
And now look 28.89 kg equal pound table for the second version of a formula:
Kilograms Pounds
28.89 63.558
As you can see, after rounding off, if it comes to how much 28.89 kilogram equals pounds, the outcomes are not different. The bigger number the more significant difference. Please note it when you want to make bigger number than 28.89 kilograms pounds conversion.
How many kilograms 28.89 pound
Now you know how to convert 28.89 kilograms how much pounds but we will show you something more. Are you curious what it is? What do you say about 28.89 kilogram to pounds and ounces calculation?
We will show you how you can convert it step by step. Start. How much is 28.89 kg in lbs and oz?
First things first - you need to multiply amount of kilograms, in this case 28.89, by 2.20462262. So 28.89 * 2.20462262 = 63.6915474918. One kilogram is equal 2.20462262 pounds.
The integer part is number of pounds. So in this case there are 2 pounds.
To calculate how much 28.89 kilogram is equal to pounds and ounces you have to multiply fraction part by 16. So multiply 20462262 by 16. It is exactly 327396192 ounces.
So final outcome is exactly 2 pounds and 327396192 ounces. You can also round off ounces, for example, to two places. Then final result is exactly 2 pounds and 33 ounces.
As you see, calculation 28.89 kilogram in pounds and ounces quite simply.
The last calculation which we are going to show you is calculation of 28.89 foot pounds to kilograms meters. Both foot pounds and kilograms meters are units of work.
To convert it you need another formula. Before we show you it, look:
• 28.89 kilograms meters = 7.23301385 foot pounds,
• 28.89 foot pounds = 0.13825495 kilograms meters.
Now have a look at a formula:
Number.RandomElement()) of foot pounds * 0.13825495 = the outcome in kilograms meters
So to calculate 28.89 foot pounds to kilograms meters you have to multiply 28.89 by 0.13825495. It is equal 0.13825495. So 28.89 foot pounds is exactly 0.13825495 kilogram meters.
It is also possible to round off this result, for example, to two decimal places. Then 28.89 foot pounds will be exactly 0.14 kilogram meters.
We hope that this conversion was as easy as 28.89 kilogram into pounds calculations.
We showed you not only how to make a conversion 28.89 kilogram to metric pounds but also two another calculations - to know how many 28.89 kg in pounds and ounces and how many 28.89 foot pounds to kilograms meters.
We showed you also other way to do 28.89 kilogram how many pounds calculations, it is using 28.89 kg en pound calculator. This is the best choice for those of you who do not like converting on your own at all or this time do not want to make @baseAmountStr kg how lbs calculations on your own.
We hope that now all of you are able to make 28.89 kilogram equal to how many pounds conversion - on your own or using our 28.89 kgs to pounds calculator.
Don’t wait! Convert 28.89 kilogram mass to pounds in the best way for you.
Do you want to make other than 28.89 kilogram as pounds calculation? For example, for 10 kilograms? Check our other articles! We guarantee that calculations for other numbers of kilograms are so easy as for 28.89 kilogram equal many pounds.
How much is 28.89 kg in pounds
At the end, we are going to summarize the topic of this article, that is how much is 28.89 kg in pounds , we prepared one more section. Here we have for you all you need to remember about how much is 28.89 kg equal to lbs and how to convert 28.89 kg to lbs . It is down below.
What is the kilogram to pound conversion? The conversion kg to lb is just multiplying 2 numbers. Let’s see 28.89 kg to pound conversion formula . Check it down below:
The number of kilograms * 2.20462262 = the result in pounds
See the result of the conversion of 28.89 kilogram to pounds. The exact result is 63.6915474918 lbs.
You can also calculate how much 28.89 kilogram is equal to pounds with second, shortened type of the formula. Let’s see.
The number of kilograms * 2.2 = the result in pounds
So in this case, 28.89 kg equal to how much lbs ? The result is 63.6915474918 lb.
How to convert 28.89 kg to lbs in a few seconds? You can also use the 28.89 kg to lbs converter , which will make whole mathematical operation for you and give you a correct answer .
Kilograms [kg]
The kilogram, or kilogramme, is the base unit of weight in the Metric system. It is the approximate weight of a cube of water 10 centimeters on a side.
Pounds [lbs]
A pound is a unit of weight commonly used in the United States and the British commonwealths. A pound is defined as exactly 0.45359237 kilograms.
| 3,476 | 13,214 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.796875 | 4 |
CC-MAIN-2021-17
|
latest
|
en
| 0.949421 |
https://numbas.mathcentre.ac.uk/question/9086/indices-positive-negative-simple-fraction.exam
| 1,721,577,103,000,000,000 |
text/plain
|
crawl-data/CC-MAIN-2024-30/segments/1720763517747.98/warc/CC-MAIN-20240721152016-20240721182016-00371.warc.gz
| 372,283,560 | 2,207 |
// Numbas version: exam_results_page_options {"name": "Indices: positive, negative, simple fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["n12", "n10", "n11", "x8", "x9", "n1", "x2", "x1", "x6", "x7", "x4", "x5", "x73", "n", "n8", "n9", "x10", "x11", "n2", "n3", "n4", "n5", "n6", "n7"], "name": "Indices: positive, negative, simple fraction", "tags": ["fractional indices", "indices", "negative indices", "powers", "roots"], "preamble": {"css": "", "js": ""}, "advice": "
Recall that a fractional power indicates a root. So $x^\\frac{1}{2}= \\sqrt x$.
\n
A negative power indicates a reciprocal. For instance $3^{-2} = \\dfrac{1}{3^2}$.
\n
See this Mathcentre leaflet for more information on fractional and negative powers.
", "rulesets": {"std": ["all", "fractionNumbers", "basic"]}, "parts": [{"prompt": "
If $x= \\var{x2} \\$, find the value of $x^{\\var{n}}$.
\n
$x^{\\var{n}}=$ [[0]]
\n
\n
\n
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{x2}^{n}", "minValue": "{x2}^{n}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "
If $y= \\var{x6}$, find the value of $y^{\\frac{1}{2}}$.
\n
\n
$y^{\\frac{1}{2}}$ = [[0]]
\n
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{x6}^0.5", "minValue": "{x6}^0.5", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "
If $z= \\var{x4}$, find the value of $z^{\\var{n2}}$.
\n
\n
$z^{\\var{n2}}$ = [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{x4}^{n2}", "minValue": "{x4}^{n2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "
Find the value of the following.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"n12": {"definition": "random(-3..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n12", "description": ""}, "n10": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "n10", "description": ""}, "n11": {"definition": "random(0,2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "n11", "description": ""}, "x8": {"definition": "random(1,3,5,7,11)", "templateType": "anything", "group": "Ungrouped variables", "name": "x8", "description": ""}, "x9": {"definition": "random(2,4,6,9,10,12)", "templateType": "anything", "group": "Ungrouped variables", "name": "x9", "description": ""}, "n1": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "x2": {"definition": "random(2,4,5,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "x1": {"definition": "random(2,3,4,5,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}, "x6": {"definition": "(random(2..10))^2", "templateType": "anything", "group": "Ungrouped variables", "name": "x6", "description": ""}, "x7": {"definition": "x73^3", "templateType": "anything", "group": "Ungrouped variables", "name": "x7", "description": ""}, "x4": {"definition": "random(2, 4, 5, 10)", "templateType": "anything", "group": "Ungrouped variables", "name": "x4", "description": ""}, "x5": {"definition": "random(2,3,4,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "x5", "description": ""}, "x73": {"definition": "random(-5..6 except[0,1,-1])", "templateType": "anything", "group": "Ungrouped variables", "name": "x73", "description": ""}, "n": {"definition": "random(2,3,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "n8": {"definition": "random(-4..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "n8", "description": ""}, "n9": {"definition": "random(1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "n9", "description": ""}, "x10": {"definition": "random(1,3,5,7)", "templateType": "anything", "group": "Ungrouped variables", "name": "x10", "description": ""}, "x11": {"definition": "random(2,4,6,9,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "x11", "description": ""}, "n2": {"definition": "random(-2,-3,-4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n3": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "n3", "description": ""}, "n4": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "n5": {"definition": "random(2..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "n5", "description": ""}, "n6": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "n6", "description": ""}, "n7": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n7", "description": ""}}, "metadata": {"description": "
Given a number evaluate simple power, negative power, to one half.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}]}], "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}]}
| 2,160 | 6,713 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.71875 | 4 |
CC-MAIN-2024-30
|
latest
|
en
| 0.380929 |
https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_12A_Problems/Problem_12&diff=prev&oldid=114212
| 1,618,110,448,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2021-17/segments/1618038060927.2/warc/CC-MAIN-20210411030031-20210411060031-00532.warc.gz
| 233,381,256 | 12,887 |
# Difference between revisions of "2012 AMC 12A Problems/Problem 12"
## Problem
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
$\textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5}$
## Solution 1
$[asy] unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real a=1; real b=2; pair O=(0,0); pair A=(-(sqrt(19)-2)/5,1); pair B=((sqrt(19)-2)/5,1); pair C=((sqrt(19)-2)/5,1+2(sqrt(19)-2)/5); pair D=(-(sqrt(19)-2)/5,1+2(sqrt(19)-2)/5); pair E=(-(sqrt(19)-2)/5,0); path inner=Circle(O,a); path outer=Circle(O,b); draw(outer); draw(inner); draw(A--B--C--D--cycle); draw(O--D--E--cycle); label("A",D,NW); label("E",E,SW); label("O",O,SE); label("s+1",(D--E),W); label("\frac{s}{2}",(E--O),S); pair[] ps={A,B,C,D,E,O}; dot(ps); [/asy]$
The circles have radii of $1$ and $2$. Draw the triangle shown in the figure above and write expressions in terms of $s$ (length of the side of the square) for the sides of the triangle. Because $AO$ is the radius of the larger circle, which is equal to $2$, we can write the Pythagorean Theorem.
\begin{align*} \left( \frac{s}{2} \right) ^2 + (s+1)^2 &= 2^2\\ \frac14 s^2 + s^2 + 2s + 1 &= 4\\ \frac54 s^2 +2s - 3 &= 0\\ 5s^2 + 8s - 12 &=0 \end{align*}
$$s = \frac{-8+\sqrt{8^2-4(5)(-12)}}{10} = \frac{-8+\sqrt{304}}{10} = \frac{-8+4\sqrt{19}}{10} = \boxed{\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}}$$
## Solution 2
Using the diagram above, we look at the top-right vertex of the square. Let us call this point $(x,y)$. Then, we that since the square is symmetrical over the y-axis, that the y value is equal to $2x+1$, since we can multiply the x value(which is half of $s$) by two to get $s$, and we add one since the square lies one unit above the origin. Now, all we must do is find the intersection of the larger circle, $x^2 + y^2 = 4$, and the line $y=2x+1$. Substituting the second equation into the first, we get:
$5x^2 +4x -3 = 0$
Using the quadratic formula, we arrive with $x=\frac{-4 \pm 2\sqrt{19}}{10}$. However, recall that the x value is only one half of the side length. Multiplying this value by $2$, then, and using only the positive root(since the top right vertex of the square has a positive x value), we get:
$\frac{-4 + 2\sqrt{19}}{5} \Rightarrow \boxed{\textbf{(D)}}$
## Solution 3
Like the previous solutions, we can say that the square is symmetrical with respect to the y-axis.
Since CD passes through point (0,1), let C be (-x,1) and D be (x,1)
Point B is $\left(-x, \sqrt(4-x^2)\right)$ and Point A is $\left(x,\sqrt{(4-x^2)}\right)$. Now, we see that AB = 2x and BC = $\sqrt{(4-x^2)}$-1
Setting those two equal, we get the same equation as the previous solutions: $5x^2 +4x -3 = 0$ and then solve the same way.
-Conantwiz2023
| 1,107 | 3,049 |
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 30, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 4.5625 | 5 |
CC-MAIN-2021-17
|
latest
|
en
| 0.565359 |
http://test-paper.info/forum/viewtopic.php?showtopic=1896
| 1,516,572,525,000,000,000 |
text/html
|
crawl-data/CC-MAIN-2018-05/segments/1516084890893.58/warc/CC-MAIN-20180121214857-20180121234857-00157.warc.gz
| 351,706,616 | 10,155 |
Welcome to Test-paper.info
Sunday, January 21 2018 @ 04:08 PM CST
Select a Forum » General Chat » Primary 1 Matters » Primary 2 Matters » Primary 3 Matters » Primary 4 Matters » Primary 5 Matters » Primary 6 Matters » Question, Feedback and Comments » Education Classified
Forum Index > Test Paper Related > Primary 6 Matters Raffles Girls 2011 SA2
| Printable Version
By: Jane Lim (offline) Wednesday, June 06 2012 @ 12:31 AM CDT (Read 3870 times)
Jane Lim
Hi there,
Could anyone please help me with the following 2 questions.
Many thanks in advance!!
Newbie
Registered: 10/01/10
Posts: 11
By: echeewh (offline) Wednesday, June 06 2012 @ 05:34 AM CDT
echeewh
Hello Jane,
For Q14(b), pls refer to attachment. The worked solution is on the left side (not the working /method on the right).
==========================================
Q17.
Refer to re-modelled diagram in paper as per attached.
Given ODBE is a trapezium, angle ODB = 45°.
In Square Y, draw a diagonal labelled OF ( angle EOF = 45° ), where OF // DB. I.e. ODBF is a parallelogram and OF=DB=11.75cm.
In Square X, draw a diagonal labelled OG, where OG=OF=11.75cm (OG, OF are radii of semicircle with centre O). Angle FOG is a right-angled triangle (with angle FOG = 90°)
Area of Triangle FOG = (11.75 x 11.75) / 2 = (1 / 2) x Rectangle EFGH
(a)
Area (X + Y) = Area of Rectangle EFGH = 2 x Area of Triangle FOG
= 2 x [(11.75 x 11.75) / 2] = 11.75 x 11.75 = 138.06 cm2 (nearest 2 dec places)
(b)
Circumference of Semicircle = (pi / 2) x d = (pi / 2) x (11.75 x 2) = 11.75 pi
Perimeter of S,T = Circumference of Semicircle + (4 x r) , where r = radius of semicircle
= 11.75 pi + (4 x 11.75) = 83.91 cm (nearest 2 dec places)
=========================================
Trust the above helps.
Cheers,
Edward
Active Member
Registered: 04/21/11
Posts: 627
By: mlteo (offline) Tuesday, June 12 2012 @ 11:19 PM CDT
mlteo
Quote by: echeewh
(b)
Circumference of Semicircle = (pi / 2) x d = (pi / 2) x (11.75 x 2) = 11.75 pi
Perimeter of S,T = Circumference of Semicircle + (4 x r) , where r = radius of semicircle
= 11.75 pi + (4 x 11.75) = 83.91 cm (nearest 2 dec places)
=========================================
Hi Edward
Shouldn't the answer for (b) be 83.93 ?
(11.75 x pi) + (4 x 11.74) = 36.92857 + 47 = 83.92857, which rounded to 2 decimal places is
83.93.
Thanks
Regards
Teo
Newbie
Registered: 12/31/06
Posts: 10
By: echeewh (offline) Wednesday, June 13 2012 @ 03:05 AM CDT
echeewh
Quote by: mlteo
Quote by: echeewh
Hi Edward
Shouldn't the answer for (b) be 83.93 ?
(11.75 x pi) + (4 x 11.74) = 36.92857 + 47 = 83.92857, which rounded to 2 decimal places is
83.93.
Thanks
Regards
Teo
Hi Teo,
If you refer to the question, you should use the calculator's pi since question does not say to use 22/7 or 3.14.
In your case, you are using 22/7 , correct?? Your answer will be correct if the question explicitly says to use 22/7.
Hope this clarifies.
Thank you & best rgds.
Edward
Active Member
Registered: 04/21/11
Posts: 627
By: parent603 (offline) Wednesday, June 13 2012 @ 04:10 AM CDT
parent603
Hi Edward,
Referring to Q14(b), your answer is correct. However I do not understand why the difference between the shaded and unshaded area = 4 * rectangle instead of 2 * rectangle. Pls enlighten.
Thanks for the solution.
Newbie
Registered: 11/07/11
Posts: 14
By: echeewh (offline) Wednesday, June 13 2012 @ 08:04 PM CDT
echeewh
Quote by: parent603
Hi Edward,
Referring to Q14(b), your answer is correct. However I do not understand why the difference between the shaded and unshaded area = 4 * rectangle instead of 2 * rectangle. Pls enlighten.
Thanks for the solution.
Hello parent603,
If I understand you correctly, you were asking why Shaded - Unshaded is not 2 x Rectangle. Correct??
Yes it is .. and I checked I have written in that attachment as 2 x Rectangle (3).
Did I see it wrongly or have I misinterpreted your query??
Do let me know pls. Thanks.
Kind regards,
Edward
Active Member
Registered: 04/21/11
Posts: 627
By: mlteo (offline) Thursday, June 14 2012 @ 02:42 AM CDT
mlteo
Oops. You are right, I used 22/7 as pi.
Thanks.
Newbie
Registered: 12/31/06
Posts: 10
By: parent603 (offline) Thursday, June 14 2012 @ 10:22 PM CDT
parent603
Quote by: echeewh
Quote by: parent603
Hi Edward,
Referring to Q14(b), your answer is correct. However I do not understand why the difference between the shaded and unshaded area = 4 * rectangle instead of 2 * rectangle. Pls enlighten.
Thanks for the solution.
Hello parent603,
If I understand you correctly, you were asking why Shaded - Unshaded is not 2 x Rectangle. Correct??
Yes it is .. and I checked I have written in that attachment as 2 x Rectangle (3).
Did I see it wrongly or have I misinterpreted your query??
Do let me know pls. Thanks.
Kind regards,
Edward
Hello Edward,
From your attachment, there are 2 rectangles ie 1.4 * 2.5 = 2 * 1.4 * 2.5.
I am wondering why the solution is based on 4 * 1.4 * 2.5. I do understand that your soution is correct. I am curious to know why.
Thanks,
Newbie
Registered: 11/07/11
Posts: 14
By: Angch (offline) Thursday, June 21 2012 @ 09:56 AM CDT
Angch
hi., anyone able to help problem sum Q16 ?
Thanks.
Newbie
Registered: 08/13/11
Posts: 1
By: echeewh (offline) Friday, June 22 2012 @ 12:12 AM CDT
echeewh
Quote by: parent603
Quote by: echeewh
Quote by: parent603
Hi Edward,
Referring to Q14(b), your answer is correct. However I do not understand why the difference between the shaded and unshaded area = 4 * rectangle instead of 2 * rectangle. Pls enlighten.
Thanks for the solution.
Hello parent603,
If I understand you correctly, you were asking why Shaded - Unshaded is not 2 x Rectangle. Correct??
Yes it is .. and I checked I have written in that attachment as 2 x Rectangle (3).
Did I see it wrongly or have I misinterpreted your query??
Do let me know pls. Thanks.
Kind regards,
Edward
Hello Edward,
From your attachment, there are 2 rectangles ie 1.4 * 2.5 = 2 * 1.4 * 2.5.
I am wondering why the solution is based on 4 * 1.4 * 2.5. I do understand that your soution is correct. I am curious to know why.
Thanks,
Hello parent603,
Sorry i missed your clarification posted here earlier.
My solution is based on:
Shaded - Unshaded = 2 x Rectangle (3)
If u see carefully , this Rectangle (3) is 2 x (given small rectangle of 2.5 x 1.4) and will have length of 2.5 and breadth of 2.8.
I believe your term 'the solution' is referring to the Answerkey, which is 4 x (small rectangle, 2.5 x 1.4)
Hence, if you were to calculate carefully, both will give you the same results.
If you still have doubts on this, pls write me an email and i will do my best to address your concerns.
Thankyou once again for clarifying..
Cheers,
Edward
Active Member
Registered: 04/21/11
Posts: 627
All times are CST. The time is now 04:08 pm.
Normal Topic Locked Topic Sticky Topic
New Post Sticky Topic w/ New Post Locked Topic w/ New Post
View Anonymous Posts Able to Post HTML Allowed Censored Content
| 2,210 | 7,089 |
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 3.765625 | 4 |
CC-MAIN-2018-05
|
latest
|
en
| 0.764783 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.