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http://mathforum.org/library/drmath/sets/elem_subtraction.html?start_at=81&num_to_see=40&s_keyid=40235447&f_keyid=40235448
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Dr. Math Home || Elementary || Middle School || High School || College || Dr. Math FAQ
TOPICS This page: subtraction Search Dr. Math See also the Dr. Math FAQ: order of operations Internet Library: subtraction ELEMENTARY Arithmetic addition subtraction multiplication division fractions/decimals Definitions Geometry 2-dimensional circles triangles/polygons 3D and higher polyhedra Golden Ratio/ Fibonacci Sequence History/Biography Measurement calendars/ dates/time temperature terms/units Number Sense/ About Numbers infinity large numbers place value prime numbers square roots Projects Puzzles Word Problems Browse Elementary Subtraction Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions: Flashcards/worksheets on the Web. Order of operations. Number sentences. Subtraction with borrowing/regrouping. "Subtraction" and "Negative"--Same Sign, Different Concepts? [12/14/2005] Why do we use "-" to mean both "subtraction" and "negative", when those are two different concepts? Subtrahend and Minuend [03/10/1997] What are subtrahend and minuend? I think they have to do with subtraction. Take Aways [03/28/2002] 56 - 18 = ? I don't know how to do it. Teaching Borrowing in Subtraction [03/12/1997] What kind of suggestions do you have for manipulatives? Teaching Children to Subtract [08/30/2005] I am having a hard time trying to teach my 7 year old how to subtract numbers in the teens minus single digits, such as 14 - 7 or 17 - 9. I just think you have to remember them by heart. Is there an easier way? Using Addition and Subtraction to Check Answers [02/25/2003] How can you use addition to check subtraction? How can you use subtraction to check addition? When in Rome, Know Your Place — Less a Written Notation for It [01/27/2015] An adult wonders how Romans could have developed their rules for subtraction absent place value. Citing their use of the abacus, Doctor Peterson distinguishes between operating with the concept and representing it. Why Is Mental Math Important? [03/25/2004] How can I convince a 14 year old girl who is in 8th grade the importance of mental math? I think that skills like mentally adding and subtracting 2 digit numbers and being able to estimate multiplying 2 digit large numbers are critical. My daughter's teacher says that such skills aren't needed because of calculators and computers. Page: [
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You are here
*Please note: you may not see animations, interactions or images that are potentially on this page because you have not allowed Flash to run on S-cool. To do this, click here.*
Quadratic equations are equations that can be written:
ax2 +bx + c = 0
where a, b and c are constants (numbers) and 'a' cannot be zero (-there must be some x2 's!) You might also see them written in the form (x + p)(x + q) = 0 but we?ll come to that later.
You solve them by finding the value(s) of x which make them equal to zero. There are several methods:
1. Factorising.
2. The quadratic formula.
3. Completing the square (most exam boards leave this to AS level).
4. Using the graph.
Factorising
This means 'putting in brackets'. (You also need to be able to multiply the brackets out to get the equation back to the form: ax2 + bx + c = 0).
You need to find a number to go in each bracket. Use the following conditions to find the two numbers you need:
1. They must multiply to give 'c'.
2. They must add to give 'b'.
3. If 'c' is positive both numbers must have the same sign (both positive or both negative). If 'c' is negative then the numbers must be of opposite sign.
There's no magic solution. You just need to practise, practise, practise!
Then, as the brackets multiply to give zero one of the brackets must be zero!
This gives you your solution(s). (Quadratics will either have 0, 1 or 2 solutions).
Here's an example:
Solve x2 + 3x - 10 = 0.
It's already written in the form: ax2 + bx + c = 0, so we don't need to rearrange anything. We need two numbers that multiply to give -10 (which means they're of opposite sign) and add to give 3.
There are 4 possibilities: -1 and 10, 1 and -10, -2 and 5, 2 and -5.
The only pair that adds to give +3 is -2 and 5 giving:
(x -2)(x + 5) = 0
If (x - 2) = 0 we get the solution x = 2
If (x + 5) = 0 we get the solution x = -5
Try solving x2 + 5x +6 = 0:
If an equation won't factorise (the solutions might be decimals or fractions) then you can try the formula. It looks really complicated but once you've practised using it, it will seem much easier (honest!). Anyway, it's based on the form: ax2 + bx + c = 0, and here it is:
You just substitute the numbers in and hey presto! One solution comes from using the + sign in front of the square root and the other solution by using the - sign.
Here's an example:
Solve x2 - 2x - 4 = 0 using the formula.
Here a = 1, b = -2, and c = -4. So we can put these values into the formula:
Simplifying:
Putting this carefully into your calculator (see 'Calculator' section) gives the solutions:
x = 3.24, and x = -1.24 (both to 2 decimal places).
Heres a worked example, follow it through:
How do you know how many solutions there are?
Well, the important bit of the formula is the bit inside the square root: b2- 4ac
If b2- 4ac is positive, there are two solutions.
If b2- 4ac is zero, there is one solution.
If b2- 4ac is negative, there are no solutions.
To look at graphs of linear and quadratic equations, go to Graphs Learn-its.
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My Math Forum Square roots question
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion
August 1st, 2007, 02:28 PM #1 Newbie Joined: Aug 2007 Posts: 1 Thanks: 0 Square roots question What is the square root of 19 to the nearest whole number? What is the square root of 171 to the nearest whole number? Solve F to the second power = 169 is square root of 36 rational, irrational, interger rational, or other? estimate b to the 2nd power = 525 to the nearest integer. square root of 17 to nearest tenth. square root of negative 102 round to the nearest tenth.. thanks.. if these could be answered by today that would help out and everything.
August 1st, 2007, 04:30 PM #2 Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 let us assume that we can't use any calculator related tools, and let us solve it both algebraically and in calculus terms. Q. What is the square root of 19 to the nearest whole number? A. let us assume that newton's method will help us solve this. let f(x)=x^2-19=0, where we want to find for x, approximation. take the derivative of f(x).. which gives us f'(x)=2x. let us assume that x_0=4, and x_1=5. let's use the linear approximation, using general equation f(x)=l(x)=f(x_k)+f'(x_k)(x-x_k), where k is the kth term of x. let's use x_0=4 first. f(x)=l(x)=f(4)+f'(4)(x-4)=-3+8(x-4)=8x-35=0 8x=35 x=35/8=4.375 is an example of an close approximation. let's use the x_1=5 for our final approximation. f(x)=l(x)=f(5)+f'(5)(x-5)=6+10(x-5)=10x-44=0 10x=44 x=44/10=22/5=4.4 is an another example of an close approximation. since the 4.375 is an approximation for x_0=4, and 4.4 is an approximation for x_1=5, and x=sqrt(19) must be between those two approximations, we can say that nearest whole number for sqrt(19) is 4. do this the same way for your second problem, third problem, fifth, sixth and seventh problem. since sqrt(36)=6, we can say that it's "positive, integer, rational, even, natural, composite number".
August 11th, 2007, 09:40 AM #3
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Quote:
Originally Posted by johnny Q. What is the square root of 19 to the nearest whole number? A. let us assume that Newton's method will help us solve this.
Why make that assumption? Newton's method may fail for problems that are only slightly more difficult.
Also, do you really do things exactly the same way for the seventh problem?
August 11th, 2007, 12:44 PM #4
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Posts: 2,140
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Quote:
Why make that assumption? Newton's method may fail for problems that are only slightly more difficult. What is your answer for the third problem? Also, do you really do things exactly the same way for the seventh problem?
Since the math problem here is generally according to basic arithmetic, is it better idea to use some basic simplifying and calculating results, instead of using Newton's Method?
August 11th, 2007, 02:27 PM #5
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I did some mental math to give pointers to the answers.
Quote:
Originally Posted by Time What is the square root of 19 to the nearest whole number?
For positive reals, the square root increases monotonically, so it's clear that the answer must be 4 or 5 (since 16 < 19 < 25). Thus all you have to see is whether 4.5 * 4.5 is more or less than 19.
Quote:
Originally Posted by Time What is the square root of 171 to the nearest whole number?
Same as above, but with 13 and 14. Actually your intuition should give you this one right away without calculation, if you don't need a proof.
Quote:
Originally Posted by Time Solve F to the second power = 169
There's a negative and a positive answer. This is easy because the answer is a whole number.
Quote:
Originally Posted by Time is square root of 36 rational, irrational, interger rational, or other?
This is easy: square roots of integers other than perfect squares are always transcendental, so all you have to see is if 36 is a perfect square or not.
Quote:
Originally Posted by Time estimate b to the 2nd power = 525 to the nearest integer.
Again, two answers. Off the top of my head 23 is close, but check that this is the closest.
Quote:
Originally Posted by Time square root of 17 to nearest tenth.
4.0 < x < 4.4, since 16 < 17 < 1.21 * 16. Just subdivide the intervals a few more times, checking each time to see what side 17 is on.
Quote:
Originally Posted by Time square root of negative 102 round to the nearest tenth..
10.1 squared is easy to calculate, since you can just treat it as a binomial. (10+0.1)^2 = 100 + 2*10*0.1 + 0.01 = 102.01. I imagine that's the answer.
August 12th, 2007, 03:25 PM #6
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Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Time square root of negative 102 round to the nearest tenth..
10.1 squared is easy to calculate, since you can just treat it as a binomial. (10+0.1)^2 = 100 + 2*10*0.1 + 0.01 = 102.01. I imagine that's the answer.
Be careful with that one. The question asked about the square root of a negative number. I assume all these questions are over the set of reals, because there is no real sensible way to define nth root over the complex numbers that will make it not multivalued. Over the reals, the square root of a negative number is undefined.
Over the set of complex numbers, the answer would be ±10.01i, where i:=√-1.
August 13th, 2007, 07:54 AM #7
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Quote:
Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by Time square root of negative 102 round to the nearest tenth..
10.1 squared is easy to calculate, since you can just treat it as a binomial. (10+0.1)^2 = 100 + 2*10*0.1 + 0.01 = 102.01. I imagine that's the answer.
Be careful with that one. The question asked about the square root of a negative number. I assume all these questions are over the set of reals, because there is no real sensible way to define nth root over the complex numbers that will make it not multivalued. Over the reals, the square root of a negative number is undefined.
Over the set of complex numbers, the answer would be ±10.01i, where i:=√-1.
No, just 10.01i; it asked for the square root of -102, not the solutions for the equation x * x = -102.
August 13th, 2007, 08:40 AM #8 Senior Member Joined: Nov 2006 From: I'm a figment of my own imagination :? Posts: 848 Thanks: 0 I suppose you can stil define square root as a function over complex numbers. The difficulty arises in odd roots, where the most natural definition over complex numbers disagrees with the most natural definition over real numbers when you examine the negative reals. For example, when dealing with roots of complex numbers, the most natural definition for the cube root of -1 (if you insist on making cube root a function) is cbrt(e^(pi*i))=e^(pi*i/3)=1/2+sqrt(3)*i/2, where as, over the set of reals, the only candidate for the cube root of -1 is -1. That is why I think it makes more sense to, once you start dealing with complex numbers, let nth root be multivalued (specifically, it has n values, as long as n is an integer).
August 13th, 2007, 08:47 AM #9
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Quote:
Originally Posted by roadnottaken For example, when dealing with roots of complex numbers, the most natural definition for the cube root of -1 (if you insist on making cube root a function) is cbrt(e^(pi*i))=e^(pi*i/3)=1/2+sqrt(3)*i/2, where as, over the set of reals, the only candidate for the cube root of -1 is -1.
I've always seen it defined as a function, never as a multivalued function. Of course conventions vary, but the ones I've seen choose the root as the one in the appropriate arc of the plane starting at the real axis and moving as far as it needs to -- so theta is in [0, 1/n) where 0 is the real line and the interval moves counterclockwise.
But yes, if it's multivalued you should have two answers, and if the range is defined as the reals you should have zero. Maybe the teacher should tell students how many answers to give...
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# to the nearest tenth, the best estimate of the square root of 171 is?
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# How To Apply For Welfare Assistance In Oregon
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Transcript, but in some cases the weights might not add. A usual average is easily calculated with the.
### Excel Tips: Use sumproduct to Calculate Weighted How to calculate weighted averages in Excel
In those cases, youll need to use the weighted mean formula.
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## Calculating a Weighted Average - Blacks Domain
H the weight of h) q the weight of q) e the weight of e) final grade (.95.30) (.86.50).20).90.285.43.2.90.715.2.90.2.90 -.715.2.185.185/.2.925.5 This grade calculator can help to calculate your grade or the grade you need to get.
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1 2, learn how your classes are weighted.
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### Weighted Average Grade Calculator - Blacks DomainHow can you calculate weighted percentages?
6 4, scale the average up or down. For example, if your homework and quizzes are one category, don't enter them as homework in one row and quiz in another. Add the results. So if your grades for a semester were A, B, and C in honors classes and an A and B in regular classes, your GPA for each class would.5,.5,.5,.0, and.0, respectively. For your advanced classes, add the extra number.
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Siegel - Maple Help
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algcurves
Siegel
use Siegel's algorithm for reducing a Riemann matrix
Calling Sequence Siegel(B)
Parameters
B - Riemann matrix
Description
• A Riemann matrix is a symmetric matrix whose imaginary part is strictly positive definite. In the context of algebraic curves, such a matrix is obtained as a normalized periodmatrix of the algebraic curve.
• A Siegel transformation is a transformation from the canonical basis of the homology of a Riemann surface to a new canonical basis of the homology on the Riemann surface such that:
1 The real part of the new Riemann matrix has entries that are less than or equal to $\left|\frac{1}{2}\right|$.
The imaginary part of B is strictly positive definite. Then it can be decomposed as $\mathrm{\Im }\left(B\right)=\mathrm{transpose}\left(T\right)T$. The columns of T generate a lattice L. Then
2 The length of the shortest element of L has a lower bound of $\sqrt{\frac{\sqrt{3}}{2}}$,
and
3 $\mathrm{max}\left(\left|{N}_{i}\right|\right)$ : {${\left|\mathrm{TN}\right|}^{2}\le {R}^{2}$, $N$ an integer vector} has an upper bound depending only on R and g (=dimension of B) (thus not on B).
• The Siegel(B) command returns a list $\left[\mathrm{s1},\mathrm{s2}\right]$ where $\mathrm{s1}$ is the new Riemann matrix, and $\mathrm{s2}$ is the symplectic transformation matrix on the canonical basis of the homology such that the Riemann matrix in the new basis is $\mathrm{s1}$. If B is a $g$ by $g$ matrix, then $\mathrm{s2}$ is a $2g$ by $2g$ matrix. If $\mathrm{s2}=\mathrm{Matrix}\left(\left[\left[a,b\right],\left[c,d\right]\right]\right)$, where $a,b,c$, and $d$ are $g$ by $g$ matrices, the new Riemann matrix is $\mathrm{s1}=\frac{aB+b}{cB+d}$.
Examples
> $\mathrm{with}\left(\mathrm{algcurves}\right):$
> $f≔{y}^{3}-{x}^{9}-2{x}^{3}y:$
> $b≔\mathrm{periodmatrix}\left(f,x,y,\mathrm{Riemann}\right)$
${b}{≔}\left[\begin{array}{ccc}{0.500000055125149}{+}{0.959847047127724}{}{I}& {-0.500000030191873}{-}{0.181985134690607}{}{I}& {0.641559302373033}{+}{0.570916110133834}{}{I}\\ {-0.500000001970352}{-}{0.181985123303614}{}{I}& {0.499999991378705}{+}{0.866025409210244}{}{I}& {-0.907603726955998}{-}{0.524005264964518}{}{I}\\ {0.641559334375792}{+}{0.570916144445057}{}{I}& {-0.907603757382622}{-}{0.524005277968408}{}{I}& {0.549162995371542}{+}{1.23866447525944}{}{I}\end{array}\right]$ (1)
> $s≔\mathrm{Siegel}\left(b\right):$
> ${s}_{1}$
$\left[\begin{array}{ccc}{0.499999991378705}{+}{0.866025409210244}{}{I}& {0.499999983918888}{-}{0.181985128997111}{}{I}& {-0.407603726088198}{-}{0.342020142469353}{}{I}\\ {0.499999983918888}{-}{0.181985128997111}{}{I}& {-0.499999944874851}{+}{0.959847047127724}{}{I}& {0.141559263249263}{-}{0.388930919838278}{}{I}\\ {-0.407603726088198}{-}{0.342020142469353}{}{I}& {0.141559263249263}{-}{0.388930919838278}{}{I}& {-0.233955586252134}{+}{1.05667926780828}{}{I}\end{array}\right]$ (2)
> ${s}_{2}$
$\left[\begin{array}{cccccc}{0}& {1}& {0}& {1}& {0}& {1}\\ {1}& {0}& {0}& {-1}& {1}& {-1}\\ {-1}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {1}\end{array}\right]$ (3)
References
Deconinck, B., and van Hoeij, M. "Computing Riemann Matrices of Algebraic Curves." Physica D Vol 152-153, (2001): 28-46.
Siegel, C. L. Topics in Complex Function Theory. Vol. 3. Now York: Wiley, 1973.
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https://thelaptopadviser.com/what-time-was-it-17-hours-ago-from-now-unraveling-the-time-puzzle/
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# What Time Was It 17 Hours Ago From Now: Unraveling the Time Puzzle
17 hours ago from now, the time would have been [current time – 17 hours]. It’s important to remember that the concept of time is constantly moving forward, so looking back at a specific moment can sometimes be challenging. However, with the help of basic calculations and an understanding of how time works, we can determine what the time was 17 hours ago.
To calculate the exact time 17 hours ago, you’ll need to consider both the current hour and date. Start by subtracting 17 from the current hour. If this results in a negative number, you’ll need to adjust it by adding 24 (since there are 24 hours in a day). Next, take into account any changes in the date if necessary.
For example, if it is currently 3:00 PM on July 1st and we want to know what time it was 17 hours ago, we would subtract 17 from 3:00 PM. This gives us a result of 10:00 AM on July 1st.
Calculating past times can be useful for various reasons such as tracking events or understanding historical context. Whether you’re trying to reminisce about a specific moment or simply curious about how time has passed, being able to determine what the time was several hours ago adds another dimension to our understanding of temporal relationships.
Please note that these calculations are based on standard clock times and may not account for factors such as daylight saving adjustments or changes due to different regions’ time zones.
## What Time Was It 17 Hours Ago From Now
Let’s dive into the fascinating world of time calculations and explore how to determine what time it was 17 hours ago from now. It might seem like a simple task, but there are a few key steps involved in obtaining an accurate answer.
To calculate the time 17 hours ago from now, follow these steps:
1. Start by noting down the current time: [insert current time].
2. Subtract 17 hours from the current hour value. For example, if it is currently 3:00 PM, subtracting 17 hours would give us 10:00 AM.
3. Take note of any changes in dates that occur during this calculation. If subtracting 17 hours takes you back to the previous day, adjust both the date and time accordingly.
It’s important to keep track of any changes in daylight saving time or other factors that may affect the accuracy of this calculation. Factors such as leap years and different time zones can also come into play when determining the exact time.
Remember, always double-check your calculations using reliable sources or tools to ensure accuracy, especially when dealing with critical timing situations.
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# Evaluate: $\int_{-1}^{1}\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}dt$
The value of $$\int_{-1}^{1}\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}dt$$ (where $$0)
is equal to
(A) $$\frac{1}{2a}\left(\ln(\frac{1-a}{1+a})\right)^2$$
(B) $$\frac{1}{2a}\ln(\frac{1+a}{1-a})$$
(C) $$\frac{1}{a}\ln(\frac{1+a}{1-a})$$
(D) $$\frac{1}{2a}\left(\ln(\frac{1+a}{1-a})\right)^2$$
(E) $$\frac{1}{a}\left(\ln(\frac{1-a}{1+a})\right)^2$$
My Attempt
$$I=\int_{-1}^{1}\frac{\ln(1+t)}{1-at}dt-\int_{-1}^{1}\frac{\ln(1-t)}{1-at}dt=\int_{-1}^{1}\frac{\ln(1+t)}{1-at}dt-\int_{-1}^{1}\frac{\ln(1+t)}{1+at}dt$$
$$I=\int_{-1}^{1}\ln(1+t)\left(\frac{1}{1-at}-\frac{1}{1+at}\right)dt=\int_{-1}^{1}\ln(1+t)\left(\frac{2at}{1-a^2t^2}\right)dt$$
After this I am not able to do. Is this approach correct I wonder
• consider two substitutions, one for $u=1+t$, and one for $u = 1-t$. Solve the integrals separately $ln(1+t) - ln(1-t)$ and then cancel terms Commented Mar 13 at 0:57
• (A) and (D) are the same.
– Gary
Commented Mar 13 at 3:58
Letting $$x=\frac{1-t}{1+t}$$ transforms the integral into \begin{aligned} I & =-2 \int_0^{\infty} \frac{\ln x}{[(1-a)+(1+a) x](1+x)} d x \\ & =-\frac{2}{1+a} \underbrace{ \int_0^{\infty} \frac{\ln x}{\left(x+k\right)(1+x)} d x}_{J} \end{aligned} where $$k= \frac{1-a}{1+a}$$.
Putting $$x\mapsto\frac{k}{x}$$ changes \begin{aligned} J & =\int_0^{\infty} \frac{\ln k-\ln x}{\left(1+x\right)(x+k)} d x \\ & =\ln k \int_0^{\infty} \frac{d x}{\left(1+x\right)(x+k)}-J\\&= \frac{ \ln k}{2} \int_0^{\infty} \frac{d x}{\left(1+x\right)(x+k)}\\&=\frac{\ln k}{2} \cdot \frac{1}{k-1} \int_0^{\infty} \left( \frac{1}{1+x}-\frac{1}{x+k}\right) d x\\&= \frac{\ln ^2 k}{2(k-1)}\\&= \frac{\ln ^2\left(\frac{1-a}{1+a}\right)}{\frac{-4a}{1+a}} \end{aligned} Hence $$\boxed{I=\frac{1}{2 a} \ln ^2\left(\frac{1-a}{1+a}\right)}$$
\begin{align} &\int_{-1}^1\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}\ \overset{t\to \frac{1-t}{1+t}}{dt}\\ =& -\frac2{1+a}\int_0^\infty \frac{\ln t}{(t+1)(t+ \frac{1-a}{1+a})} \overset{t\to \frac{1-a}{1+a}\frac1t}{dt}\\ =& -\frac2{1+a}\int_0^\infty \frac{\ln \frac{1-a}{1+a}-\ln t}{(t+1)(t+ \frac{1-a}{1+a})} {dt}\\ =& -\frac{\ln \frac{1-a}{1+a}}{1+a}\int_0^\infty \frac{1}{(t+1)(t+ \frac{1-a}{1+a})} {dt}\\ =& -\frac{\ln \frac{1-a}{1+a}}{1+a} \bigg( -\frac{1+a}{2a}{\ln\frac{1-a}{1+a}}\bigg)= \frac{1}{{2a}} \ln^2\frac{1-a}{1+a} \end{align}
\begin{aligned} \int_{-1}^1\ln\left(\frac{1+t}{1-t}\right)\frac{1}{1-at}\mathrm dt&=\int_{-1}^1\ln(1+t)\frac{2at}{1-a^2t^2}\mathrm dt\\ &=-2at\int_{-1}^1\sum_{k\geq 1}\frac{(-1)^kt^{k}}{k}\sum_{n\geq 0}a^{2n}t^{2n}\mathrm dt\\ &=-2\sum_{k\geq 1}\sum_{n\geq 0}\frac{(-1)^ka^{2n+1}}{k}\int_{-1}^1t^{2n+k+1}\mathrm dt\\ &=2\sum_{k\geq 0}\sum_{n\geq 0}\frac{(-1)^ka^{2n+1}}{k+1}\int_{-1}^1t^{2n+k+2}\mathrm dt \end{aligned} In view that \int_{-1}^1t^{2n+k+2}=\left\{ \begin{aligned} &0,\ 2m+k+2\ \text{odd}\\ &\frac{2}{2m+2k+3},\ 2m+k+2\ \text{even} \end{aligned} \right. , this means that only the terms with even $$k$$ will "survive" because $$2n+2+k=$$even$$+k=$$even. So we'll transform $$k\to 2k$$: \begin{aligned} 2\sum_{k\geq 0}\sum_{n\geq 0}\frac{a^{2n+1}}{2k+1}\int_{-1}^1t^{2(n+k+1)}\mathrm dt&=4\sum_{k\geq 0}\sum_{n\geq 0}\frac{a^{2n+1}}{(2k+1)(2k+2n+3)}\\ &= (...)\\ &=2\sum_{i\geq 0}\sum_{j\geq 0}\frac{a^{2i+2j+1}}{(2i+1)(2j+1)}\\ &=\frac{1}{2a}\left(\ln\left(\frac{1+a}{1-a}\right)\right)^2\implies \mathbf{(A)=(D)}\ \text{is the solution} \end{aligned}
Though I kinda cheated in this last part, I'd need help on finding the steps in $$(...)$$ to convert the double summation I got into the other one since I don't really know how to do that...
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# Sandra Oh Under The Microscope (11/03/2019)
How will Sandra Oh fare on 11/03/2019 and the days ahead? Let’s use astrology to undertake a simple analysis. Note this is of questionable accuracy – do not take this too seriously. I will first work out the destiny number for Sandra Oh, and then something similar to the life path number, which we will calculate for today (11/03/2019). By comparing the difference of these two numbers, we may have an indication of how well their day will go, at least according to some astrology people.
PATH NUMBER FOR 11/03/2019: We will analyze the month (11), the day (03) and the year (2019), turn each of these 3 numbers into 1 number, and add them together. What does this entail? We will show you. First, for the month, we take the current month of 11 and add the digits together: 1 + 1 = 2 (super simple). Then do the day: from 03 we do 0 + 3 = 3. Now finally, the year of 2019: 2 + 0 + 1 + 9 = 12. Now we have our three numbers, which we can add together: 2 + 3 + 12 = 17. This still isn’t a single-digit number, so we will add its digits together again: 1 + 7 = 8. Now we have a single-digit number: 8 is the path number for 11/03/2019.
DESTINY NUMBER FOR Sandra Oh: The destiny number will calculate the sum of all the letters in a name. Each letter is assigned a number per the below chart:
So for Sandra Oh we have the letters S (1), a (1), n (5), d (4), r (9), a (1), O (6) and h (8). Adding all of that up (yes, this can get tedious) gives 35. This still isn’t a single-digit number, so we will add its digits together again: 3 + 5 = 8. Now we have a single-digit number: 8 is the destiny number for Sandra Oh.
CONCLUSION: The difference between the path number for today (8) and destiny number for Sandra Oh (8) is 0. That is lower than the average difference between path numbers and destiny numbers (2.667), indicating that THIS IS A GOOD RESULT. But this is just a shallow analysis! As mentioned earlier, this is just for fun. If you want to see something that people really do vouch for, check out your cosmic energy profile here. Go ahead and see what it says for you – you’ll be glad you did.
### Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
#### Latest posts by Abigale Lormen (see all)
Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
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Teachers Pay Teachers
Multiplication Centers Multiplication Games
Subjects
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Product Rating
4.0
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PDF (Acrobat) Document File
63.65 MB | 40 pages
PRODUCT DESCRIPTION
I know your students will love these centers as much as mine have!
I designed these centers to be engaging and purposeful. If students are engaged in meaningful practice I am free to run small math groups that maximize student learning.
In this download you will find 10 centers. These centers will help students practice and apply skills involving properties of multiplication (distributive, associative, and commutative properties) as well as fact families, problem solving, and using place value to solve multiplication problems. Check out the preview for a closer look.
You might also be interested in:
Multiplication Centers: The Basics
Multiplication Centers: The Bundle
Multiplication Exit Slips
Each center has student friendly directions right on the game board or recording sheet. The materials for each center are minimal. Students may need a paperclip, pencil, dice, any item that will serve as a game piece, and cards (included).
Centers:
Properties of Multiplication: Students place the cards in a stack. They choose a card and determine the property shown between the two equations and record their answer on the recording sheet. Then, they solve the highlighted equation on each card.
Flip, Break, Solve: Practice double digit multiplication using the distributive property. The recording sheet guides students through the process step by step. Students choose a card, and find the product step by step.
Multiplication BIG Time: In this center students will put their knowledge of multiplication and place value to work. Students roll a dice to get their single digit factor. Then they spin the spinner to get the double digit factor ending in zero and solve. For a bonus, and for students needing to extend learning students divide the product by 2 or in half.
Double Digit X Race: With a partner, students race around the game board solving equations with one single digit factor and another double digit factor. Who will have the most symbols on the game board when the race is finished?
Zero Multi-digit Memory: In this game of memory partners practice matching equations ending in zero with the product. The player with the most matches at the end of the game wins!
Smartie Products: With a partner, students practice solving two double digit factors ending in zero. Spin the spinner to determine which equation on the game board to solve. The student who collects the most game pieces wins!
Associative Property Solve & Compare: Using the spinner, students create two problems that use the associative property. After solving each equation compare their products. (INDEPENDENT)
Distribute and Solve: Students choose a card that demonstrates the distributive property. Then using their knowledge of the distributive property and the step by step directions on the recording sheet students change the equation and solve. (INDEPENDENT)
ZING! Match or Solve?: Students match up pairs of cards using the distributive property and record them on the recording sheet. If the card says ZING! Follow the directions to solve. (INDEPENDENT)
Problem Solving Roll n’ Race: Students take turns drawing a card and either solving the story problem or writing an equation using a letter for the unknown variable. If the other player agrees student rolls a dice and moves on the game board. The first player to reach the finish line wins! (PARNTER)
Check out the preview for a closer look!
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Factoring Perfect Cube Trinomials // jonathanescapes.com
# How to Factor a Perfect Cube Sciencing.
A perfect cube is a number that can be written as a^3. When factoring a perfect cube, you would get a a a, where "a" is the base. Two common factoring procedures dealing with perfect cubes are factoring sums and differences of perfect cubes. To do this, you will need to factor. 21/11/2016 · This algebra video tutorial focuses on factoring perfect square trinomials. This video provides a formula that will help to do so. It contains plenty of examples and practice problems for you to work on. Factoring Other Types of Trinomials Worksheets for factoring Perfect Square Trinomials In this lesson, we will learn how to factor perfect square trinomials. The following diagrams show the factoring and expanding of Perfect Square Trinomials. Scroll down the page for examples and solutions of factoring Perfect Square Trinomials. 18/05/2014 · This video by Fort Bend Tutoring shows the process of finding and factoring perfect square trinomials. Ten 10 examples showing how to factor a perfect square trinomial are displayed in this video. This math concept is. 24/11/2016 · This algebra video tutorial focuses on factoring special cases such as factoring different forms of binomials and special products of polynomials specifically perfect square trinomials, difference of perfect squares, and.
27/07/2015 · This feature is not available right now. Please try again later. Sum of Two Perfect Cubes. You will need to know how to factor the sum of perfect cubes for your math test. An algebraic expression for the sum of perfect cubes is as follows: x 3y 3. The form for factoring the sum of perfect cubes is: x 3y 3 = xyx 2 – xyy 2 You should also know the above above form by heart for your math test. Factoring perfect square trinomials Before we explain the straightforward way of factoring perfect square trinomials, we need to define the expression perfect square trinomial.
24/01/2013 · Factoring the difference and sum of cubes 1.Video Highlights: 00:00 Example of factoring the sum of cubes 02:15 Example of factorin. Note: The quadratic portion of each cube formula does not factor, so don't waste time attempting to factor it. Yes, a 2 – 2abb 2 and a 22abb 2 factor, but that's because of the 2 's on their middle terms. These sum- and difference-of-cubes formulas' quadratic terms do not have that "2", and thus cannot factor. Learn perfect cubes factoring square trinomials with free interactive flashcards. Choose from 500 different sets of perfect cubes factoring square trinomials flashcards on Quizlet. Perfect Square trinomial. Chart of Squares & Cube s. Learn these perfect squares and perfect cubes!!!! Perfect Squares Perfect Cubes. Factoring Special Binomials: Difference of Cubes & Sum of Cubes. Difference fo cubes: Pattern. Sum of Cubes: The difference or sum of two perfect cube terms have factors of a binomial times a trinomial.
Learn how to factor quadratics that have the "perfect square" form. For example, write x²6x9 as x3². If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains. and. are unblocked. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers. ©B W2L0W1l4L WKGuftSaV fS`o]fltowrarrLe] RLjL[Ck.R G wAZlfli CrBiqg^hAtOs[ Er\e`sfejrMvLeUdF.Q ^ WMka^dReb UwniCtthz nIunyftiTnAiGt^em CAslJgxembRrjaC U2Y.
Learn perfect cubes factoring square trinomials sum with free interactive flashcards. Choose from 116 different sets of perfect cubes factoring square trinomials sum flashcards on Quizlet. Factoring a Non-perfect Square Trinomial. Trinomials with leading coefficient one Trinomials with leading coefficient some number other than one. Trinomials with Leading Coefficient One Consider xaxb = x²bxaxab = x²axbaab Combine like terms. 17/10/2009 · To factor a cubic polynomial, start by grouping it into 2 sections. Then, find what's common between the terms in each group, and factor the commonalities out of the terms. If each of the 2 terms contains the same factor, combine them. Finally, solve for the variable in the roots to get your solutions. Learn perfect cubes trinomials factoring polynomials difference with free interactive flashcards. Choose from 51 different sets of perfect cubes trinomials factoring polynomials difference flashcards on Quizlet.
## Factoring Special Cases, and Forms of Binomials..
21/12/2019 · In this lesson, you will learn how to factor sums and differences of perfect cube binomials. You will learn to use the formula that gives you the factors through a few examples. Learn perfect cubes factoring polynomials trinomials sum with free interactive flashcards. Choose from 30 different sets of perfect cubes factoring polynomials trinomials sum flashcards on Quizlet. Learn perfect cubes square trinomials with free interactive flashcards. Choose from 500 different sets of perfect cubes square trinomials flashcards on Quizlet. ©K P2 T0I1 G2X CKsu Dt3aa OSlo uflt gw ga yroe 5 rL 9LnCw.3 s dAqlrl e Gr5iRgJhCtHs0 7rFelsOear tvNeMdM.L K aM 8a 5d FeQ pwxiGtih K tI snIf si 5n 1ibtfe u aA Tl 0g secb 5rPa 9 X2t.
### Factoring perfect square trinomials - Basic.
Factoring Perfect Cubes Perfect Cubes: The cube of a number is that number raised to the third power. The number that results is called a perfect cube. Example: 23 =2×2×2 =8 The cube of 2 is 8 and 8 is the perfect cube of 2. 33 =3×3×3 =27 The cube of 3 is 27 and 27 is the perfect cube of3. There is one "special" factoring type that can actually be done using the usual methods for factoring, but, for whatever reason, many texts and instructors make a big deal of treating this case separately. "Perfect square trinomials" are quadratics which are the results of squaring binomials. Remember that "trinomial" means "three-term.
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## Tuesday, 3 May 2011
### Explain the law of Diminishing Marginal utility and discuss its limitations and importance
LAW OF DIMINISHING MARGINAL UTILITY :- It is a common experience of every consumer that as he gets more units of a particular, commodity the marginal utility goes on diminishing. This tendency on the part of a marginal utility to diminish with every increase in the stock of a thing is called law of diminishing marginal utility.
MARSHALL SAYS, " The additional benefit which a person derives from an increase of his stock of a thing diminishes with every increase in stock that he already has."
EXAMPLE OF DIMINISHING MARGINAL UTILITY :- This law can be explained by the following example. Suppose in the month of June a person start drinking water. First glass of water has a great utility for him. If he takes the second glass of water, the utility will be less than the first. If he drinks the third glass , the utility of third will be less than the second, and so on. The utility goes on diminishing with the consumption of every next unit and it drops down to zero. If the consumer is forced further, the utility will become negative. This law can also be explained by the following table :
EXPLANATION :- The above table show that first glass of water gives units of utility to the thirsty man. When he takes second the marginal utility drops down to 8. When he consumes the 6th glass the marginal utility drops down to zero and by the use of 7th it becomes negative.
EXPLANATION :- Along "OX" we measure the units of commodity consumed along "OY" utility derived from them. The utility of the first glass of water is represented by the first rectangle and second glass by the second rectangle and so on. FF' curve is the diminishing utility curve.
ASSUMPTIONS OF DIMINISHING MARGINAL UTILITY
1. NATURE OF THE COMMODITY :- There should be no change in the nature of the commodity. For example, If first mango taken is not better, while the second is better, then the utility will not decrease and the utility of second will be greater than first.
2. REASONABLE UNITS :- It is assumed that the units of a commodity which are used should be suitable and reasonable if the units are too small then this law will not operate.
3. CONTINUOUS USE :- It is also assumed that the units of the commodity should be used continuously. If there is interval between the consumption the same two units then the law will not be applicable.
4. NO CHANGE IN INCOME :- It is also assumed that the income of the consumer should not change, otherwise the law may not operate.
5. NO CHANGE IN FASHION AND CUSTOMS :- If there is a sudden change in fashion or customs of a consumer, the law may not operate.
6. RARE COLLECTIONS :- If there are two diamonds in the world the possession of the second diamond will push up the marginal utility.
7. NO CHANGE IN THE STOCK OF OTHER PEOPLE :- Sometimes an increase in the stock of a commodity increases the marginal utility. For example the number of telephone increase in the city, but the utility of our telephone increases.
8. STATE OF MIND SHOULD NOT CHANGE :- If a consumer has been told that mango is a tonic for his health, then marginal utility will increase instead of falling.
EXCEPTIONS OR LIMITATIONS
1. DESIRE OF MONEY :- This law is not applicable in case of money with an increase in wealth man wants to get more and more.
2. DESIRE OF KNOWLEDGE :- Some experts say that man wants to get more and more knowledge so the law can not be applied in this case.
3. USE OF LIQUOR :- With the additional use of liquor like wine marginal utility also goes on increasing.
4. PERSONAL HOBBY :- In case of hobby also this law can not operate. For example , as the collection of tickets increases, its utility also increases.
5. FASHION :- Utility also depends upon fashion. If the fashion of any commodity changes, its utility drops down to zero. On the other hand if fashion exists then utility increases.
#### 7 comments:
Anonymous,
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# Proportions and Ratios
### Definition of Ratio
A ratio is a relationship between two values. For instance, a ratio of 1 pencil to 3 pens would imply that there are three times as many pens as pencils. For each pencil there are 3 pens, and this is expressed in a couple ways, like this: 1:3, or as a fraction like 1/3. There do not have to be exactly 1 pencil and 3 pens, but some multiple of them. We could just as easily have 2 pencils and 6 pens, 10 pencils and 30 pens, or even half a pencil and one-and-a-half pens! In fact, that is how we will use ratios -- to represent the relationship between two numbers.
### Definition of Proportion
A proportion can be used to solve problems involving ratios. If we are told that the ratio of wheels to cars is 4:1, and that we have 12 wheels in stock at the factory, how can we find the number of cars we can equip? A simple proportion will do perfectly. We know that 4:1 is our ratio, and the number of cars that match with those 12 wheels must follow the 4:1 ratio. We can setup the problem like this, where x is our missing number of cars:
$$\frac{4}{1}=\frac{12}{x}$$
To solve a proportion like this, we will use a procedure called cross-multiplication. This process involves multiplying the two extremes and then comparing that product with the product of the means. An extreme is the first number (4), and the last number (x), and a mean is the 1 or the 12.
To multiply the extremes we just do $$4 * x = 4x$$. The product of the means is $$1 * 12 = 12$$. The process is very simple if you remember it as cross-multiplying, because you multiply diagonally across the equal sign.
You should then take the two products, 12 and 4x, and put them on opposite sides of an equation like this: $$12 = 4x$$. Solve for x by dividing each side by 4 and you discover that $$x = 3$$. Reading back over the problem we remember that x stood for the number of cars possible with 12 tires, and that is our answer.
It is possible to have many variations of proportions, and one you might see is a double-variable proportion. It looks something like this, but it easy to solve.
$$\frac{16}{x}=\frac{x}{1}$$
Using the same process as the first time, we cross multiply to get $$16 * 1 = x * x$$. That can be simplified to $$16 = x^2$$, which means x equals the square root of 16, which is 4 (or -4). You've now completed this lesson, so feel free to browse other pages of this site or search for more lessons on proportions.
## Ratios and Proportions Calculator
Use the tool below to convert between fractions and decimal, or to take a given ratio expression and solve for the unknown value.
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# Cavalieri's quadrature formula
Cavalieri's quadrature formula computes the area under the cubic curve, together with other higher powers.
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral
${\displaystyle \int _{0}^{a}x^{n}\,dx={\tfrac {1}{n+1}}\,a^{n+1}\qquad n\geq 0,}$
and generalizations thereof. This is the definite integral form; the indefinite integral form is:
${\displaystyle \int x^{n}\,dx={\tfrac {1}{n+1}}\,x^{n+1}+C\qquad n\neq -1.}$
There are additional forms, listed below. Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials.
The term "quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve y = xn. Traditionally important cases are y = x2, the quadrature of the parabola, known in antiquity, and y = 1/x, the quadrature of the hyperbola, whose value is a logarithm.
## Forms
### Negative n
For negative values of n (negative powers of x), there is a singularity at x = 0, and thus the definite integral is based at 1, rather than 0, yielding:
${\displaystyle \int _{1}^{a}x^{n}\,dx={\tfrac {1}{n+1}}(a^{n+1}-1)\qquad n\neq -1.}$
Further, for negative fractional (non-integer) values of n, the power xn is not well-defined, hence the indefinite integral is only defined for positive x. However for n a negative integer the power xn is defined for all non-zero x, and the indefinite integrals and definite integrals are defined, and can be computed via a symmetry argument, replacing x by −x, and basing the negative definite integral at −1.
Over the complex numbers the definite integral (for negative values of n and x) can be defined via contour integration, but then depends on choice of path, specifically winding number – the geometric issue is that the function defines a covering space with a singularity at 0.
### n = −1
There is also the exceptional case n = −1, yielding a logarithm instead of a power of x:
${\displaystyle \int _{1}^{a}{\frac {1}{x}}\,dx=\ln a,}$
${\displaystyle \int {\frac {1}{x}}\,dx=\ln x+C,\qquad x>0}$
(where "ln" means the natural logarithm, i.e. the logarithm to the base e = 2.71828...).
The improper integral is often extended to negative values of x via the conventional choice:
${\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C,\qquad x\neq 0.}$
Note the use of the absolute value in the indefinite integral; this is to provide a unified form for the integral, and means that the integral of this odd function is an even function, though the logarithm is only defined for positive inputs, and in fact, different constant values of C can be chosen on either side of 0, since these do not change the derivative. The more general form is thus:[1]
${\displaystyle \int {\frac {1}{x}}\,dx={\begin{cases}\ln |x|+C^{-}&x<0\\\ln |x|+C^{+}&x>-0\end{cases}}}$
Over the complex numbers there is not a global antiderivative for 1/x, due this function defining a non-trivial covering space; this form is special to the real numbers.
Note that the definite integral starting from 1 is not defined for negative values of a, since it passes through a singularity, though since 1/x is an odd function, one can base the definite integral for negative powers at −1. If one is willing to use improper integrals and compute the Cauchy principal value, one obtains ${\displaystyle \int _{-c}^{c}{\frac {1}{x}}\,dx=0,}$ which can also be argued by symmetry (since the logarithm is odd), so ${\displaystyle \int _{-1}^{1}{\frac {1}{x}}\,dx=0,}$ so it makes no difference if the definite integral is based at 1 or −1. As with the indefinite integral, this is special to the real numbers, and does not extend over the complex numbers.
### Alternative forms
The integral can also be written with indexes shifted, which simplify the result and make the relation to n-dimensional differentiation and the n-cube clearer:
${\displaystyle \int _{0}^{a}x^{n-1}\,dx={\tfrac {1}{n}}a^{n}\qquad n\geq 1.}$
${\displaystyle \int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}+C\qquad n\neq 0.}$
More generally, these formulae may be given as:
${\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {1}{ax+b}}dx={\frac {1}{a}}\ln \left|ax+b\right|+C}$
More generally:
${\displaystyle \int {\frac {1}{ax+b}}\,dx={\begin{cases}{\frac {1}{a}}\ln \left|ax+b\right|+C^{-}&x<-b/a\\{\frac {1}{a}}\ln \left|ax+b\right|+C^{+}&x>-b/a\end{cases}}}$
## Proof
The modern proof is to use an anti-derivative: the derivative of xn is shown to be nxn−1 – for non-negative integers. This is shown from the binomial formula and the definition of the derivative – and thus by the fundamental theorem of calculus the antiderivative is the integral. This method fails for ${\displaystyle \int {\frac {1}{x}}\,dx,}$ as the candidate antiderivative is ${\displaystyle {\frac {1}{0}}\cdot x^{0}}$, which is undefined due to division by zero. The logarithm function, which is the actual antiderivative of 1/x, must be introduced and examined separately.
The derivative ${\displaystyle (x^{n})'=nx^{n-1}}$ can be geometrized as the infinitesimal change in volume of the n-cube, which is the area of n faces, each of dimension n − 1.
Integrating this picture – stacking the faces – geometrizes the fundamental theorem of calculus, yielding a decomposition of the n-cube into n pyramids, which is a geometric proof of Cavalieri's quadrature formula.
For positive integers, this proof can be geometrized:[2] if one considers the quantity xn as the volume of the n-cube (the hypercube in n dimensions), then the derivative is the change in the volume as the side length is changed – this is xn−1, which can be interpreted as the area of n faces, each of dimension n − 1 (fixing one vertex at the origin, these are the n faces not touching the vertex), corresponding to the cube increasing in size by growing in the direction of these faces – in the 3-dimensional case, adding 3 infinitesimally thin squares, one to each of these faces. Conversely, geometrizing the fundamental theorem of calculus, stacking up these infinitesimal (n − 1) cubes yields a (hyper)-pyramid, and n of these pyramids form the n-cube, which yields the formula. Further, there is an n-fold cyclic symmetry of the n-cube around the diagonal cycling these pyramids (for which a pyramid is a fundamental domain). In the case of the cube (3-cube), this is how the volume of a pyramid was originally rigorously established: the cube has 3-fold symmetry, with fundamental domain a pyramids, dividing the cube into 3 pyramids, corresponding to the fact that the volume of a pyramid is one third of the base times the height. This illustrates geometrically the equivalence between the quadrature of the parabola and the volume of a pyramid, which were computed classically by different means.
Alternative proofs exist – for example, Fermat computed the area via an algebraic trick of dividing the domain into certain intervals of unequal length;[3] alternatively, one can prove this by recognizing a symmetry of the graph y = xn under inhomogeneous dilation (by d in the x direction and dn in the y direction, algebraicizing the n dimensions of the y direction),[4] or deriving the formula for all integer values by expanding the result for n = −1 and comparing coefficients.[5]
## History
Archimedes computed the area of parabolic segments in his The Quadrature of the Parabola.
A detailed discussion of the history, with original sources, is given in Template:Harv; see also history of calculus and history of integration.
The case of the parabola was proven in antiquity by the ancient Greek mathematician Archimedes in his The Quadrature of the Parabola (3rd century BCE), via the method of exhaustion. Of note is that Archimedes computed the area inside a parabola – a so-called "parabolic segment" – rather than the area under the graph y = x2, which is instead the perspective of Cartesian geometry. These are equivalent computations, but reflect a difference in perspective. The Ancient Greeks, among others, also computed the volume of a pyramid or cone, which is mathematically equivalent.
In the 11th century, the Islamic mathematician Ibn al-Haytham (known as Alhazen in Europe) computed the integrals of cubics and quartics (degree three and four) via mathematical induction, in his Book of Optics.[6]
The case of higher integers was computed by Cavalieri for n up to 9, using his method of indivisibles (Cavalieri's principle).[7] He interpreted these as higher integrals as computing higher-dimensional volumes, though only informally, as higher-dimensional objects were as yet unfamiliar.[8] This method of quadrature was then extended by Italian mathematician Evangelista Torricelli to other curves such as the cycloid, then the formula was generalized to fractional and negative powers by English mathematician John Wallis, in his Arithmetica Infinitorum (1656), which also standardized the notion and notation of rational powers – though Wallis incorrectly interpreted the exceptional case n = −1 (quadrature of the hyperbola) – before finally being put on rigorous ground with the development of integral calculus.
Prior to Wallis's formalization of fractional and negative powers, which allowed explicit functions ${\displaystyle y=x^{p/q},}$ these curves were handled implicitly, via the equations ${\displaystyle x^{p}=ky^{q}}$ and ${\displaystyle x^{p}y^{q}=k}$ (p and q always positive integers) and referred to respectively as higher parabolae and higher hyperbolae (or "higher parabolas" and "higher hyperbolas"). Pierre de Fermat also computed these areas (except for the exceptional case of −1) by an algebraic trick – he computed the quadrature of the higher hyperbolae via dividing the line into equal intervals, and then computed the quadrature of the higher parabolae by using a division into unequal intervals, presumably by inverting the divisions he used for hyperbolae.[9] However, as in the rest of his work, Fermat's techniques were more ad hoc tricks than systematic treatments, and he is not considered to have played a significant part in the subsequent development of calculus.
Of note is that Cavalieri only compared areas to areas and volumes to volumes – these always having dimensions, while the notion of considering an area as consisting of units of area (relative to a standard unit), hence being unitless, appears to have originated with Wallis;[10][11] Wallis studied fractional and negative powers, and the alternative to treating the computed values as unitless numbers was to interpret fractional and negative dimensions.
The exceptional case of −1 (the standard hyperbola) was first successfully treated by Grégoire de Saint-Vincent in his Opus geometricum quadrature circuli et sectionum coni (1647), though a formal treatment had to wait for the development of the natural logarithm, which was accomplished by Nicholas Mercator in his Logarithmotechnia (1668).
## References
1. "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012
2. See Rickey.
3. Template:Harv
4. Template:Harv
5. Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163–174 [165–9 & 173–4]
6. Template:Harv
7. Template:Harv – see Informal pedagogical synopsis of the Analysis chapter for brief form
8. See Rickey reference for discussion and further references.
9. Ball, 281
10. Britannica, 171
### History
• Cavalieri, Geometria indivisibilibus (continuorum nova quadam ratione promota) (Geometry, exposed in a new manner with the aid of indivisibles of the continuous), 1635.
• Cavalieri, Exercitationes Geometricae Sex ("Six Geometrical Exercises"), 1647
• in Dirk Jan Struik, editor, A source book in mathematics, 1200–1800 (Princeton University Press, Princeton, New Jersey, 1986). ISBN 0-691-08404-1, ISBN 0-691-02397-2 (pbk).
• Mathematical expeditions: chronicles by the explorers, Reinhard Laubenbacher, David Pengelley, 1998, Section 3.4: "Cavalieri Calculates Areas of Higher Parabolas", pp. 123–127/128
• A short account of the history of mathematics, Walter William Rouse Ball, "Cavalieri", p. 278–281
• "Infinitesimal calculus", Encyclopaedia of Mathematics
• The Britannica Guide to Analysis and Calculus, by Educational Britannica Educational, p. 171 – discusses Wallace primarily
### Proofs
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# Force as a function of velocity as $t$ goes to infinity, strange result
Here is the question:
"A particle with mass m is given an initial velocity $$v_0$$ so that it moves in a straight line (you can consider it positive). It is subject only to a force that is inversely proportional to the square of its speed, as $$F = −c_xv^2$$.
(a) Write down Newton’s 2nd law for the particle, and the corresponding differential equation for the velocity as a function of time.
(b) Integrate the equation of motion to find out v(t).
(c) Integrate again to find out x(t).
(d) What is the total distance the particle will travel? Why? For full marks, you need to explain clearly why the answer makes physical sense. Hint: What is the limit of x(t) when t → ∞?.
I solved the differential equation and got:
$$v(t)=\frac{1}{\frac{ct}{m} + \frac{1}{v_0}}$$
And
$$x(t)=\frac{m}{c}ln(ct/m+v_0^-1)+x_0$$
But as t goes to infinity, v approaches 0, suggesting the distance is finite, and x goes to infinity which is contradictory. Also, given that the object as an initial positive velocity and is subject to a retarding force, won't v quickly reach 0 and the object will stop moving? That's not reflected in the equations of motion.
Ok, so here's what I'm thinking. The retarding force is proportional to the square of velocity. Eventually the velocity will be $$0 and every change in velocity will lead to a smaller and smaller acceleration that approaches 0 but never reaches it (or only reaches it at $$x=\infty$$.
• what is cx here? Oct 17, 2020 at 6:14
• @sheltonBenjamim it's some constant. Oct 17, 2020 at 6:28
• It occurs to me, isn't this a non linear differential equation? Oct 17, 2020 at 6:29
• This question might help you out: math.stackexchange.com/questions/3808999/… Oct 17, 2020 at 9:05
• This is similar to the sum $\sum_n^\infty 1/n$. $1/n$ is decreasing but it's not decreasing fast enough for $S_n$ to converge, the sum is still divergent. This $\sum_n^\infty 1/{n^2}$ sum however converges to $\pi^2 /6$. Oct 17, 2020 at 19:39
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Last updated on 6/23/22
## Test a Hypothesis
Hypothesis testing is a statistical method that helps decide if the observation of a variable can be trusted. Observation of a variable means making a measurement or assuming some property of the variable based on a finite number of samples.
The observation can be:
• A measure: mean, median, standard deviation, percentile, etc.
• comparison between two populations: correlation, difference in means, etc.
• Some intrinsic characteristic of the variable: is normally distributed, stationary, etc.
Hypothesis testing is commonly used across many industries and drives a wide range of statistical tests. It's also at the center of model selection and interpretation in linear and logistic regression!
In this chapter, we start with an overview of hypothesis testing. We then apply different statistical tests to our school and auto-mpg datasets to illustrate hypothesis testing and understand its strength and shortcomings
The school dataset has the height, weight, and age of 237 school children but also their gender with 111 girls and 126 boys. It would be interesting to compare the height of boys and girls and see if the difference is statistically significant.
If you take the height average from the sample dataset, you have:
Boys seem taller on average than girls, though not by much. The difference is only 4 cm. Is that difference statistically significant?
Asking this statistically significant question implies that although you observe a difference, that observation may be just a random effect caused by the samples in the dataset. Can that observation be trusted, is it reliable, or is it just a fluke?
This is where hypothesis testing can help. The general idea is to calculate the probability that your observation is due to some random factor. When that probability is small, you conclude that the observation can be trusted.
Imagine that you want to assess that girls and boys don't have the same average height ( ) using the schoolchildren example. Therefore, you want to be able to reject the null hypothesis that girls and boys have the same average height.
You have the null hypothesis:
You would like to reject in favor of the alternative hypothesis:
To compare the means of two different populations, use the t-test. The Python SciPy library has a t-test function. The documentation indicates that:
scipy.stats.ttest_ind(a, b, ...)
Calculate the t-test for the means of two independent samples of scores.
This is a two-sided test for the null hypothesis that two independent samples have identical average values.
Let's open the dataset and collect the heights of girls and boys in an array:
import pandas as pd
G = df[df.sex == 'f'].height.values
B = df[df.sex == 'm'].height.values
Applying the t-test to the two populations boils down to these two lines of code:
from scipy import stats
result = stats.ttest_ind(G,B)
print(result)
The result is:
Ttest_indResult(statistic=-3.1272160497574646, pvalue=0.0019873593355122674)
Since the p-value is smaller than the standard threshold of p=0.05, you can reject the null hypothesis of equal averages. The conclusion is that, regarding the schoolchildren population, boys are not the same height as girls.
##### Did You Notice?
You probably noticed that we did not conclude that boys are taller than girls because we did not define the null and alternative hypothesis as:
#### Another Example
Instead of looking at heights, let's compare the weight of boys versus girls in that school. The question is:
• Is the difference in mean weight between boys and girls statistically significant?
As an exercise, you can apply the same t-test to the weights of boys and girls and determine if the difference in average is statistically significant.
Once you're done, check out the answer below!
Here's the code:
average_weight_girls = df[df.sex == 'f'].weight.mean()
You may have noticed that the t-test returns values with two values: statistic and p-value. Let's take a look at both of these below.
#### The P-Value
The p-value is the probability of the observation, assuming that the null hypothesis is true.
This is graphically illustrated by:
Graphic by Chen-Pan Liao from wikimedia.org
This graph shows the probability distribution of observation for a given statistical test, assuming the null hypothesis in the special case, where that probability distribution is normal. When the probability distribution is known, it possible to calculate its percentiles and confidence intervals. The standard 0.05 p-value threshold corresponds to the 95% confidence interval of the probability distribution.
#### The T-Test Statistic
The t-test returns another value: the statistic. What is that?
In statistics, a statistic is just another name for a measure or calculation. The average, the correlation, and the median are all statistics.
In practice, the way hypothesis testing works is a bit of a mind twister:
1. Start by choosing the adequate statistical test (we chose the t-test).
2. Then define the hypothesis you want to reject, calling it the null hypothesis.
3. Calculate the p-value, which is the probability of the observation assuming this null hypothesis.
4. When the p-value is small, reject the null hypothesis.
The hypothesis that what you observe happened by chance is the null hypothesis and is noted .
The opposite hypothesis is called the alternative hypothesis and is noted .
The probability of the observation, assuming the null hypothesis is true, is called the p-value.
If the p-value is below a certain threshold, usually p=0.05, you can conclude that the null hypothesis is unlikely to be true and reject it; thus, adopting the alternative hypothesis when both are mutually exclusive.
Need a review of the t-test? Check out this screencast below! If you want to explore further, you can find a good explanation of the different types of t-tests on the Investopedia website.
What if the observation is some intrinsic characteristic of the variable?
Good question! Let's look at that next.
Here's another example of hypothesis testing; this time, assessing the nature of the distribution of a variable.
Plotting the histogram and probability distribution of the variable acceleration from the auto-mpg dataset is demonstrated on the following graph:
Compared to the distribution of a normal distribution, the two plots look very similar. So it looks like the acceleration variable is normally distributed, but is it? How confident of that conclusion are you just looking at the graph? This is a question that hypothesis testing can help answer.
#### Normal Distribution
But first, a quick word on the normal distribution.
A variable is said to follow a normal distribution when it is shaped like a bell curve:
A more formal mathematical definition is that the probability density of the normal distribution is:
Where is the mean or expectation of the distribution (and also its median and mode), and is the standard deviation. A variable that follows a normal distribution of mean and variance is noted:
The normal distribution is omnipresent in statistical and machine learning. It possesses many elegant and efficient properties.
#### Normality Test
A common way to test the normality of a variable is the Kolmogorov-Smirnov test for goodness of fit. The Kolmogorov-Smirnov tests the null hypothesis that two distributions are identical.
Here we will test the hypothesis that the acceleration variable and a normal distribution of similar mean and variance are identical.
So, in this case, the null hypothesis is the observation that we want to verify. If the data shows that we should reject the null hypothesis, we conclude that our data is not normally distributed.
In the following Python script, open the auto-mpg dataset and apply the Kolmogorov-Smirnov test (KS) from the SciPy library (documentation). The KS test takes the empirical data and the name of a standard distribution in our case. The test assumes that the sample distribution is centered and of variance 1, so you need to center and normalize the data to be able to compare it to an N(0,1) distribution.
You do so with the following operation:
from scipy import stats
The result is:
Since the p-value is well above the 0.05 threshold, you cannot reject the null hypothesis.
The conclusion is that the p-value, i.e., the probability of observing our data under the assumption that the data follows a normal distribution, is high enough to be trusted.
Take another variable from the auto-mpg dataset and use the KS test to test if that variable is normally distributed. Also, consider the data from the school dataset. Is the height or the weight of the school children normally distributed?
Check out the solution in the following screencast:
Once you get around the logic behind hypothesis testing, the method seems simple enough. Depending on the p-value, you reject the null hypothesis or not. But there are several shortcomings of that method mostly due to a misinterpretation of the results.
#### Population Bias
In the first part of this chapter, we concluded that on average, boys are taller than girls based on a dataset of 237 samples of schoolchildren. This conclusion only applies to our dataset, and cannot be directly generalized to a more global population without more statistical evidence and analysis.
For instance, if the boys in this school were all basketball players, and taller than the normal population, then our dataset would be biased. Hidden biases in the dataset are always something to look for and test for when possible.
#### Truth and Rejecting the Null Hypothesis
One common abuse of hypothesis testing is that rejecting the null hypothesis directly implies that the observation is true.
Rejecting the null hypothesis means that every other possibility may be true. For instance, if you test that a variable follows a normal distribution centered of variance 1.
The null hypothesis is: H0:X∼(0,1)
If is rejected, that means that several things could be true:
• The mean is not 0.
• The variance is not 1.
• The distribution is not normal.
#### What Is the Correct Interpretation of the P-Value?
The p-value is often interpreted as the probability of the null hypothesis ( ), which is not entirely true. The p-value is the probability of the observation assuming the null hypothesis ( ). In probability theory land, you write that the p-value
#### P-Hacking
But there's more, and it's called p-hacking!
By running repeated experiments on the same datasets but different chunks of the dataset, there's bound to be a subset of samples for which the p-value is lower than 0.05. This is called p-hacking: running an experiment many times until the p-value is low enough so you can reject the null hypothesis and claim that the observation is real.
P-hacking is superbly illustrated in this comic from XKCD. A scientist tests the impact of jelly beans on acne and defines the null hypothesis as, "There's no link between jelly beans and acne." He then tests all the possible colors, each time with a p-value > 0.05, until one day, testing green jelly beans, the p-value is below 0.05. The scientist happily concludes that "green jelly beans cause acne."
There's a push back by many scientists on hypothesis testing and the concept of statistical significance. The claim is that it is too easy to hack and often leads to erroneous conclusions.
However, hypothesis testing is widely used for all types of problems and data analysis and is far from being discarded as a reliable method for statistical analysis.
In this chapter, you learned about hypothesis testing, a key method in statistics.
The key takeaways from this chapter are:
• Hypothesis testing involves four steps:
• Select the right statistical test and rejection threshold.
• Define a null hypothesis and an alternative one.
• Apply the test on the data.
• Decide to reject the null hypothesis or not based on the p-value.
• Use the t-test to compare means of two samples of the same data.
• Apply the Kolmogorov-Smirnov test to assess that a variable is normally distributed.
• A statistic is a calculation based on the data.
• The p-value is the probability of our observation, assuming the null hypothesis and is not the probability of the null hypothesis.
This concludes the first part of the course. You are now ready to build linear regression models.
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https://socratic.org/questions/a-solution-with-a-volume-of-0-25-liters-contains-20-grams-of-hydrogen-fluoride-h
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# A solution with a volume of 0.25 liters contains 20 grams of hydrogen fluoride, HF. What is the molarity of the solution?
Oct 22, 2016
${\text{4 mol L}}^{- 1}$
#### Explanation:
In order to find a solution's molarity, you must determine how many moles of solute you get per liter of solution.
To determine the number of moles of hydrofluoric acid present in your sample, use the compound's molar mass. Hydrofluoric acid has a molar mass of ${\text{20.01 g mol}}^{- 1}$, which means that one mole of this compound has a mass of $\text{20.01 g}$.
Since your sample has a mass of $\text{20 g}$, you can say that it contains
20 color(red)(cancel(color(black)("g"))) * "1 mole HF"/(20.01color(red)(cancel(color(black)("g")))) = "0.9995 moles HF"
Now, you know that this many moles are being dissolved in $\text{0.25 L}$, so you can say that $\text{1 L}$ of this solution will contain
1 color(red)(cancel(color(black)("L solution"))) * "0.9995 moles HF"/(0.25color(red)(cancel(color(black)("L solution")))) = "3.998 moles HF"
Rounded to one significant figure, the molarity of the solution will be
$\textcolor{g r e e n}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\text{molarity HF solution" = "4 mol L}}^{- 1}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
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https://www.naukri.com/code360/problem-details/merge-sort_920442
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Last Updated: 2 Dec, 2020
# Merge Sort
Easy
## Problem statement
#### Example :
``````Merge Sort Algorithm -
Merge sort is a Divide and Conquer based Algorithm. It divides the input array into two-parts, until the size of the input array is not ‘1’. In the return part, it will merge two sorted arrays a return a whole merged sorted array.
``````
``````The above illustrates shows how merge sort works.
``````
##### Note :
``````It is compulsory to use the ‘Merge Sort’ algorithm.
``````
##### Input format :
``````The first line of input contains an integer ‘T’ denoting the number of test cases.
The next 2*'T' lines represent the ‘T’ test cases.
The first line of each test case contains an integer ‘N’ which denotes the size of ‘ARR’.
The second line of each test case contains ‘N’ space-separated elements of ‘ARR’.
``````
##### Output Format :
``````For each test case, print the numbers in non-descending order
``````
##### Note:
``````You are not required to print the expected output; it has already been taken care of. Just implement the function.
``````
##### Constraints :
``````1 <= T <= 50
1 <= N <= 10^4
-10^9 <= arr[i] <= 10^9
Time Limit : 1 sec
``````
## Approaches
### 01 Approach
The basic idea is that we divide the given ‘ARR’ into two-part call them ‘leftHalves’ and ‘rightHalves’ and call the same function again with both the parts. In the end, we will get sorted ‘leftHaves’ and sorted ‘righthalves’ which we merge both of them and return a merged sorted ‘ARR’.
We implement this approach with a divide and conquer strategy.
Here is the algorithm :
1. Divide ‘ARR’ into two-part ‘leftHalves’ and ‘rightHalves’ and the size of both parts are almost equal means ‘leftHalves’ can have one size extra comparing to ‘rightHalves’
• Recursively solve for ‘leftHalves’
• Recursively solve for ‘rightHalves’
2. In the recursive part, every time we will get some part of ‘ARR’. Then divide it into two parts until the size of each subarray is not equal to 1.
3. In the return part, we get two sorted arrays ‘leftHalves’ and ‘rightHalves’ using recursion.
4. After getting both sorted parts, we merge both of them in such a way so that we get a merged sorted array.
MERGE() function :
1. Suppose we have two sorted arrays ‘leftHalves’ and ‘rightHalves’ then we merge both of them into ‘mergedArr’
2. Currently, we have two pointers ‘ptrLeft’ and ‘ptrRight’, and both are pointing to starting indices of ‘leftHalves’ and ‘rightHalves’.
• If ‘leftHalves[ptrLeft] < rightHalves[ptrRight]’ then add ‘leftHalves[ptrLeft]’ in ‘mergeArr’ and increase ‘ptrLeft’ by one.
• Else add ‘rightHalves[ptrRight]’ in ‘mergeArr’ and increase ‘ptrRight’ by one.
3. Add remaining elements from ‘leftHalves’ and ‘rightHalves’.
4. Copy ‘mergeArr’ elements to ‘ARR’.
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# find maclaurin series
• May 19th 2009, 02:24 PM
cm3pyro
find maclaurin series
integral of 1/(2+x^3)dx bounds:0 to .5
• May 19th 2009, 03:03 PM
skeeter
Quote:
Originally Posted by cm3pyro
integral of 1/(2+x^3)dx bounds:0 to .5
$\frac{1}{2+x^3} = \frac{1}{2} \cdot \frac{1}{1 - \left(-\frac{x^3}{2}\right)} = \frac{1}{2}\left(1 - \frac{x^3}{2} + \frac{x^6}{4} - \frac{x^9}{8} + ... \right)$
integrate ...
$\frac{1}{2}\left[x - \frac{x^4}{8} + \frac{x^7}{28} - \frac{x^{10}}{80} + ... \right]_0^{0.5}$
using the first three terms should get you within 0.00001 of the actual value
• May 19th 2009, 03:10 PM
cm3pyro
Quote:
Originally Posted by skeeter
$\frac{1}{2+x^3} = \frac{1}{2} \cdot \frac{1}{1 - \left(-\frac{x^3}{2}\right)} = \frac{1}{2}\left(1 - \frac{x^3}{2} + \frac{x^6}{4} - \frac{x^9}{8} + ... \right)$
integrate ...
$\frac{1}{2}\left[x - \frac{x^4}{8} + \frac{x^7}{28} - \frac{x^{10}}{80} + ... \right]_0^{0.5}$
using the first three terms should get you within 0.00001 of the actual value
i need a maclaurin series
• May 19th 2009, 04:18 PM
skeeter
Quote:
Originally Posted by cm3pyro
i need a maclaurin series
what do you think I posted?
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https://www.school-for-champions.com/SCIENCE/temperature_limits_equations.htm
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SfC Home > Physics > Thermal Energy >
# Equations for Temperature Limits
by Ron Kurtus
The lower and upper temperature limits can be approached but not physically reached. There is a relationship between kinetic energy, speed of the particles and temperature. Absolute zero is the coldest possible temperature. The limit for the highest temperature is when the particles reach the speed of light.
Questions you may have include:
• What is relationship between kinetic energy, speed and temperature?
• What happens when a material is heated?
• What is the upper temperature limit?
This lesson will answer those questions. Useful tool: Units Conversion
## Relationships
There are equations that determine the relationship between kinetic energy of an ideal gas, temperature and velocity of the atoms or molecules in the gas.
Note: An ideal gas is a theoretical gas composed of randomly-moving point particles that interact only through elastic collisions. It is useful in determining simple equations, as opposed to the highly complex ones of the real world.
### Kinetic energy and temperature
The relationship between the kinetic energy of the molecules or atoms in an ideal gas and temperature is:
KE = 2kT/3
where
• KE = the kinetic energy of particles in an ideal gas in joules (J)
• k = Boltzmann's constant (a number that relates energy and temperature)
k = 1.38*10−23 joule/kelvin
• T = temperature in degrees kelvin (K)
Kinetic energy-temperature relationship equations for real-world gases, liquids and solids are too complex to work with at this level of study.
### Kinetic energy and velocity
The kinetic energy of a moving mass of particles is:
KE = ½mv²
where
• KE = kinetic energy in joules or kg-m²/s²
• m = mass in kilograms (kg)
• v = velocity in meters/second (m/s)
• = velocity squared or v*v in m²/s²
• ½mv² is ½ times m times
### Temperature and velocity
You can find the relationship between the temperature and the velocity of the particles in an ideal gas.
Since KE = 2kT/3 and KE = ½mv², you can substitute for KE to get 2kT/3 = ½mv². Then, you can multiply by 3 and divide by 2k to get:
T = 3mv²/4k
where
• T is measured in degrees kelvin (K)
• m is the mass of in kilograms
• v is in meters/second
• k is in joule/kelvin or kg-m²/s²-kelvin
## Absolute zero
It can easily be seen from T = 3mv²/4k that when T = 0 kelvin, the velocity of the particles v = 0. Thus the kinetic energy due to linear movement is zero. But the atoms still possess spin, which means they still have some energy.
Another fact is that the equation is really an approximation, since we are dealing with an ideal gas. A real-world gas would not be able to reach T = 0.
## Temperature and the speed of light limit
The greatest temperature possible is limited by how fast its atoms can travel. The upper limit that anything can travel is at the speed of light.
Although kinetic energy is KE = ½mv², the limiting energy is defined by Einstein's Theory of Relativity equation
E = mc²
where;
• m = the resting mass
• = the speed of light (c) squared
Thus, in theory, the highest possible temperature is defined by:
T = 3mc²/2k
You can calculate that temperature by substituting the appropriate values. This equation may not fit into the Theory of Relativity, since the mass of a particle increases dramatically as the particle approaches the speed of light. But, at the very least, it is an interesting exercise.
## Summary
The lower and upper temperature limits can be approached but not physically reached. The relationship between kinetic energy, speed of the particles and temperature determines that value of absolute zero and the limit for the highest possible temperature.
## Resources and references
Ron Kurtus' Credentials
### Websites
Kinetic Temperature - HyperPhysics
Physics Resources
### Books
(Notice: The School for Champions may earn commissions from book purchases)
## Students and researchers
www.school-for-champions.com/science/
temperature_limits_equations.htm
Please include it as a reference in your report, document, or thesis.
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School for Champions
Physics topics
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# 35.5 cm in inches
Come friends, today we will know how much is 35.5 cm in inches. let’s start
## 35.5 cm, How many feet and inches are there?
35.5 centimeters is equal to 1 foot and 1.976 inches. There are easy methods to calculate how many feet and inches are in 35.5 centimeters. The first method is to perform simple arithmetic calculations which are not difficult. To perform arithmetic calculations, you need to be aware of length measurement units and how they work. Since we are going to do the conversion from centimeters to feet and inches, you should know that 1 foot is equal to 12 inches and 1 centimeter is equal to 0.3937 inches. In case of large numbers, it is not possible to calculate manually. That’s when online height conversion tools come into play. We have explained both the methods in detail below.
#### 35.5 Centimeters to Feet and Inches using a simple formula
We are going to do the calculation which is easy to understand using simple maths. As you know, 1 foot is equal to 12 inches and 1 centimeter is equal to 0.3937 inches, so we need to multiply 0.3937 by 35.5 to get the output in inches. Multiplying 0.3937 by 35.5, we have 13.98 inches. Now, 12 is equal to 1 foot so the result is 1 foot 1.976 inches.
#### 35.5 centimeters to feet and inches using the conversion tool
You can also use our online height conversion tool to find out how many feet and inches to 35.5 cm. Simply enter the value 35.5 in the Centimeters input field. Our tool will instantly convert 35.5 centimeters to feet and inches and display the result for 1 feet 1.976 inches in the Feet and inches field. You may also want to convert 35.5 cm to other length units described above in the tool. You don’t need to do anything. Our tool will take care of this and do the simultaneous instant conversion and show the resulting value in the respective fields.
Calculations of centimeters to feet and inches are generally used when you want to calculate the height or length of an object. The basic questions surrounding such calculations are how long is 35.5 centimeters in feet and inches, how wide is 35.5 centimeters in feet and inches, how big is 35.5 centimeters in feet and inches, how much is 35.5 centimeters in feet and inches, etc. . The answer to all your questions is 1 feet 1.976 inches. We have already talked about how to convert 35.5 centimeters to feet and inches.
It doesn’t matter what kind of unit conversion you are going to do, but in every unit conversion, the conversion factor plays an important role. Feet, inches and centimeters are part of the length measurement units, so it’s not that hard to convert 35.5 centimeters to feet and inches and 1 foot 1.976 inches to centimeters. The conversion is quite smooth and accurate. The conversion factor between 2 units mostly involves division or multiplication. The unit measurement mark for feet is ft and inches is in. This means you can write 1 foot and 1.976 inches as 1 foot 1.976 inches. The unit measurement symbol for centimeter is cm. This means that you can write 35.5 cm as 35.5 cm.
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# Ex.10.1 Q2. CIRCLES Solution - NCERT Maths Class 10
Go back to 'Ex.10.1'
## Question
Fill in the blanks:
(i) A tangent to a circle intersects it in _________ point (s).
(ii) A line intersecting a circle in two points is called a ____________.
(iii) A circle can have _________ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called _________ .
Video Solution
Circles
Ex 10.1 | Question 2
## Text Solution
Steps:
(i) A tangent to a circle intersects it in One point (s).
Reasoning:
A tangent to a circle is a line that intersects the circle at only one point.
(ii) A line intersecting a circle in two points is called a Secant .
Reasoning:
Secant is a line that intersects the circle in two points.
(iii) A circle can have Two parallel tangents at the most.
Reasoning:
Tangent at any point of a circle is perpendicular to the radius through the point of contact. Extended radius is a diameter which has two end points and hence two tangents which are parallel to themselves and perpendicular to the diameter.
Center $$O,$$ diameter $$AB,\text{ tangents}\, PQ, RS$$ and, $${PQ}\; \| \;{RS}$$
$$A$$ and $$B$$ are called as point of contact.
(iv) The common point of a tangent to a circle and the circle is called Point of contact .
Reasoning:
A tangent to a circle is a line that intersects the circle at only one point and that point is called as point of contact.
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https://www.jiskha.com/questions/466522/analyze-the-function-ln-x-cx-2-to-find-the-unique-value-of-c-such-that-there-is-exactly
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# calc
analyze the function ln x = cx^2 to find the unique value of c such that there is exactly one solution to the equation. to do this, find the value of c such that both sides of the equation have equivalent slopes at some point; this will give you a proper x-coordinate to work with.
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## Similar Questions
1. ### Calculus AB
Analyze the function ln x=cx^2 to find the unique value of c such that there is exactly one solution to the equation. To do this find the value of c such that both sides of the equation have equivalent slopes at some point; this
asked by jessica on December 13, 2010
2. ### College alg
Analyze the graph of the following function as follows: (a) Find the x- and y-intercepts. (b) Determine the end behavior: find the power function that the graph of f resembles for large values of |x|. (c) Find the maximum number
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https://mayankpathak.hashnode.dev/discover-max-element-of-each-column-in-matrix
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# Discover Max element of each column in matrix
Mayank Pathak
·Jun 12, 2021·
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Hey guys👋, In this post we will be discuss the Program to find maximum element of each column in a matrix i.e., to find the maximum value in each column of the given matrix. Since it is considered as an important problem to solve while practicing, hence thought to share🤝 with you all.
### Problem Description
In a family, the people are arranged in rows and columns. Male persons in the families are arranged in a row and females are arranged in a column. Find the eldest women in each column. (Write a program to find the maximum element in each column of the matrix.)
You can find the same set of problem in different way on the various coding platform.
Input Format:
The input consists of (m*n+2) integers.
The first integer corresponds to m, the number of rows in the matrix and the second integer corresponds to n, the number of columns in the matrix.
The remaining integers correspond to the elements in the matrix.
The elements are read in row-wise order, the first row first, then second row and so on.
Assume that the maximum value of m and n is 10.
Output Format:
Refer to the sample output for details.
Sample Input:
3
2
4 5
6 9
0 3
Sample Output:
6
9
### Explaination :
In this we will be discussing the Program to find maximum element of each column in a matrix i.e., to find the maximum value in each column of the given matrix. This can be achieved by simple loop and conditional statement. Initialize the max variable to first element of each column. If there is only one element present in each column of the matrix then the loop did not execute and max hold the only present value in the matrix, thus that element becomes the maximum of each column. If matrix has more than one element, than loop executes and if any element found bigger than the previously assigned value, then that element becomes the largest.
### Logic to follow to come-up with the solution :
1. Declare the required sets of variables to use in the code.
2. Initialize the max variable to first element of each column.
3. If there is only one element present in each column of the matrix then the loop did not execute and max hold the only present value in the matrix, thus that element becomes the maximum of each column.
4. If matrix has more than one element, than loop executes and if any element found bigger than the previously assigned value, then that element becomes the largest.
5. At last maximum value of each column is displayed as the result output.
### Coding Time 👨💻
``````#include<iostream>
#include <bits/stdc++.h>
using namespace std;
void largestInColumn(int mat[10][10], int rows, int cols)
{
for (int i = 0; i < cols; i++)
{
int maxm = mat[0][i];
for (int j = 1; j < rows; j++)
{
if (mat[j][i] > maxm)
maxm = mat[j][i];
}
cout << maxm << endl;
}
}
int main()
{
int n,m;
cin>>n>>m;
int mat[10][10];
for(int i=0;i<n;i++)
{
for(int j=0;j<m;j++)
{
cin>>mat[i][j];
}
}
largestInColumn(mat, n, m);
return 0;
}
``````
``````Input :
3
2
4 5
6 9
0 3
``````
``````Output
6
9
``````
Hence with the above set of logic and code you can easily understand and solve the problem to find maximum number in each column of a matrix.
Hope with this you learned and acquired some basic knowledge of C++ Programming.
Drop a Love❤ if you liked👍 this post, then share 🤝this with your friends and if anything is confusing or incorrect then let me know in the comment section.
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https://merchsprout.com/the-importance-of-sample-size/
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# The Importance of sample size
##### George Tewson
co-founder Merchsprout.com
We spoke in our past blog about what a sample is, and what are the best way of obtaining random sampling subgroups for testing. In this post, we will be discussing the importance of sample size.
## Sample Size
We briefly said in our previous post that Bigger = Better and this is the case. A greater number of samples gives you a greater accuracy of the total population. But that doesn’t mean that we need a very large sample size.
We can use the average of a sample and the data that is away from the mean. Bear with me here as we get to delve into some statistical analysis geekiness.
## Standard Deviation
We talk a lot about standard deviation in sample testing, and with good reason. Standard deviation is the average distance between the data point and the mean. Because all random data can be normalised, we can determine the probability that data points will fall in a certain point away from the mean of a data set or subset of values.
If we have a big enough data set we can quantify the uncertainty around the distribution of given values and the range of the values.
But first, let’s refresh ourselves with what a mean average is:
## Mean
I always remember the mean because it was the meanest to calculate at school, and if you are an 8-year-old boy named George, calculating maths was not on the agenda. Pokemon cards were though… Some things never change.
So mean is an average worked out by adding up all the numbers in a sample and dividing them by the number of data points. An example:
So as we can see the mean is 5.22. This independently does not mean much. But its a very important number when we need to calculate the standard deviation.
So as we said before the standard deviation is a distance from the mean. To be more precise its the squared distance from the mean. Don’t be too scared by that, or the below calculation below. It will all make sense soon:
Again, don’t get scared at the above, its merely showing us that we can work out very large populations of data from potentially much smaller populations of data.
## Big Data from Small Data
We can see from the above, 68% of the data falls in the first standard deviations of mean (-1 and +1). 95% of the data within 2 standard deviations from the mean and 99.7% of all data should fall within 3 standard deviations of mean.
Each of these standard deviation points are 1 Sigma. Meaning that 99.7% of our data should span across Six Sigma points, or 6 standard deviations.
Can you see why its a good tool now?
Now the caveat to the above is that this only works when we have the well-centred bell-shaped curve.
We know that if selecting samples the majority of our data should fall within 3 standard deviations of our mean. If not, we know that we have problems in manufacturing defects.
And the best thing is to figure this out we can do some clever statistical analysis to determine if we have potential outliers, how many we have and if these are within the permitted accepted quality limit.
## How this can help me sample data?
We will go into how this can help in a later blog post, but the fundamentals are, using standard deviation we can calculate an acceptance criteria based upon a mean of data. From this mean we can determine where 99.7% of data should lie. If we have data points that are outside of these points, we have a situation where we possibly need to reject or rework a batch of parts.
## Don’t worry though
If the above seems a bit far fetched, don’t worry, luckily there is some pretty trick software that can work out sampling data for us. What I wanted to try and allow you to understand is that we can generate confidence levels from very small sample data. Hence the importance of sample size and what we can generate from it.
## Conclusion
When we are sampling individual components for QC testing the first job is to ensure that there is a good sample selection that is free from bias.
That bias can come from a lot of individual factors. Even things such as ease of access to a product can influence the selection of samples. So it’s important to ensure you select samples in an unbiased, isolated way.
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## Tuesday 19 March 2024
### On This Day in Math - March 19
Pearls of Sluze, *Mathworld Wolfram
There is no reason why the history and philosophy of science should not be taught in such a way as to bring home to all pupils the grandeur of science and the scope of its discoveries.
~Prince Louis-Victor de Broglie
The 78th day of the year; 78 is the smallest number that can be written as the sum of 4 distinct squares in 3 ways. *What's Special About This Number
In his pamphlet, "The Thousand Yard Model," Guy Ottewell creates a sacale model universe with the sun as a bowling ball. 78 feet away, the Earth is represented by a peppercorn.
In his doctoral thesis in the early 60's, Ron Graham proved that 78, and every number greater than 78 can be partitioned into distinct numbers so that the sum of their reciprocals is one. 78=2+6+8+10+12+40, and the reciprocals of all these distinct integers add up to one. There are at least two smaller numbers for which this is true. Can you find them?
78 is the sum of the first twelve integers, and thus a triangular number.
90 = 21+22+23+24, 78= 25+26 + 27, but 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2 = 2030
The cube of 78 is equal to the sum of three distinct cubes, 783 = 393 + 523 + 653
(Historically, it seems Ramanujan was inspired by a much smaller such triplet 63 = 33 + 43 + 53
77 and 78 form the fourth Ruth-Aaron pair, named for the number of home runs hit by Babe Ruth, 714, and the number when Aaron broke the record, 715 (he hit more afterward). They are consecutive numbers that have the same sums of their prime factors (77 = 7*11, 78 = 2*3*13, and 7+11 = 2+3+13).
EVENTS
In 1474, the Venetian Patent Law, the first of its kind in the world, declared that “each person who will make in this city any new and ingenious contrivance, not made heretofore in our dominion, as soon as it is reduced to perfection... It being forbidden to any other in any territory and place of ours to make any other contrivance in the form and resemblance thereof, without the consent and licence of the author up to ten years.” The law was intended to attract inventors and investors to Venice and stimulate new economic activities. *TIS
*Mark Jardine,
1681 Last observation of C/1680 V1, also called the Great Comet of 1680, Kirch's Comet, and Newton's Comet. It has the distinction of being the first comet discovered by telescope. Discovered by Gottfried Kirch on 14 November 1680, New Style, it became one of the brightest comets of the 17th century--reputedly visible even in daytime--and was noted for its spectacularly long tail. Passing only 0.4 AUs from Earth on 30 November, it sped around an incredibly close perihelion of .006 AU (898,000 km) on 18 December 1680, reaching its peak brightness on 29 December as it rushed outward again. It was last observed on 19 March 1681. As of December 2010 the comet was about 252.1 A.U. from the Sun. While the Kirch Comet of 1680-1681 was discovered and subsequently named for Gottfried Kirch , credit must also be given to the Jesuit, Eusebio Kino, who charted the comet’s course. During his delayed departure for Mexico, Kino began his observations of the comet in Cadíz in late 1680. Upon his arrival in Mexico City, he published his Exposisión astronómica de el [sic] cometa (Mexico City, 1681) in which he presented his findings. Kino’s Exposisión astronómica is among one of the earliest scientific treatises published by a European in the New World. Aside from its brilliance, it is probably most noted for being used by Isaac Newton to test and verify Kepler's laws. *Wik
1706 Advertisement in English Tabloid for William Jones's Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics. This is the book in which Jones introduces the symbol pi for the ratio of the circumference to diameter of a circle.
*Review of the State of the English Nation (Cumulation) (London, England), Tuesday, March 19, 1706; Issue 34.
1752 Following the death of her father on March 19, 1752, a new phase of Maria Agnesi’s life began that lasted until her death. She restricted her study to theology and gave her time, effort, and money to devotional and charitable activities. Although continuing to live with her family, she kept a separate apartment, where she cared for a few poor, sick people. From 1759 she lived in a rented house with four of her poor people; and when money was needed for her charitable activity, she sold her gifts from the Empress Maria Theresa to a rich Englishman. Besides caring for the sick and indigent, she often taught catechism to working-class people. *Hubert Kennedy, Eight Mathematical Biographies, Pg 8
*Selected witch of Agnesi curves
1791 Prior to 1784, when Jefferson arrived in France, most if not all of his drawings were made in ink. In Paris, Jefferson began to use pencil for drawing, and adopted the use of coordinate, or graph, paper. He treasured the coordinate paper that he brought back to the United States with him and used it sparingly over the course of many years. He gave a few sheets to his good friend David Rittenhouse, the astronomer and inventor:
"I send for your acceptance some sheets of drawing-paper, which being laid off in squares representing feet or what you please, saves the necessity of using the rule and dividers in all rectangular draughts and those whose angles have their sines and cosines in the proportion of any integral numbers. Using a black lead pencil the lines are very visible, and easily effaced with Indian rubber to be used for any other draught."
A few precious sheets of the paper survive today. *Monticello.org
Jefferson was widely interested in Science. For those who wish to know more about his scientific interest, I can recommend this book
1791 Report made to the Paris Academy of Sciences advocating the metric system, including the decimal subdivision of the circle. The committee consisted of J. C. Borda, J. Lagrange, P. S. Laplace, G. Monge, and de Condorcet. [Cajori, History of Mathematics 266] See April 14, 1790. *VFR
A metric system of angles was brought in, with 400 degrees in a full turn (100 degrees in a right angle). Now the earth would rotate 40 degrees in an hour and, since the metre had been designed so that one quarter meridian was 10 million metres, each degree of latitude would be 100 kilometres long. It was certainly a rational system but its introduction would require all watches, all clocks, all trigonometric tables, all charts etc. to be changed. Condorcet proposed that teams of out of work wig makers should be used to recalculate new mathematical tables with the new units. Why, one might ask, were the wig makers out of work? Well they had been employed by the aristocrats who, following the Revolution, no longer required their services! *SAU
The resolution found some traction in angle measures."In 1857, Mathematical Dictionary and Cyclopedia of Mathematical Science has: "The French have proposed to divide the right angle into 100 equal parts, called grades, but the suggestion has not been extensively adopted." In 1987 Mathographics by Robert Dixon has: “360° = 400 gradians = 2π radians.” And for those who have, or had, one, "The Texas Instruments TI-89 Titanium calculator has three modes, radians, degrees, and gradians."
*Jeff Miller "Angle: Units in which angle values are interpreted and displayed: RADIAN, DEGREE or GRADIAN* (* not available on the TI-92 family). *TI Knowledge Base web page
1797 The date of the entry in Gauss’s scientific diary showing that he had already discovered the double periodicity of certain elliptic functions. *VFR Gauss was investigating the lemniscate. Two days later he would show how to divide the lemniscate into five equal parts by ruler and compass. This means he must have had some sense of complex multiplication of elliptic functions. Abel would generalize this in 1826.
Lemniscate of Bernoulli
1892 E. Hastings Moore, of Northwestern University, was elected professor of mathematics by the Board of Trustees of the new University of Chicago. *T. W. Goodspeed, The Story of the University of Chicago
1915 The first image of Pluto was taken by astronomer Thomas Gill at Lowell Observatory in 1915 using a nine-inch telescope borrowed from Swarthmore College. Percival undertook a passionate search for what he called “Planet X.” He took photographs of the sky where Planet X was predicted to be lurking, but failed to recognize Pluto because it was much fainter than expected. Percival died suddenly in 1916, not knowing he had in fact taken an image of Pluto. Only with the lens of history can we look back and recognize those photographs as containing some of the first images of Pluto. The calculations for the place to search for the undiscovered planet were directed by Elizabeth Williams, the head human computer, performing mathematical calculations on where Lowell should search for an unknown object and its size based on the differences in the orbits of Neptune and Uranus. Her calculations led to predictions for the location of the unknown planet. Lowell died unexpectedly in 1916 and the search was discontinued. In 1930 the search would resume, leading to the recognition of Pluto as a planet. Williams and her husband were then dismissed from their positions at the observatory by Percival Lowell's widow, Constance, because it was considered inappropriate to employ a married woman.
1918 "An Act to preserve daylight and provide standard time for the United States" was enacted on March 19, 1918. It both established standard time zones and set summer DST to begin on March 31, 1918. *WebExhibits
1937 John von Neumann gave a popular lecture at Princeton on the game of poker. Game Theory became one of his substantial contributions to mathematics. [A. Hodges, Alan Turing. The Enigma, p. 550]The Book that inspired the movie.
In 1921, Emile Borel, a French mathematician, published several papers on the theory of games. He used poker as an example and addressed the problem of bluffing and second-guessing the opponent in a game of imperfect information. Borel envisioned game theory as being used in economic and military applications. Borel's ultimate goal was to determine whether a "best" strategy for a given game exists and to find that strategy. While Borel could be arguably called as the first mathematician to envision an organized system for playing games, he did not develop his ideas very far. For that reason, most historians give the credit for developing and popularizing game theory to John Von Neumann, who published his first paper on game theory in 1928, seven years after Borel.
For Von Neumann, the inspiration for game theory was poker, a game he played occasionally and not terribly well. Von Neumann realized that poker was not guided by probability theory alone, as an unfortunate player who would use only probability theory would find out. Von Neumann wanted to formalize the idea of "bluffing," a strategy that is meant to deceive the other players and hide information from them.
In his 1928 article, "Theory of Parlor Games," Von Neumann first approached the discussion of game theory, and proved the famous Minimax theorem. From the outset, Von Neumann knew that game theory would prove invaluable to economists. He teamed up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory.
I'm "All IN" on this hand.
1949 The American Museum of Atomic Energy opened for the public in an old WWII cafeteria in Oak Ridge, Tennessee. The site had been part of the US projects to develop atomic bombs by processing U235. A new facility was opened in 1975. *Lucio Gelmini
In 1958, Britain's first planetarium, the London Planetarium, opened in the west wing of Madame Tussaud's. It is one of the world's largest. The site used was that of the former Cinema and Restaurant added in 1929, that had been destroyed by a German bomb in 1940.*TIS
1953 Frances Crick writes a letter to his son. "Dear Michael, Jim Watson and I have probably made a most important discovery.” This was only two weeks after Crick solved the DNA puzzle and may well be the first written description of the code. The letter, was auctioned at Christie’s on April 10, 2013 for six million dollars. *NY Times Science
Crick letter *NBC
2008 GRB 080319B was a gamma-ray burst (GRB) detected by the Swift satellite at 06:12 UTC on March 19, 2008. The burst set a new record for the farthest object that was observable with the naked eye: it had a peak visual apparent magnitude of 5.7 and remained visible to human eyes for approximately 30 seconds. The magnitude was brighter than 9.0 for approximately 60 seconds. If viewed from 1 AU away, it would have had a peak apparent magnitude of −67.57 (21 quadrillion times brighter than the Sun seen from Earth) *Wik
*artist's impression of gamma-ray burst GRB 080319B
2019 One of the top prizes in mathematics has been given to a woman. The Norwegian Academy of Science and Letters announced it has awarded this year’s Abel Prize to Karen Uhlenbeck, an emeritus professor at the University of Texas at Austin. The award cites “the fundamental impact of her work on analysis, geometry and mathematical physics.” *NY Times
BIRTHS
1782 Baron Wilhelm von Biela (19 Mar 1782, 18 Feb 1856 at age 73) Austrian astronomer who was known for his measurement (1826) of a previously known comet as having an orbital period of 6.6 years. Subsequently, known as Biela's Comet, it was observed to break in two (1846), and in 1852 the fragments returned as widely separated twin comets that were not seen again. However, in 1872 and 1885, bright meteor showers (known as Andromedids, or Bielids... current Andromedids are only weakly represented by displays of less than three meteors per hour around November 14. ) were observed when the Earth crossed the path of the comet's known orbit. This observation provided the first concrete evidence for the idea that some meteors are composed of fragments of disintegrated comets.*TIS
1799 William Rutter Dawes (19 Mar 1799, 15 Feb 1868 at age 68) English amateur astronomer who set up a private observatory and made extensive measurements of binary stars and on 25 Nov 1850 discovered Saturn's inner Crepe Ring (independently of American William Bond). In 1864, he was the first to make an accurate map of Mars. He was called "Eagle-eyed Dawes" for the keenness of his sight with a telescope (though otherwise, he was very near-sighted). He devised a useful empirical formula by which the resolving power of a telescope - known as the Dawes limit - could be quickly determined. For a given telescope with an aperture of d cm, a double star of separation 11/d arcseconds or more can be resolved, that is, be visually recognized as two stars rather than one. *TIS
1862 Adolf Kneser (19 March 1862 in Grüssow, Mecklenburg, Germany - 24 Jan 1930 in Breslau, Germany (now Wrocław, Poland)) He is remembered most for work mainly in two areas. One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general. He wrote an important text on integral equations. The second main area of his work was the calculus of variations. He published Lehrbruch der Variationsrechnung (Textbook of the calculus of variations) (1900) and he gave the topic many of the terms in common use today including 'extremal' for a resolution curve, 'field' for a family of extremals, 'transversal' and 'strong' and 'weak' extremals *SAU
*Wik
1885 Margaret Harwood (March 19, 1885 – February 6, 1979) was born in Littleton, Massachusetts, became the first woman – and for a long time the only woman – to serve as director of an independent astronomical observatory. She took charge of the Maria Mitchell Observatory on Nantucket Island in 1916, and remained in that post for forty-one years.
Miss Harwood had planned to study physics, chemistry and math when she entered Radcliffe College in 1903, but her choice of lodgings turned her to astronomy. She boarded with the family of Arthur Searle, a genial fixture at the Harvard College Observatory. Soon she was trailing him up Observatory Hill, learning to use the telescopes, earning the friendship and mentoring of other staff members, from Edward Pickering to Annie Jump Cannon and Henrietta Leavitt. By the time of Miss Harwood’s graduation, she was ready to step into a paid position as an assistant. The position didn’t pay much, however, and she supplemented her income of about $500 per year by teaching science in the mornings at a couple of local schools. In 1912, the Maria Mitchell Association awarded Miss Harwood a new fellowship in astronomy worth$1,000. It came with a new opportunity: From June to December of that year, she took up residence in the old Mitchell homestead on Nantucket, where she curated a small museum and library, used the telescope in the next-door dome to further her own research on asteroids, and lectured on astronomy to the locals every Monday night.
She received an offer from Wellesley College to begin teaching astronomy there upon completion of her graduate studies. But the Maria Mitchell Association, keen to keep her and see her continue her own research, matched the Wellesley salary and made her director of the Nantucket observatory . She was only thirty years old.
In 1957, with considerable reluctance, Miss Harwood retired from her post at Nantucket. In 1961 she accepted the Annie Jump Cannon Prize, which had been established by its namesake in the 1930s, and first conferred on Cecilia Payne. The prize is still awarded today by the American Astronomical Society to a young woman at the start of her career, but it no longer comes with a custom-designed piece of astronomically themed jewelry . Instead, the winner is invited to lecture about her research at the Society’s annual meeting. No doubt Miss Harwood would approve.*LH
Custom-made pin in the shape of a galaxy, designed for the occasion of the award of the Annie Jump Cannon Prize to Margaret Harwood, 1961 (Schlesinger Library, Radcliffe Institute, Harvard Institute)
1900 Frederic Joliot-Curie (19 Mar 1900; 14 Aug 1958 at age 58) French physicist and physical chemist who became personal assistant to Marie Curie at the Radium Institute, Paris, and the following year married her daughter Irène (who was also an assistant at the institute). Later they collaborated on research, and shared the 1935 Nobel Prize in Chemistry "in recognition of their synthesis of new radioactive elements." For example, they discovered that aluminium atoms exposed to alpha rays transmuted to radioactive phosphorus atoms. By 1939 he was investigating the fission of uranium atoms. After WW II he supervised the first atomic pile in France. He succeeded his wife as head of the Radium Institute upon her death in 1956. *TIS
Frédéric and Irène Joliot-Curie | Nobel Prize-Winning French
1910 Jacob Wolfowitz (March 19, 1910 – July 16, 1981) was a Polish-born American statistician and Shannon Award-winning information theorist. He was the father of former Deputy Secretary of Defense and World Bank Group President Paul Wolfowitz.
While a part-time graduate student, Wolfowitz met Abraham Wald, with whom he collaborated in numerous joint papers in the field of mathematical statistics. This collaboration continued until Wald's death in an airplane crash in 1950. In 1951, Wolfowitz became a professor of mathematics at Cornell University, where he stayed until 1970. He died of a heart attack in Tampa, Florida, where he was a professor at the University of South Florida.
Wolfowitz's main contributions were in the fields of statistical decision theory, non-parametric statistics, sequential analysis, and information theory.*Wik
1910 Jerome Namias (19 Mar 1910, 10 Feb 1997 at age 86) American meteorological researcher most noted for having pioneered the development of extended weather forecasts and who also studied the Dust Bowl of the 1930s and the El Niño phenomenon. *TIS In 1971 he joined the Scripps Institution and established the first Experimental Climate Research Center. His prognosis of warm weather during the Arab oil embargo of 1973 greatly aided domestic policy response.*Wik
1927 Allen Newell (March 19, 1927 – July 19, 1992) was a researcher in computer science and cognitive psychology at the RAND Corporation and at Carnegie Mellon University’s School of Computer Science, Tepper School of Business, and Department of Psychology. He contributed to the Information Processing Language (1956) and two of the earliest AI programs, the Logic Theory Machine (1956) and the General Problem Solver (1957) (with Herbert A. Simon). He was awarded the ACM's A.M. Turing Award along with Herbert A. Simon in 1975 for their basic contributions to artificial intelligence and the psychology of human cognition *Wik
1951 Arthur T. Benjamin (March 19, 1961; ) is an American mathematician who specializes in combinatorics. Since 1989 he has been a Professor of Mathematics at Harvey Mudd College.
He is known for mental math capabilities and mathemagics performances. These have included shows at the Magic Castle and TED. He is also the first mathematician to have been featured on the Colbert Report.
The Mathematical Association of America gave him a regional award for distinguished teaching in 1999 and a national one in 2000. He was the Mathematical Association of America's George Pólya Lecturer for 2006-8. In 2012 he became a fellow of the American Mathematical Society.
Benjamin was one of the performers at the inaugural San Diego Science Festival on April 4, 2009. He also won the American Backgammon Tour in 1997. *Wik A video of his "mathmagic" is here
And his book, The Magic of Math: Solving for x and Figuring Out Why, is delightful,
DEATHS
1406 Ibn Khaldūn or Ibn Khaldoun Al-Ḥaḍrami, May 27, 1332 AD/732 AH – March 19, 1406 AD/808 AH) was a Muslim historiographer and historian who is often viewed as one of the fathers of modern historiography,sociology and economics.
He is best known for his Muqaddimah (known as Prolegomenon in English), which was discovered, evaluated and fully appreciated first by 19th century European scholarship, although it has also had considerable influence on 17th-century Ottoman historians like Ḥajjī Khalīfa and Mustafa Naima who relied on his theories to analyze the growth and decline of the Ottoman Empire. Later in the 19th century, Western scholars recognized him as one of the greatest philosophers to come out of the Muslim world. *Wik
Ibn Khaldun Statue and Square, Mohandessin, Cairo
1862 John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) is remembered for the Campbell-Baker-Hausdorff theorem which gives a formula for multiplication of exponentials in Lie algebras. *SAU His 1903 book, Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups, popularized the ideas of Sophus Lie among British mathematicians.
He was elected a Fellow of the Royal Society in 1905, and served as President of the London Mathematical Society from 1918 to 1920. *Wik
1685 René François Walter de Sluse (2 July 1622 in Visé, Principality of Liège (now Belgium) - 19 March 1685 in Liège, Principality of Liège (now Belgium)) a French mathematician, intellectual and clergyman who wrote many books about mathematics and contributed to the development of mathematics.
Plague in Église Saint-Martin
He studied at a university in Rome, and later moved to Liège. His position in the church prevented him from visiting other mathematicians, but he corresponded with the mathematicians and intellectuals of the day.
He studied calculus and his work discusses spirals, tangents, turning points and points of inflection.
There is a family of curves named after him called the Pearls of Sluze: the curves represented by the following equation with positive integer values of m, n and p:
yn = k(a - x)pxm *Wik
This group of curves was studied by de Sluze between 1657 and 1698. It was Blaise Pascal who named the curves after de Sluze.
1922 George Ballard Mathews, FRS (February 23, 1861 — March 19, 1922) was a London born mathematician who specialized in number theory.
After receiving his degree (as Senior Wrangler) from St John's College, Cambridge in 1883, he was elected a Fellow of St John's College. *Wik Mathews also wrote Algebraic equations (1907) which is a clear exposition of Galois theory, and Projective geometry (1914). This latter book develops the subject of projective geometry without using the concept of distance and it bases projective geometry on a minimal set of axioms. The book also treats von Staudt's theory of complex elements as defined by real involutions. The book contains a wealth of information concerning the projective geometry of conics and quadrics. *SAU
1930 Henry Faulds (1 Jun 1843, 19 Mar 1930 at age 86) Scottish physician who, from 1873, became a missionary in Japan, where he worked as a surgeon superintendent at a Tokyo hospital, taught at the local university, and founded the Tokyo Institute for the Blind. In the late 1870s, his attention was drawn to fingerprints of ancient potters remaining on their work that he helped unearth at an archaeological dig site in Japan. He commenced a study of fingerprints, and became convinced that each individual had a unique pattern. He corresponded on the subject with Charles Darwin, and published a paper about his ideas in Nature (28 Oct 1880). When he returned to Britain in 1886, he unsuccessfully offered his fingerprinting identification scheme for forensic uses to Scotland Yard. Undeserved confusion on priority for the discovery with Francis Galton and Sir William J. Herschel lasted until 1917. *TIS
1978 Gaston Maurice Julia (February 3, 1893 – March 19, 1978) was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related.*Wik A report of his bravery during WWI during which he lost his nose:
January 25, 1915, showed complete contempt for danger. Under an extremely violent bombardment, he succeeded despite his youth (22 years) to give a real example to his men. Struck by a bullet in the middle of his face causing a terrible injury, he could no longer speak but wrote on a ticket that he would not be evacuated. He only went to the ambulance when the attack had been driven back. It was the first time this officer had come under fire.
When only 25 years of age, Julia published his 199 page masterpiece Mémoire sur l'iteration des fonctions rationelles which made him famous in the mathematics centres of his day. The beautiful paper, published in Journal de Math. Pure et Appl. 8 (1918), 47-245, concerned the iteration of a rational function f. Julia gave a precise description of the set J(f) of those z in C for which the nth iterate f n(z) stays bounded as n tends to infinity. (These are the Julia Sets popularized by Mandelbrot) *SAU
1984 Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics. A Bellman equation, also known as a dynamic programming equation, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory. The "Curse of dimensionality", is a term coined by Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space.*Wik
1987 Louis Victor Pierre Raymond duc de Broglie (15 Aug 1892,19 Mar 1987 at age 94) was a French physicist best known for his research on quantum theory and for his discovery of the wave nature of electrons. De Broglie was of the French aristocracy - hence the title "duc" (Prince). In 1923, as part of his Ph.D. thesis, he argued that since light could be seen to behave under some conditions as particles (photoelectric effect) and other times as waves (diffraction), we should consider that matter has the same ambiguity of possessing both particle and wave properties. For this, he was awarded the 1929 Nobel Prize for Physics. *TIS
He is buried in the Cimetière de Neuilly-sur-Seine (Ancien),Hauts-de-Seine, Ile-de-France Region, France. (Just outside Paris)
1933 Chen Jingrun (Chinese: 陳景潤; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime.
His work on the twin prime conjecture, Waring's problem, Goldbach's conjecture and Legendre's conjecture led to progress in analytic number theory. In a 1966 paper he proved what is now called Chen's theorem: every sufficiently large even number can be written as the sum of a prime and a semiprime (the product of two primes) – e.g., 100 = 23 + 7·11. Despite being persecuted during the Cultural Revolution, he expanded his proof in the 1970s.
After the end of the Cultural Revolution, Xu Chi wrote a biography of Chen entitled Goldbach's Conjecture). First published in People's Literature in January 1978, it was reprinted on the People's Daily a month later and became a national sensation. Chen became a household name in China and received a sackful of love letters from all over the country within two months.
Chen died of complications of pneumonia on March 19, 1996, at the age of 63 years
===============================================================
2011 J(ames) Laurie Snell, (January 15th, 1925, Wheaton, Illinois; March 19, 2011, Hanover, New Hampshire) was an American mathematician.
A graduate of the University of Illinois, he taught at Dartmouth College until retiring in 1995. Among his publications was the book "Introduction to Finite Mathematics", written with John George Kemeny and Gerald L. Thompson, first published in 1956 and in multiple editions since.
The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating the price process. Snell has published the related theory 1952 in the paper Applications of martingale system theorems.*Wik
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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Home > Standard Error > Linear Regression Standard Error Vs Standard Deviation
# Linear Regression Standard Error Vs Standard Deviation
## Contents
T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. When this occurs, use the standard error. The important thing about adjusted R-squared is that: Standard error of the regression = (SQRT(1 minus adjusted-R-squared)) x STDEV.S(Y). In the special case of a simple regression model, it is: Standard error of regression = STDEV.S(errors) x SQRT((n-1)/(n-2)) This is the real bottom line, because the standard deviations of the check my blog
For the same reasons, researchers cannot draw many samples from the population of interest. Learn MATLAB today! You can see that in Graph A, the points are closer to the line than they are in Graph B. The sample standard deviation of the errors is a downward-biased estimate of the size of the true unexplained deviations in Y because it does not adjust for the additional "degree of http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-to-interpret-s-the-standard-error-of-the-regression
## Standard Error Of Regression
First we need to compute the coefficient of correlation between Y and X, commonly denoted by rXY, which measures the strength of their linear relation on a relative scale of -1 Of the 2000 voters, 1040 (52%) state that they will vote for candidate A. Jim Name: Nicholas Azzopardi • Wednesday, July 2, 2014 Dear Mr.
But remember: the standard errors and confidence bands that are calculated by the regression formulas are all based on the assumption that the model is correct, i.e., that the data really Journal of the Royal Statistical Society. In fact, if we did this over and over, continuing to sample and estimate forever, we would find that the relative frequency of the different estimate values followed a probability distribution. Standard Error In Excel Fitting so many terms to so few data points will artificially inflate the R-squared.
However, there are certain uncomfortable facts that come with this approach. Standard Error Of Regression Formula Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Standard Error of the Estimate Author(s) David M. Of course deriving confidence intervals around your data (using standard deviation) or the mean (using standard error) requires your data to be normally distributed. Thanks for writing!
Technically, this is the standard error of the regression, sy/x: Note that there are (n − 2) degrees of freedom in calculating sy/x. Standard Error Calculator The uncertainty in the regression is therefore calculated in terms of these residuals. The only difference is that the denominator is N-2 rather than N. Notice that it is inversely proportional to the square root of the sample size, so it tends to go down as the sample size goes up.
## Standard Error Of Regression Formula
In multiple regression output, just look in the Summary of Model table that also contains R-squared. http://stattrek.com/estimation/standard-error.aspx?Tutorial=AP Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the Standard Error Of Regression In this scenario, the 2000 voters are a sample from all the actual voters. Standard Error In R Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation".
To see the rest of the information, you need to tell Excel to expand the results from LINEST over a range of cells. http://cdbug.org/standard-error/linear-regression-standard-error-of-coefficients.php In fact, the confidence interval can be so large that it is as large as the full range of values, or even larger. Thus, larger SEs mean lower significance. We need a way to quantify the amount of uncertainty in that distribution. Difference Between Standard Deviation And Standard Error
R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, S is 3.53399, which tells us that the average distance of the data points from the fitted line is about 3.5% body fat. For a given set of data, polyparci results in confidence interval with 95% (3 sigma) between CI = 4.8911 7.1256 5.5913 11.4702So, this means we have a trend value between 4.8911 http://cdbug.org/standard-error/linear-regression-and-standard-error.php The standard error of the mean can provide a rough estimate of the interval in which the population mean is likely to fall.
Therefore, ν = n − 2 and we need at least three points to perform the regression analysis. Standard Error Definition For example if the 95% confidence intervals around the estimated fish sizes under Treatment A do not cross the estimated mean fish size under Treatment B then fish sizes are significantly Log In to answer or comment on this question.
## So, for models fitted to the same sample of the same dependent variable, adjusted R-squared always goes up when the standard error of the regression goes down.
The mean age was 23.44 years. Hence, it is equivalent to say that your goal is to minimize the standard error of the regression or to maximize adjusted R-squared through your choice of X, other things being That's what the standard error does for you. Standard Error Of Proportion Assumptions and usage Further information: Confidence interval If its sampling distribution is normally distributed, the sample mean, its standard error, and the quantiles of the normal distribution can be used to
more than two times) by colleagues if they should plot/use the standard deviation or the standard error, here is a small post trying to clarify the meaning of these two metrics For example, the U.S. For example, a correlation of 0.01 will be statistically significant for any sample size greater than 1500. http://cdbug.org/standard-error/linear-regression-standard-error.php Adjusted R-squared can actually be negative if X has no measurable predictive value with respect to Y.
See unbiased estimation of standard deviation for further discussion. The standard error, .05 in this case, is the standard deviation of that sampling distribution. As the sample size gets larger, the standard error of the regression merely becomes a more accurate estimate of the standard deviation of the noise. Jim Name: Olivia • Saturday, September 6, 2014 Hi this is such a great resource I have stumbled upon :) I have a question though - when comparing different models from
If the Pearson R value is below 0.30, then the relationship is weak no matter how significant the result. The two most commonly used standard error statistics are the standard error of the mean and the standard error of the estimate. Play games and win prizes! The same phenomenon applies to each measurement taken in the course of constructing a calibration curve, causing a variation in the slope and intercept of the calculated regression line.
asked 4 years ago viewed 31326 times active 3 years ago 11 votes · comment · stats Linked 1 Interpreting the value of standard errors 0 Standard error for multiple regression? Therefore, which is the same value computed previously. The standard error is computed solely from sample attributes. item is installed, selecting it will call up a dialog containing numerous options: select Regression, fill in the fields in the resulting dialog, and the tool will insert the same regression
share|improve this answer answered Nov 10 '11 at 21:08 gung 74.2k19160309 Excellent and very clear answer! For example, if we took another sample, and calculated the statistic to estimate the parameter again, we would almost certainly find that it differs. This statistic measures the strength of the linear relation between Y and X on a relative scale of -1 to +1. The forecasting equation of the mean model is: ...where b0 is the sample mean: The sample mean has the (non-obvious) property that it is the value around which the mean squared
price, part 2: fitting a simple model · Beer sales vs.
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The mid-segment of a triangle (also called a midline) is a segment joining the midpoints of two sides of a triangle.
"Mid-Segment Theorem": The mid-segment of a triangle, which joins the midpoints of two sides of a triangle, is parallel to the third side of the triangle and half the length of that third side of the triangle.
Examples:
1 Given M, N midpoints. MN = 12 Find DF. ANSWER: 2 Given D, E midpoints. DE = 3x = 5 AB = 26 Find x. ANSWER: 3 Given right ΔRST. G, N, J midpoints. ST = 6; RS = 8 Find perimeter of ΔGNJ. ANSWER:
Proof of Mid-Segment Theorem - Using Coordinate Geometry
For this proof, the diagram has been positioned in the first quadrant with one side on the x-axis to keep the algebraic computations as simple as possible, without losing the general positioning of the triangle. Be aware that other positionings are also possible.
Coordinate Geometry formulas needed for this proof: Midpoint Formula: Distance Formula:
Proof:
Proof of Mid-Segment Theorem - Using Similar Triangles
For this proof, we will prove ΔMFN is similar ΔDFE, by SAS for similar triangles, to obtain corresponding angles for parallel lines and establish a pair of proportional sides.
Statements Reasons 1. 1. Given 2. 2. A mid-segment joins the midpoints of two sides of a triangle. 3. 3. Midpoint of a segment divides a segment into 2 congruent segments. 4. DM = MF; FN = NE 4. Congruent segments are segments of = length. 5. DM + MF = DF; FN + NE = FE 5. Segment Addition Postulate (or Whole Quantity) 6. MF + MF = DF; FN + FN = FE 6. Substitution 7. 2MF = DF; 2FN = FE 7. Addition (or Combine Like Terms) 8. ; 8. Multiplication (or Division) property of equality. [This step establishes the ratio of similitude between the two triangles.] 9. 9. Reflexive Property (or Identity Property) 10. 10. SAS for Similar Triangles: If an ∠ of one Δ is congruent to the corresponding ∠ of another Δ and the lengths of the sides including these ∠s are in proportion, the Δs are similar. 11. 11. Corresponding angles in similar triangles are congruent. 12. 12. If 2 lines are cut by a transversal such that the corresponding angles are congruent, the lines are parallel. 13. 13. Corresponding sides of similar triangles are in proportion. QED.
Proof of Mid-Segment Theorem - Using Parallelogram
For this proof, we will utilize an auxiliary line, congruent triangles and the properties of a parallelogram.
Statements Reasons 1. 1. Given 2. 2. A mid-segment joins the midpoints of two sides of a triangle. 3. Through E draw line parallel to . Extend to intersect at M1. 3. Through a point not on a line, only one line can be drawn parallel to the given line. Parallel Postulate. 4. 4. Midpoint of a segment divides a segment into 2 congruent segments. 5. ∠DFE ∠FEM1 5. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent. 6. ∠FNM ∠M1NE 6. Vertical angles are congruent. 7. ΔFNM ΔM1NE 7. ASA - If 2∠s and the included side of one Δ are congruent to the corresponding parts of another Δ, the Δs are congruent. 8. 8. CPCTC - corresponding parts of congruent triangles are congruent. 9. 9. Substitution (or Transitive property) 10. DMM1E is a parallelogram 10. A quadrilateral with one pair of sides both || and congruent is a parallelogram. 11. 11. A parallelogram is a quad. with 2 pair of opposite sides parallel. 12. 12. Opposite sides of a parallelogram are congruent. 13. 13. Congruent segments have = measure. 14. MN + M1N = MM1 14. Segment Addition Postulate (or whole quantity) 15. MN + MN = DE 15. Substitution 16. 2MN = DE 16. Addition (or combine like terms) 17. MN = ½DE 17. Division (or Multiplication) of Equalities
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Examveda
# A number is first decreased by 10% and then increased by 10%. The number so obtained is 50 less than the original number. The original number is :
A. 5000
B. 5050
C. 5500
D. 5900
Let the original number be $$x$$
= 110% of (90% of $$x$$)
= $$\left( {\frac{{110}}{{100}} \times \frac{{90}}{{100}} \times x} \right)$$
= $$\frac{{99x}}{{100}}$$
\eqalign{ & \therefore x - \frac{{99x}}{{100}} = 50 \cr & \Rightarrow \frac{x}{{100}} = 50 \cr & \Rightarrow x = 50 \times 100 \cr & \Rightarrow x = 5000 \cr}
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gravity
# gravity - 22 Experiment II Determination of g from...
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22 Experiment II: Determination of g from Newton’s Laws Goals Experimentally determine the acceleration ( a ) of a two-body system, and the friction force ( f ) Calculate g , the acceleration due to gravity, from a and f using Newton’s laws of motion Introduction and Background The acceleration due to gravity, g , is an important quantity that is most commonly used to calculate the weight (W) of an object from its mass (m): W = mg. The value of g depends on the location. For example, the g value on the Moon is much smaller than that on Earth. Even on the surface of the Earth g varies slightly from location to location, although it is a good approximation to use g = 9.80 m/s 2 as an average. In this lab, however, we will pretend that we know nothing about the value of g , and we will experimentally determine the value of g by measuring the acceleration of a two-mass system and the friction force associated with the motion. Theory : The set-up for the experiment is shown schematically in Fig. 2-1. Obviously the acceleration of the two masses depends on the weight of the mass m 1 , which in turn depends on the value of g . Mass m 2 is a cart, able to roll on the lab table with a small but finite friction. m 2 is connected to the hanging mass m 1 by a light (negligible mass) string passing over a pulley. When the system is released from rest, m 1 will accelerate downward and m 2 will accelerate forward with an acceleration of the same magnitude since the string is unstretchable. This acceleration can be derived by applying Newton’s second law individually to both masses. The free-body force diagrams for both masses are shown in Figure 2-2. Now derive g using Newton’s second law and the force diagrams and show the derivation in your lab report . You should arrive at the equation: 1 1 2 1 ) ( m f a m m m g + + = (2-1) Therefore, we can determine the value of g if we can experimentally measure a and f since m 1 and m 2 are easily determined. Measuring a : The acceleration of the two-body system can be regarded as constant if the friction is constant. This is one of the assumptions we will make in this experiment. For motion with constant acceleration we have Figure 2.1 – Experimental arrangement.
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https://www.physicsforums.com/threads/volume-of-water-required-to-cool-thermal-nuclear-plants.906514/
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# Volume of water required to cool thermal/nuclear plants?
Tags:
1. Mar 5, 2017
### kokodile
1. The problem statement, all variables and given/known data
In the year 2004 the USA produced 1787 TWh of electrical energy in conventional thermal plants and 476 TWh in nuclear plants. Assuming 30% efficiency for nuclear plants and 40% for conventional thermal plants, determine the (annual) volume of cooling water required to cool these plants in once-through cooling if the cooling water undergoes a temperature increase of 10°C. (Neglect the heat lost up the chimney in conventional plants, and assume two significant digits in given data.)
ηthermal=.4
Ethermal=1787 TWh
ηnuclear=.3
Enuclear=476 TWh
2. Relevant equations
η=Useful (electrical) energy/total energy in
Q=mcΔT
Volume=mass/ρ
1 TWh=10^9 KWh
1 kWh = 3.6*10^6 J
3. The attempt at a solution
I did each plant one separately (of course) and converted the 1787 TWh of the thermal plant to Joules so that I could use it in the specific heat equation. When I converted it to Joules I got 9.5*10^24 Joules. I took that number and plugged it into my specific heat equation to get 2.3*10^20 kg. I plugged that mass into the Volume equation and got a volume of 2.3*10^17 m^3
I did the same steps for the nuclear plant and got 4*10^16 m^3.
Because the book only has one answer, I assumed they added the two up and got 2.7*10^17 m^3. The answer in the book is 2.7*10^11 m^3. So I'm close, but my exponent is off. I'm pretty sure I'm converting it correctly from TWh to Joules.
I also found that when I converted the TWh to MJ instead of Joules, I get the right answer, but I can't have it in MJ because the specific heat equation uses 4186 Jkg-1K-1
Last edited: Mar 5, 2017
2. Mar 5, 2017
### Staff: Mentor
Let's see more details of your calculation.
3. Mar 5, 2017
### kokodile
I think I also forgot to type out another step I took. Using the efficiency equation, I did 1787 TWh/.4 and got 4467.5 TWh.
I used 4467.5 TWh and 1787 TWh to find the waste heat energy. So I did 4467.5-1787 and got 2680.5 TWh. Because this is still in TWh, I converted it to Joules. (Should I convert it to Joules before I find the waste heat energy?)
Okay, here is my conversion from TWh to Joules.
2680.5 TWh x 1012W/1 TW x 1 J/s/1 W x 3600s/1 h = 9.65*1024 J (I originally wrote that I got 9.5*10^24 J, but it's actually 9.65*10^24. I still get the same answer though)
9.65*1024J/4186J*10K = 2.3*1020 kg
2.3*1020 kg/1000 = 2.3*1017m3
For the second plant, I took the exact same steps.
Using the efficiency equation, I did 476 TWh/.3 and got 1586.7 TWh.
I used 1586.7 TWh and 476 TWh to find the waste heat energy. So I did 1586.7-476 and got 1110.7 TWh. This is still in TWh, so I converted it to Joules.
1110.7 TWh x 1012W/1 TW x 1 J/s/1 W x 3600s/1 h = 4*1024 J
4*1024J/4186J*10K = 9.6*1019
9.6*1019 kg/1000 = 9.56*1016
If I add those two up, I get (2.3*1017) + (9.56*1016) = 3.25*1017m3
I did realize that I forgot to find the waste heat energy first the second time around, but even with finding the waste heat energy, I still get the wrong answer.
4. Mar 6, 2017
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Calculator search results
Formula
List all divisors
$$18 \times 6$$
$1 , 2 , 3 , 4 , 6 , 9 , 12 , 18 , 27 , 36 , 54 , 108$
Find all divisors
$\color{#FF6800}{ 18 } \color{#FF6800}{ \times } \color{#FF6800}{ 6 }$
Do prime factorization
$\color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ 2 } \times 2 \times 3 ^ { 2 } \times 3$
If the exponent is omitted, the exponent of that term is equal to 1
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \times 2 \times 3 ^ { 2 } \times 3$
$2 ^ { 1 } \times \color{#FF6800}{ 2 } \times 3 ^ { 2 } \times 3$
If the exponent is omitted, the exponent of that term is equal to 1
$2 ^ { 1 } \times \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \times 3 ^ { 2 } \times 3$
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \times \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \times 3 ^ { 2 } \times 3$
Add the exponent as the base is the same
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \times 3 ^ { 2 } \times 3$
$2 ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \times 3 ^ { 2 } \times 3$
Add $1$ and $1$
$2 ^ { \color{#FF6800}{ 2 } } \times 3 ^ { 2 } \times 3$
$2 ^ { 2 } \times 3 ^ { 2 } \times \color{#FF6800}{ 3 }$
If the exponent is omitted, the exponent of that term is equal to 1
$2 ^ { 2 } \times 3 ^ { 2 } \times \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } }$
$2 ^ { 2 } \times \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \times \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } }$
Add the exponent as the base is the same
$2 ^ { 2 } \times \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } }$
$2 ^ { 2 } \times 3 ^ { \color{#FF6800}{ 2 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } }$
Add $2$ and $1$
$2 ^ { 2 } \times 3 ^ { \color{#FF6800}{ 3 } }$
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } }$
List divisors of factors
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \\ \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } }$
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \\ \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } }$
Find all divisors by combining factors which is possible for the reduction of fraction
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } }$
$\color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 0 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 1 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 0 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 1 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } , \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 3 } }$
Calculate the product of all divisors
$\color{#FF6800}{ 1 } , \color{#FF6800}{ 2 } , \color{#FF6800}{ 3 } , \color{#FF6800}{ 4 } , \color{#FF6800}{ 6 } , \color{#FF6800}{ 9 } , \color{#FF6800}{ 12 } , \color{#FF6800}{ 18 } , \color{#FF6800}{ 27 } , \color{#FF6800}{ 36 } , \color{#FF6800}{ 54 } , \color{#FF6800}{ 108 }$
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The number of significant figures in 0.00060 m is
A
1
B
2
C
3
D
4
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Generated By DoubtnutGPT
To determine the number of significant figures in the measurement 0.00060 m, we can follow these steps:1. Identify the Leading Zeros: The number 0.00060 m has leading zeros (the zeros before the first non-zero digit). These leading zeros do not count as significant figures. 2. Find the First Non-Zero Digit: The first non-zero digit in 0.00060 is '6'. 3. Count the Significant Figures: - The digit '6' is significant. - The trailing zero after '6' (the zero in '60') is also significant because it indicates precision in the measurement. - Therefore, we have two significant figures from '6' and '0'.4. Total Count of Significant Figures: - The leading zeros (0.000) do not count. - The significant figures are '6' and '0'. - Thus, the total number of significant figures in 0.00060 m is 3.Final Answer:The number of significant figures in 0.00060 m is 3.---
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Updated on:7/8/2024
Knowledge Check
• Question 1 - Select One
The number of significant figures in 0.10200 is
A6
B5
C3
D2
• Question 2 - Select One
The number of significant figures in 0.030050L is
Afive
Bfour
Ctwo
Dsix
• Question 3 - Select One
The number of significant figures in 0.0009 is
A4
B3
C2
D1
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http://oeis.org/A246646
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A246646 Irregular triangle T(n,m) with sieved modified Collatz sequences for k = A246647(n), n >= 1, m = 1, ..., A248154(n). 3
2, 1, 3, 5, 8, 4, 2, 6, 3, 7, 11, 17, 26, 13, 20, 10, 5, 9, 14, 7, 12, 6, 15, 23, 35, 53, 80, 40, 20, 16, 8, 18, 9, 19, 29, 44, 22, 11, 21, 32, 16, 24, 12, 25, 38, 19, 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103, 155, 233, 350, 175, 263, 395, 593, 890, 445, 668, 334, 167, 251, 377, 566, 283, 425, 638, 319, 479, 719, 1079, 1619, 2429, 3644, 1822, 911, 1367, 2051, 3077, 4616, 2308, 1154, 577, 866, 433, 650, 325, 488, 244, 122, 61, 92, 46, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS The row length sequence for this irregular triangle is A248154. The (modified or Terras) Collatz map is T(k) = (3*k +1)/2 if k is odd and T(k) = k/2 if k is even. See the array A070168. The present irregular array starts with row n=1 for k=2 with 2, 1 and ends because the next number would be 2 which appeared already in this row (this is the trivial cycle). Row n=2 for k=3 is then 3, 5, 8, 4, 2 and stops with 2 which is the first number in this row which appeared already in row k=1. A row for k=4 does not show up because 4 already appeared in the row for k=3. Also no row for k=5 appears. Row n=3 is for k=6 with 6,3, etc. In this way a 'minimal' Collatz table is build. The Collatz conjecture is that every positive integer is present (the end numbers in each row n >= 2 appear exactly twice). LINKS Eric Weisstein's World of Mathematics, Collatz Problem, EXAMPLE The irregular triangle T(n,m) begins: n, k \ m 1, 2: 2 1 2, 3: 3 5 8 4 2 3, 6: 6 3 4, 7: 7 11 17 26 13 20 10 5 5, 9: 9 14 7 6, 12: 12 6 7, 15: 15 23 35 53 80 40 20 8, 16: 16 8 9, 18: 18 9 10, 19: 19 29 44 22 11 11, 21: 21 32 16 12, 24: 24 12 13, 25: 25 38 19 ... Row n=14, k=27: 27 41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23; Row n=15, k=28: 28 14; Row n=16, k=30: 30 15; ... The complete modified Collatz iteration until 1 is reached is obtained, for example for k=19, as follows: 19 29 44 22 11, (11) 17 26 13 20 10 5, (5) 8 4 2, (2) 1, that is 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1, which is row n=19 of A070168. CROSSREFS Cf. A246647, A248154, A070168. Sequence in context: A286390 A135017 A070168 * A198094 A263047 A021828 Adjacent sequences: A246643 A246644 A246645 * A246647 A246648 A246649 KEYWORD nonn,tabf,easy AUTHOR Wolfdieter Lang, Oct 02 2014 STATUS approved
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https://officialsavanarae.com/dermatology/frequent-question-how-many-moles-are-in-a-ml.html
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Frequent question: How many moles are in a mL?
Contents
What unit is mol mL?
In chemistry, the most commonly used unit for molarity is the number of moles per liter, having the unit symbol mol/L or mol⋅dm3 in SI unit. A solution with a concentration of 1 mol/L is said to be 1 molar, commonly designated as 1 M.
Is mol L the same as mL?
mol/L↔umol/mL 1 mol/L = 1000 umol/mL.
How many moles are in 1 mL of water?
of moles of water in 1 ml = 1/18 mol.
How do you find moles with molarity and mL?
To calculate the number of moles in a solution given the molarity, we multiply the molarity by total volume of the solution in liters.
Is mM MOL mL?
How many Mole/Milliliter are in a Millimolar? The answer is one Millimolar is equal to 0.000001 Mole/Milliliter.
How do you convert Millimolar to mg mL?
So, to convert mg/mL to mM, I divided the concentration in mg/mL (20 mg/mL) by the molecular weight of the sample (232,278 g/mole) and multiplied by 1,000.
How many moles are in a Nanomole?
Nanomole is a unit of measurement for amount of substance. Nanomole is a decimal fraction of amount of substance unit mole. One nanomole is equal to 1e-9 moles.
Does mL mean microliter?
One megaliter (ML) equals one million liters. One microliter (µL, lowercase mu) equals one millionth of a liter.
What is a 1 molar solution?
Molarity is another standard expression of solution concentration. … A 1 molar (M) solution will contain 1.0 GMW of a substance dissolved in water to make 1 liter of final solution. Hence, a 1M solution of NaCl contains 58.44 g. Example: HCl is frequently used in enzyme histochemistry.
How many liters are in a mole?
As long as the gas is ideal, 1 mole = 22.4L.
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# How Many Horsepower Does A Horse Have? (Question)
While it is true that the maximum output of a horse is around 15 horsepower, when you average the output of a horse over the course of a work day it ends up being around a horsepower. Watt defined this amount as “the amount of work required from a horse to pull 150 pounds out of a hole that was 220 feet deep”.
## How many horses are in 1 horsepower?
Each horse pushed with a force that Watt estimated at 180 pounds. From this, Watt calculated that one horsepower was equivalent to one horse doing 33,000 foot-pounds of work in one minute.
## How many HP is the average horse?
An average work horse achieves a maximum of just under 15 hp at a sprint, while a human at peak production achieves around five horsepower. Interestingly, a horse exerting 1 horsepower can lift 330 pounds of coal 100 feet per minute, 33 pounds of coal 1000 feet in one minute, or 1000 pounds 33 feet in one minute.
## Why do horses have 15 horsepower?
His improved design for a steam engine was much more efficient than previous designs, requiring much less fuel. So, he developed the horsepower as a way to demonstrate to customers who hadn’t yet switched from horses to steam engines that it was a good investment.
## How many horsepower does a human have?
When considering human-powered equipment, a healthy human can produce about 1.2 hp (0.89 kW) briefly (see orders of magnitude) and sustain about 0.1 hp (0.075 kW) indefinitely; trained athletes can manage up to about 2.5 hp (1.9 kW) briefly and 0.35 hp (0.26 kW) for a period of several hours.
## How fast is 1hp?
horsepower, the common unit of power; i.e., the rate at which work is done. In the British Imperial System, one horsepower equals 33,000 foot-pounds of work per minute —that is, the power necessary to lift a total mass of 33,000 pounds one foot in one minute.
## How much horsepower does an elephant have?
One estimate I have seen is that an elephant can pull up to 9 tons = 18000 lb. However, to calculate horsepower, the speed of pulling has to be factored in. Let us postulate that the elephant can pull 18000 lb, against a coefficient of friction of 0.5, at a rate of 5 feet/second. so 18000 x 0.5 x 5 / 550 = 81.8 HP.
## How much horsepower does a duck have?
It dictates that an animal’s rate of consumption, or metabolic rate, is about the same as its mass, multiplied to the power of 3/4. The next step is to divide the mass of a duck by the mass of a horse and complete Kleiber’s law. This gives a value of 131.2 duckpower to one horsepower.
## How much horsepower does Usain Bolt have?
The data shows that Bolt’s force peaks during the first second of the race, and after that remains relatively consistent. His speed peaked around 7 seconds into the race, at over 27 mph. The researchers calculated a maximum power output of about 3.5 horsepower.
## How much horsepower does a reindeer have?
We sometimes use horsepower, so taking that number and dividing by 8, then divide by 745.7 to get a horsepower per reindeer of 6.11 sextillion horsepower (a bit more than your car). The heat of vaporization of water, and thus most humans and reindeer, is about 2.26 million Joules per kilogram.
## How much horsepower does a donkey have?
A unit of power equal to 250 watts; it is approximately ⅓ horsepower.
## How much power does a dog have?
In dogs, it is 1:125 across all breeds, according to a study published in the Intelligence issue of Popular Science. This means that while dogs don’t have as much brain power as us, they have a lot more compared to other animals (the ratio for great white sharks is 1:2,550).
## How much horsepower does a bear have?
Packing up to 4,500 horsepower in a muscular 8,700-pound package, Rare Bear is a radically modified, one-of-a-kind version of a Grumman F8F-2 Bearcat.
## Is 300 a lot of horsepower?
300 horsepower is actually very little. Consider that some engines for large ships have 100,000 horsepower at around 100 rpm or less. Large power plants have even more power than that. By comparison, 300 horsepower is trivial.
## Horsepower – Energy Education
Figure 1. The engine from the Koenigsegg One1 supercar, which produces 1341 horsepower (one megawatt). Horsepower is a measure of power. Power is defined as the rate at which energy is exchanged; it is defined as the amount of energy used divided by the amount of time it takes to consume that energy. As a result, the measurement of horsepower relates to how much power an engine can produce over a continuous period of time. Inventor James Watt, who made substantial advancements to the steam engine, is credited with coining the term horsepower.
## Power of a horse
Although it may appear reasonable to believe that one horsepower is the maximum amount of power a horse is capable of producing at any given time, this is not the accurate assumption. Horses can produce up to 15 horsepower at their peak, but the greatest output of a person is slightly more than a single horsepower at its peak. For elite athletes, this production may be significantly greater, with Tour de France cyclists producing around 1.2 horsepower for approximately 15 seconds and little less than 0.9 horsepower for almost one minute.
It is possible that Watt intended to be deceptive when he coined the term “horsepower,” but he had good cause for doing so.
• Suppose he claimed that his pricey engine produced the same short-term production as a horse.
• It was instead calculated in terms of the amount of work a horse could complete in a day, giving it the equivalent output of 10 horses rather than just one.
• Watt may have “skewed the facts” in order to make his engine more enticing, but he was not deceiving anyone.
• It was determined by Watt to be “the amount of effort necessary by a horse to draw 150 pounds out of a hole that was 220 feet deep.” In a similar line, the typical human’s output over the course of a day is 100 watts, which is where the phrase “energy servant” originates from.
## Horsepower vs. torque
When acquiring a new automobile, the crib sheet will give two output values for the engine: horsepower and torque, which are both important. These values will be presented in the units of SAE certified Horsepower (which is the same as horsepower, but certified by the SAE) andpound-feet if you live in the United States or Canada, respectively. If you live in Europe, the outputs will be indicated in Watts and Newton meters, as opposed to the United States. There is a fundamental difference between these two sets of units in that, whereas horsepower is a measurement of power, which refers to an amount of energy transfer over time, Newton-meters are a measurement of torque, which is a measurement of rotational force that has no unit of time attached to it.
Another distinction between horsepower and torque is that horsepower is defined as energy output over time, whereas torque is defined as immediate energy production.
In a similar vein, a car with a lot of torque will be able to accelerate more quickly than a car with less torque, because its immediate output will be greater than that of the latter.
## Conversions
• Energy, end-use energy, primary energy, and energy conversion technology are all terms that can be used to refer to energy. Alternatively, visit a random page. More information about horsepower may be found here.
## References
Ethan Boechler, Allison Campbell, Jordan Hanania, James Jenden, Kailyn Stenhouse, Dayna Wiebe, and Jason Donev are among the actors that have appeared in the film. The most recent update was made on September 27, 2021. Obtain a Citation
## How Much Horsepower Does a Horse Have?
When it comes to automobiles, the phrase “horsepower” is one that many of us are acquainted with. However, how much horsepower does a horse have is a mystery. A horse does not have the equivalent of one horsepower, as is commonly believed, and this is truly incorrect. Horses can produce a maximum power of 14.9 horsepower on average, according to statistics. This number, on the other hand, will vary from horse to horse. The greatest power produced by an average individual is 1.2 horsepower. It was customary to use a draft horse to calculate horsepower in the past.
You may also learn more about horse strength by checking out our guide to the strongest horse breeds.
## What is Horsepower?
A horsepower is a unit of measurement that is commonly used to refer to the output of engines or motors in terms of power. Approximately 33,000 foot-pounds of effort each minute, or 550 foot-pounds per second, is equal to one unit of horsepower (745.7 watts). James Watt, a Scottish engineer, is credited with inventing the horsepower measuring system. When Watt invented the unit of measure in the 18th century, he was looking for a means to compare the power of draft horses and steam engines. After making improvements to the design of steam engines, he utilized his calculations to demonstrate that they were more fuel-efficient than they had previously been.
• He deduced from this that the amount of power and energy applied by the horse to turn the wheel is equal to 33,000 foot-pounds per minute.
• Watt was the inventor of the watt, which is a measure of power equal to one joule of work produced per second and named after him.
• courtesy of Gertan / Shutterstock.com Horsepower quickly gained popularity as a standard unit of measurement for motors and engines.
• Also see: How much weight can a horse tow at a time?
## Why is it Called Horsepower?
During the late 18th century, when James Watt created the term “horsepower,” horses were the most important source of power in the globe, according to historians. He came up with it as a means to compare the power of an engine to the power of horses. Horses were utilized for a variety of tasks all throughout the world, including transportation, farming, spinning mill wheels, towing barges, and many other activities. When machines were first invented, many people were cautious to place their trust in these man-made things.
To demonstrate the superiority of the steam engine over a draft horse, Watt devised a series of comparisons between the two machines’ respective capacities. He was able to demonstrate the advantages of operating the steam engine as a result of his actions.
## How Much Horsepower Does a Human Have?
In a limited period of time, a healthy individual can generate around 1.2 horsepower on average. A person is capable of maintaining 0.1 horsepower indefinitely. An individual can generate around 0.27 horsepower on average. An athlete who is in incredible form can generate up to 2.5 horsepower for a few seconds at a time. The exact amount of horsepower a person possesses will vary depending on their physical condition and the type of physical activity they engage in.
## How is Horsepower Calculated?
horsepower is calculated using the equation hp= Fd / t. HP is an abbreviation for horsepower, F is an abbreviation for force in pounds, d is an abbreviation for distance in feet, and t represents time in minutes.
## How Many CC’s are in One Horsepower?
In general, one horsepower is equal to 14 to 17 cubic centimeters of capacity. The most typical response is that one horsepower is equal to 15 CC, while the actual truth is more complicated. The figure will fluctuate based on a variety of parameters such as engine tune, engine size, fuel, and boosting levels.
## Does Horsepower Make a Car Faster?
It is generally accepted that greater engine horsepower results in improved acceleration. Essentially, this implies that it will have superior overall performance, and more horsepower may translate into higher speed. The link between speed and horsepower, on the other hand, is not linear. There are other considerations to make, such as the size, weight, torque, and aerodynamics of the vehicle.
## Does one horsepower really equal the power of one horse?
It is possible to measure the power output of your car’s engine in a variety of different ways, but the most common is in horsepower. Upon first appearance, you would believe that horsepower refers to a horse’s ability to produce power. Is one horsepower, on the other hand, truly equivalent to one horse’s power?
### What is horsepower?
In electrical engineering, horsepower is a unit of measurement for power, which is defined as the rate at which work is completed. If you envision ‘work’ as the act of moving a heavy object up a hill, then ‘power’ is the rate at which you do it. In the beginning, the word was coined by Scottish engineer James Watt to contrast the output of steam engines with the power of horses. Power outputs of piston engines, such as those found in your automobile, together with turbines, electric motors, and other machinery were eventually included in the definition.
### Does one horsepower equal one horse?
No, not at all. It is a frequent fallacy that one horsepower is equivalent to the maximum power production of a horse, which has a maximum power output of around 14.9 horsepower. This is not true. A human individual, on the other hand, is only capable of producing roughly five horsepower at its maximum power output. As a result, Watt defined horsepower as the amount of power that a horse can maintain continuously for a lengthy amount of time. There are many various types of horsepower, on the other hand.
### How can horsepower have different definitions?
In contrast to the measurement of other quantities such as time, the precise definition of horsepower might vary based on geographical variances and the individual equipment whose power is being measured, for example. Mechanical horsepower, also known as imperial horsepower, is a unit of measurement established by James Watts that is approximately comparable to 745.7 watts of power. Mechanical horsepower is a unit of measurement invented by James Watts that is about equivalent to 745.7 watts of power.
It is equivalent to 735.5 watts and was intended to quantify the power of horses.
While one horsepower is about similar to 746 watts for electric motors, one boiler horsepower, which is used to measure the output of steam boilers, is much more disparate, with one boiler horsepower being equivalent to 9,810 watts for steam boilers.
### Anything else I should know?
When it comes to your automobile, the brake horsepower, or bhp, is the most often used metric of engine power in the United Kingdom. However, while your engine may be capable of producing 100 horsepower, the different mechanical components of your car’s driveline will ensure that not all of that power will be transferred to the wheels at some point.
See also: How Much Banamine To Give A Horsehow Far Can You Ride A Horse In A Day? (Correct answer)
#### Find out more about power outputs and what they meanhere
With the term “horsepower,” visions of exotic Italian roadsters, souped-up American muscle cars, and screamin’ Formula One machines immediately come to mind. One item that, maybe unexpectedly, does not frequently come to mind while thinking about horses is, you know, a real horse. When we take the time to think about it, though, we find ourselves with more questions than answers: Is it true that every horse has roughly the same amount of power? What is it about the twenty-first century that we are still comparing vehicles to horses?
Afterwards, there is the most important issue of them all: What is the maximum amount of horsepower a horse can produce?
As we gallop our way toward the solution, we’ll say “neigh” to befuddlement and confusion.
## What is horsepower?
Before we get started, let’s clarify what a horsepower is. A horsepower is a unit of power measurement that is most commonly used to quantify the output power of motors and engines. Approximately 746 Watts are required to produce one mechanical horsepower (also known as imperial horsepower). More information may be found at: The Dunce Cap Has Undergone an Extraordinary Transformation
## A bit of history
By the late 18th century, renowned Scottish engineerJames Watt was having difficulty marketing his improved steam engine to the public. Draft horses were utilized by industrialists to power their machinery (think plows, wagons, and mills). Watt employed a marketing approach, emphasizing the strength of horses, because they were set in their ways and saw no obvious disadvantages to having an all-equine workforce. The typical horse could turn the mill wheel around 2.4 times per minute, according to his calculations.
His demonstration that his steam engine could surpass that work rate many times over simplified the comparison and made the advantages more clear to the audience.
## The power of one horse
The daily labor output of a healthy draft horse is nearly comparable to one horsepower; hence, Watt’s estimations were precise enough to be within a tolerable margin of error, given the limitations of the technology. A Belgian draft horse that is trotting. The ordinary horse has the ability to maintain a maximum output of roughly 14.9 horsepower for a short length of time. (Photo courtesy of Wikimedia/Vkarel) Horses, on the other hand, are not machines.
They work harder at some times of the day than at others; they create more power during short, intense bursts of exertion than than gradual, protracted labor; and there’s also consideration for the variances between various horse breeds to take into consideration.
## Not all horses are alike
During the 1925 Iowa State Fair, it was discovered that the average horse can maintain a maximum output of roughly 14.9 horsepower for a very short length of time on average. As a result, although this is puzzling, the typical adult horse produces substantially more than one horsepower regardless of breed. The phrase has nothing to do with veterinary science, rather it relates to the amount of force necessary to move 33,000 foot-pounds of mass every minute in a straight line. Horse breeds differ from one another in terms of biology.
There is no one-size-fits-all solution, and there is no ballpark figure that applies to all horses.
## Power outputs and the average human
That’s a difficult topic to answer since, like horses, individuals come in a variety of forms, sizes, and physical capacities, making it difficult to answer. It is estimated that the typical healthy and physically active adult individual produces around 0.27 HP, however this varies depending on their activity level. At their peak, the same individual can exert around 1.2 HP. Humans have a limited ability to endure around 0.1 HP forever. Usain Bolt, the Jamaican sprinter who won the 100m final at the Rio Olympics in 2016, in the aftermath of his victory.
The image is courtesy of Wikimedia/Fernando Frazo/Agência Brasil.
Usain Bolt, a former sprinter, was able to generate a maximum of 3.5 horsepower in a burst.
The metric system will be discussed in this section. Measurement is a complicated subject, and there are many different approaches of taking measurements (or in this instance, horses). Mechanical horsepower, often known as imperial horsepower, is the term used today to refer to the initial measurement of horsepower, to which we mentioned above. Your prediction about whether the presence of metric horsepower implies the existence of the metric horsepower unit was right. In engineering, metric horsepower is defined as the amount of force necessary to raise a 75 kg (165 lb) weight one meter in exactly one second.
The 746 Watts of imperial horsepower are conveniently similar enough to one other that the two standards are frequently used interchangeably.
## Don’t forget steam engines
It is unfortunate that the muddle does not stop there. There is also a unit known as boiler horsepower, which quantifies the ability of a boiler to deliver steam. The equivalent of one unit of boiler horsepower is 9,812 Watts. Then there’s electrical horsepower, which measures the power output of electrical motors, hydraulic horsepower, which measures the power of hydraulic machinery, drawbar horsepower, which is used in locomotives, and so on.
Electrical horsepower is one type of horsepower, while hydraulic horsepower is another. The list is not exhaustive, but it would take a significant amount of HP to get through it all.
## Horsepower in cars
Matters do not become any less complicated when looking at the power of a vehicle engine. The output of electric motors or automobile engines can be represented by a variety of different forms of horsepower, depending on the application or step in the transmission process at which the motor or engine is used. More information may be found at: The Bundling Bag: Managing the Hormones of Teenage Girls in the 17th Century The speed of the pistons is represented by the number of horsepower. The term “indicated horsepower” refers to the theoretical power output of an engine.
Don’t make the mistake of conflating shaft horsepower (well, get up!).
Finally, effective horsepower is the result of combining the factors listed above to obtain a median average after deducting losses due to inefficiencies from the total.
## In conclusion
Despite its various variations, horsepower is simply the average amount of power created by an average horse when it is asked to complete an average amount of work over an average duration of time. That may seem a little ambiguous, but it’s the best answer you can get without memorizing a whole engineering textbook. However, if you’re under time constraints on quiz night, try 14 HP. The majority of horses are capable of producing and maintaining 14 horsepower.
## How Much Horse Power Does a Horse Have?
When someone mentions the word “horsepower,” most people immediately think of race vehicles. Unfortunately, we will not be discussing racing automobiles today. Fortunately, we’ll be discussing horses and how much horse power can be produced by a single horse. This is exciting news! If you’re anything like the average person, you’d probably answer: Horsepower of a horse equals one, which is why the unit was termed “horse power” in the first place — this is common sense. However unexpected it may appear, the reality is that the horsepower of each horse is never quite the same for all of them.
In addition, the pace at which the horses are achieving their peak performance is a significant consideration.
The output of other engines, electric motors, and turbines was later measured with it since it was a handy way to do so.
In this post, we will look into some intriguing issues such as: How much horsepower does a horse have, and why does it have it? What is the formula for calculating a horse’s horsepower? As an added bonus, we’ll be looking at how much horsepower people actually possess as well.
## What Is Horsepower?
Horsepower is merely a measure of the amount of power available. However, in order to comprehend it thoroughly, we must go back to the period when this unit of measurement for power and energy was created. An engineer from Scotland, James Watt, came up with a better design for steam engines in the 18th century, stating that his version is more fuel-efficient. For the sake of demonstrating his argument and inventing something new, he was the one who established the horsepower as a unit of measurement.
It was discovered that an average draft horse could turn a 24 ft mill wheel around 2.5 times per minute without exerting excessive effort.
This value is equivalent to 746 Watts or Joules per second of electricity.
Besides that, it is a frequently used unit for determining the power consumption of both electrical and mechanical motors.
## What Is the Horsepower of One Horse?
A draft horse has an average power of 14.9 Horsepower. This figure, however, is likely to fluctuate depending on the quantity of energy it expends in a particular period of time. Horses, on the other hand, are often different in terms of breed and overall physical fitness. Horses of some breeds, such as the American Quarter Horse, will have more horsepower than the ordinary horse. As a result, horses are likely to demonstrate a range of degrees of strength. In order to comprehend how I arrived at this figure, it is necessary to recognize that one horsepower does not always equate to the horse’s maximum power output.
In other words, 1 HP is required to lift 33,000 foot-pounds of mass per minute on average, which is equivalent to 1 horsepower.
From the Crimean War to the Second World War, the horse was referenced in a paper titled “Delayed Obsolescence: The Horse in European and American Welfare from the Crimean War to the Second World War.”
## How Many Horsepower Does a Human Have?
A physically fit human being has the ability to exert around 0.27 horsepower (HP). This, once again, is largely dependent on the type of activity that they are engaged in. For example, an average bicycle rider may generate 0.2 horsepower when riding for pleasure, but a professional rider is more likely to sustain approximately 0.57 horsepower while competing in a competition. Furthermore, labor that requires more energy, such as lifting and carrying large things, running, trekking, and other activities, can increase the horsepower of the person performing the action.
An typical human can maintain 0.1 HP for an unlimited period of time, whereas s/he can maintain 0.27 HP for a period of little more than an hour.
## FAQs Related to Horsepower in a Horse
Understanding horsepower may be difficult, especially for non-technical individuals. Let’s get right to the point of clearing up the most frequently encountered misunderstandings about horsepower.
### Is 1 HP equal to a horse?
Because horses are capable of producing up to 15 horsepower on average, one horsepower does not equate to one horsepower of power. As a result, the greatest power that an equine may exert is not determined by one horsepower. In its place, it specifies the amount of energy necessary to move 33,000 foot-pounds of mass every minute.
### Why do they call it horsepower?
The term “horsepower” was coined as a mathematical means of equating engine power to the number of horses on a horse. Watt contrasted the power of a horse with the power of a steam engine in order to demonstrate that his invention was a viable alternative to animals in order to establish his point. This was due to the fact that horses were commonly utilized by farmers to undertake difficult jobs during that time period. Furthermore, horses were used to spin mill wheels, which was a common practice at the time.
### How many cc’s are in 1 horsepower?
One horsepower is equal to around 14 to 17 cubic centimeters of displacement. When compared to today’s fuel-efficient engines, this figure is likely to change significantly. In general, you may say that every 15 cc produces one horsepower.
### How much horsepower does a draft horse have?
Draft horses have an average power of 15 horsepower. Horses used for draft work are often used for labor-intensive jobs such as carrying big loads. They have the ability to maintain this level of strength for extended periods of time.
### Why does a horse have 15 horsepower?
When performing laborious work, a healthy draft horse may produce a maximum of 15 horsepower at its peak performance. The horsepower of an equine, on the other hand, fluctuates depending on the breed, health, and amount of labor that they are subjected to.
### How fast is one horsepower in mph?
In terms of speed, one horsepower is about equal to 0.163 miles per hour (in the United States). You must also know the thrust of the engine in order to compute the value of horsepower in miles per hour. Using this information, you may calculate your answer using the formula: speed = power/force.
### What kind of horse is horsepower based on?
Draft horses are used to calculate horsepower since they are bigger and stronger than other horse breeds, resulting in more overall horsepower.
## Conclusion
Despite the fact that it is a unit of measurement for power, one horsepower does not represent the highest amount of energy that a horse can expend. One horsepower is equivalent to 746 Watts or Joules per second, according to the laws of physics and general computations. Watt is also the surname of the Scottish engineer who came up with the concept to construct this unit of measurement for power in the first place.
His innovation, the steam engine, was extremely effective in replacing horses for labor-intensive tasks. Today, horsepower is often used to describe the maximum energy output of a variety of machines and equipment. Horses are capable of producing and maintaining around 15 horsepower.
## What Is Horsepower?
What is the definition of horsepower? Look away now, but the vehicle expert in your life — the one who has an opinion on everything that moves – is trying to locate a Wikipedia page right about now, so don’t bother looking. However, despite the fact that horsepower is a generally recognized and frequently marketed metric of engine power—as well as the foundation for many automobile purchases and fuel for the fire of automotive bragging rights—very few people are aware of what is actually going on in their engines.
See also: What Does It Mean When A Horse Is Green? (Solution)
## The Genesis of Horsepower
In order to understand the origins of this peculiar measure, we must go back to the 18th century, when the introduction of steam power was going to result in the early retirement of a large number of hard-working horses. James Watt (1736-1819), a Scottish-born inventor and engineer, was attempting to pique interest in his new and improved version of the steam engine when he sent this letter. The trouble was that he had no idea how to market a product that was so new that there was no reliable method to assess the advantage it offered.
Photographs courtesy of Getty Images Watt was drawn to working horses because they pulled, pushed, and lifted huge loads in factories, mills, and mines, and he wanted to learn more about them.
### Calculating the Power of a Single Horse
He began by observing mill horses at work in the millyard. They spun a mill’s central shaft by walking around a circle 24 feet in diameter, about 144 times each hour, while lashed to spokes emanating from the center shaft of the mill. Watt calculated that each horse pushed with a force of 180 pounds per horse. Watt concluded that one horsepower was comparable to one horse performing 33,000 foot-pounds of effort in one minute based on this information. Imagine a lone horse lifting a 33-pound pail of water from the bottom of a 1000-foot-deep well in 60 seconds, and you’ll have a better understanding of this—and less regrets about arithmetic class.
1. An early Watt steam engine, which was designed in 1785, is depicted here as an illustration.
2. It was all he needed to work his marketing magic and persuade corporations that the newfangled technology was the superior power source.
3. A century ago, the steam-powered locomotive was the backbone of the railroad industry, and it controlled the tracks for over 100 years.
4. Users of light bulbs are included as well.
5. Contrary to popular belief, one “watt” does not equal one horsepower.
6. At the end of the day, effective salesmanship was just as important as engineering expertise in the development of horsepower.
7. Read about these white-hot 2019 American muscle vehicles to get a taste of the rubber-melting exhilaration that comes with a lot of horsepower.
This material was generated and maintained by a third party and imported onto this website in order to assist users in providing their email addresses for further consideration. You may be able to discover further information on this and other related items at the website piano.io.
## How Many Horsepower Does A Horse Have?
When we talk about horsepower, the majority of us immediately think about automobiles! The term “horsepower” was initially used to characterize our equine pals and their incredible strength, which is no surprise. A horse, on the other hand, has how many horsepower? The majority of people believe that one horse equals one horsepower, however the truth is that it is not nearly that easy! Horsepower is defined as the average amount of labor performed by a single draft horse over the course of a typical working day.
## What Is Horsepower And How Is It Calculated?
Horsepower is a unit of measurement for the amount of power available. Power is defined as the pace at which work is completed – therefore it is not just about how powerful something is, but also about how quickly the task is completed. The term “horsepower” was used to contrast the power of steam engines with the power of working horses in the early 20th century. The goal was to demonstrate that the newly enhanced steam engines were more powerful than their predecessors. James Watts, an engineer and inventor who lived in the late 18th century, devised a unit of measurement known as horsepower.
• At this period, draft horses were used to carry out the majority of the hard lifting.
• They were able to labor for lengthy periods of time at a steady pace.
• He then calculated the amount of force the horse would have to exert to spin the wheel, as well as the total power the animal would exert over the course of a working day.
• James Watt’s methods have been enhanced by modern ones, but his calculations were quite near to the mark!
• The fact that Mr.
## How Many Horsepower Does A Horse Have?
We may easily assume that one horse has one horsepower, but as we’ve previously seen, this isn’t always the case. The labor rate of a draft horse was used to derive the measurement of horsepower. These horses are extremely strong and powerful, and they are capable of carrying out heavy work for extended periods of time. Take a look at the original measurement – a huge draft horse turning a millstone roughly 2.5 times every minute for the duration of a complete working day – for context. Do you believe your favorite horse or pony would be able to manage that amount of labor all day?
All horses are strong and muscular, and their musculature is outstanding, but none of them have the sheer bulk of physical mass that a draft horse possesses, regardless of their breed.
We may deduce from this that the horsepower of a horse changes depending on the horse’s size, body weight, muscle mass, and overall fitness.
## What Is The Average Horsepower Of A Horse?
Because there are so many factors to consider, calculating the average horsepower of a horse is quite difficult:
• Horse size, muscle mass, body weight, fitness levels, and athletic ability are all factors to consider.
The time span during which the task was completed is taken into consideration when calculating the amount of horsepower required. When you take all of these different elements into consideration, it becomes clear that calculating the average horsepower of a horse is nearly impossible. To do this, we would need a large number of various sorts of horses to perform the same work while measuring the time it takes each horse to complete the task. We would then be able to compute the average horsepower of a horse based on the information provided.
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## What Is The Maximum HP Of A Horse?
Despite the fact that modern-day horses are unlikely to be able to labor for extended periods of time like the draft horses utilized in the mills, they are nonetheless enormously strong beasts. In fact, a single horse may generate a significant amount of horsepower when operating at maximum capacity! It has been demonstrated in studies that a single horse can exert up to roughly 15 horsepower over a shorter length of time. This is due to the fact that power is a measurement of both speed and effort/force.
Horses, on the other hand, have up to 10 times the motive (moving) power of humans.
• Petra, a Belgian Draft Horse, holds the record for being the world’s strongest horse. In order to pull logs, Petra was taught as part of a team, and her owner quickly recognized that this massive horse possessed astonishing levels of horsepower. A jet engine can produce more than 1000 horsepower
• The mare Winning Brew holds the world record for the fastest horse speed, having run two furlongs in less than 21 seconds at a pace of 43.97 miles per hour in 2011. That is a significant amount of power
• It is possible to create an amazing 109000 horsepower with the world’s most powerful engine. In fact, one of the world’s tiniest pony breeds, the Shetland Pony, is really one of the world’s most powerful! This is enough electricity to run an entire suburban village! This small stocky type can pull an astonishing amount of weight pound for pound, and if it were the same size as the larger draft breeds, it would exceed them.
As we’ve taught, the horsepower of a horse is the amount of labor that one draught horse can accomplish in a single day in regular working conditions. Because it is improbable that our modern-day riding horses are as powerful as this, our horses are most likely less than one horsepower in power! We’d be interested in hearing about your horses and ponies, as well as how much horsepower you believe they may have. Please share your thoughts in the comments section below!
## How Much Horsepower Does a Horse Have?
Horsepower appears to be something that is very self-explanatory on the surface. The horse portion, on the other hand, is the four-legged agricultural companion with whom we are all accustomed. When you combine the power bit with the horse’s name, it’s obvious that we’re talking about the horse’s ability. To put it another way, how much horsepower does a horse have? It turns out that it’s not quite that straightforward. Horses can sprint at 14.9 horsepower when they are at their top capability.
As a result, it’s nearly 15 times what you would expect.
It turns out that behind the hood of your automobile lies a treasure trove of historical information!
## Where Did Horsepower Come From?
It is necessary to go back to the late 18th century in order to get the solution. James Watt, a Scottish inventor, was hard at work improving upon the steam engine. Despite the fact that Watt did not develop the steam engine (that distinction goes to Thomas Newcomen), he was responsible for inventing the separate condenser, which was critical to the efficiency and cost-effectiveness of the steam engine. It is possible that you are familiar with his last name, Watt, as a unit of power, such as a 60-watt lightbulb.
• Watt was more than simply the inspiration for the watts that were named after him.
• When Watt first started working on improving the steam engine, it was difficult to persuade anyone to purchase one of his inventions.
• Why would they spend money on a machine to accomplish work that their draft horses were already doing for them?
• When Watt invented horsepower, he was trying to describe how much labor a horse could accomplish on average during the course of a day.
• Horses are also not all capable of producing the same amount of production.
• Steam engines were able to deliver steady performance over a much longer period of time.
Therefore, it became appropriate to develop a unit of measurement for their wattage (or equivalent). Because, after all, it isn’t like folks could purchase a horse with a higher-rated engine than another horse that was available for purchase.
## What is Horsepower?
As a result, we’ve established that horsepower is not the same as one horse’s efforts. So, what precisely is horsepower? Watt discovered how much workhorses could do by visiting a mill where horses were employed to spin a grinding wheel and observing the results. He was able to construct his formula for horsepower by recording how many times per hour they rotated the wheel and calculated how much force they put on the wheel. One horsepower is equal to one horse putting out 33,000 foot pounds of effort in one minute, or one horsepower.
The unit of horsepower is denoted by the letter hp.
## Different Kinds of Horsepower
Mechanical horsepower, often known as Imperial horsepower, is the unit of horsepower that Watt invented. It differs somewhat from metric horsepower in terms of measurement. Different nations will use one or both of these units, and some countries may employ variants of these units that are unique to them. There are a variety of additional types of horsepower available, including electrical horsepower, hydraulic horsepower, and tax horsepower, to name a few examples. Machine performance may be assessed in a variety of ways using these instruments.
## How Many Horses Does My Car Really Have Under Its Hood?
Take your automobile’s horsepower rating and divide it by 15, which is approximately equal to the value of one horse’s horsepower, rounded up. This will give you an approximate number of horses worth of labor that your car is performing. In accordance with Autolist, the typical mainstream vehicle has 200 horsepower. When we divide this number by 15, we get 13.33, or thirteen and a third. So you’ll have to share one of your horses with two of your buddies as a result of this. Sorry, but I’m not the one who sets the rules!
Some of the world’s most powerful automobiles now have horsepower ratings ranging between 1200 and 1500 horsepower.
However, horsepower is not necessarily a straight correlation to speed because there are several other aspects in the car’s construction that might influence its eventual performance.
## What Breeds of Horses Have the Most Horsepower?
Because horsepower is frequently mentioned in connection with high-performance automobiles, you could assume that the quickest horses provide the most horsepower. However, horsepower is a measure of a horse’s capacity to perform labor rather than its speed. The ability of an engine to perform labor can translate into a speedier automobile. Fast and light breeds such as Thoroughbreds and Arabians, which you see winning races, aren’t the monarchs of horsepower in the horse world, as is the case with automobiles.
The fact that they are not bred to take on additional weight explains why jockeys must be so careful about maintaining their own weight.
When it comes to horsepower, it turns out that draft horses, with their strength and strong structure, are the most powerful breeds on the market.
Shires, Percherons, Belgians, and the well identifiable Clydesdales are among the draft breeds that excel at performing labor duties. And that is, after all, what horsepower is all about: performance!
## Workhorses Replaced
In order to account for this, horsepower was first estimated with draft horses in mind. A draft horse could spin the heaviest grindstone in the mill 144 times an hour, or roughly 2.5 times per minute, at the mill where Watt was doing his research. The draft horses were able to maintain that level of performance for the full day’s workload! It’s just wonderful! However, it is still not as productive as Watt’s upgraded steam engine in terms of output. After a while, other machinery, like as tractors, began to take the place of draft horses on the farm.
Aside from larger farms, draft horses are also frequently used by smaller or specialist farms where their ability to overcome obstacles is advantageous.
## How Much Horsepower Does a Human Have?
It should come as no surprise that people do not generate a great deal of horsepower. A healthy individual at optimum performance can generate around 1.2 horsepower for a brief amount of time. Athletes are capable of more. Usain Bolt holds the world record for the 100-meter dash at a maximum of 3.5 horsepower for less than one second during his world-record-breaking performance in 2009. The rest of us simple mortals, on the other hand, must make do with a less impressive output. A healthy human being can produce approximately.1hp, which is one-tenth of a horsepower, on a continuous basis.
## Final Thoughts
As it turns out, a horse is far more powerful than most of us realized! Horses in peak condition have the ability to create roughly 15 horsepower. And it’s possible that your beloved automobile has even more horse-equivalents chugging around under its hood than you were previously aware. So, whether you’re planning a lengthy road trip with your 131horses or simply want to dominate your next trivia game night with all of the new things you’ve learned, you may go forth and conquer the world. It’s all down to James Watt and his groundbreaking work with the steam engine and units of measuring power throughout the early nineteenth century.
## How Much Horsepower Does a Horse Have?
Although it has been attempted multiple times, there remains a prevalent fallacy that the power of a horse is comparable to the horsepower of a car, which continues to exist. This fallacy has been with us from the time people transitioned from riding horses or camels to driving automobiles, buses, and other modes of transportation. And, like so many of us, you’ve come to obtain a speedy response to your question. What is the maximum amount of horsepower a horse can produce? Historically, the efficiency of an object has been defined by the horsepower (power measurement unit) it produces.
Prior to the invention of engines, automobiles, machines, and other similar pieces of equipment, horses were responsible for the majority of the world’s labor.
They were stealing things, working in industries, and therefore assisting people in a variety of ways. As the horses were being phased out and replaced by machines, we began to doubt the efficiency of horses.
## How much horsepower does a horse have?
Horses have a maximum power output of 14.9 or 15 horsepower at their best. A horse can lift 330 pounds of weight 100 feet per minute, 33 pounds of weight 1000 feet per minute, or 1000 pounds of weight 33 feet in a minute provided the weight is distributed evenly. In a horsepower unit, the capacity of a horse is measured for the purpose of determining its performance. Although a lot has changed in the twentieth century, the horsepower unit has remained unchanged from the 18th century and continues to operate now.
Horsepower is the determining factor.
• Concerning the complete operating capability of a thing
• How much weight it is capable of lifting or transporting
To put it another way, the aim of horsepower is to characterize the capabilities of an item in an understandable manner.
### What is the horsepower of workhorses?
Workhorses are often regarded as the most powerful of the several kinds of horses. In order to account for the performance of workhorses, the horsepower of the horses indicated below has been determined. As a result, the workhorses’ maximum output is 15 horsepower. Let us compare the horsepower of a horse to that of a human to have a better idea. When it comes to executing jobs that need more physical power, humans lag considerably below horses in terms of efficiency. Compared to horses, who have over 15 hp, a healthy, physically active human has 5 hp, or five times the power.
In spite of the fact that horses have three times the horsepower of humans, humans have far more endurance: Horses are sprinters; they go shorter distances but quicker, and they can carry more weight for a shorter amount of time than other animals.
## Why do we measure the output of horses in the “horsepower” unit?
This widely known formula is easily understood by everyone and is used to define the output of all working machines, including horses, and is thus applicable to them as well. Since the invention of the horsepower unit, we have been defining the output of horses in terms of this unit of measurement. The efficiency of an object could not be measured or defined in the 18th century since there were no formulae or other tools available. The term “horsepower” was used to contrast the labor done by horses with that done by a contemporary steam engine.
And then there are those who
## Factors that affect the horsepower of a horse
Horses may have more horsepower than humans and just a few other animals that are employed for this reason, but they are still the most common. However, there are certain elements that influence the horsepower of a horse.
### Weather
Horses are not designed for hot weather because their large, broad bodies are unable to withstand the heat.
Temperatures between 18 and 59 degrees Fahrenheit are more comfortable for them while they are completing difficult jobs. Depending on the weather conditions, they may not be able to function at the same level as the 15 horsepower they are capable of.
### Health
The health of all mammals on the planet has a significant impact on their performance. Only a physically strong and healthy horse will be able to keep up with such a fast rate of operation.
### Breed
There are certain horse breeds that are a little slower and weaker than their fellows in terms of speed and strength. It is possible that they will not be able to meet their horsepower target. Racers and horses that have been taught to operate under difficult conditions are the exceptions to this general rule. Expecting 15 horsepower from a pet horse, for example, is a cliché.
### Sex
As a result of the rule of nature, male horses are larger and stronger than female horses, and as a result, they have a greater proclivity to perform than female horses.
### Age
Horses that are deemed to be under the age of maturity may not be able to operate with the same amount of horsepower as horses who have attained maturity. As a rule, horses begin to perform better when they are 2 years old, and they achieve their best performance when they reach 4.45 years of age.
## List of horse breeds that are proven to provide 15 horsepower output
Draft horse breeds are frequently referred to as “workhorses,” and they have been shown to provide the highest possible output (15 hp) even under the most demanding conditions. In spite of their incredible power and patience, they are still completing the tasks that machinery and labor used to do even in the world’s most industrialized nations. American cream draft, Belgian, Ardennes, Breton, Boulonnais, Comtois, Clydesdale, Dutch draft, Dole, Fjord, Finnhorse, Freiberger, Haflinger, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Friesian, Frie Italian heavy draft, Irish draft, Latvian, Jutland, Noriker, Murakoz, and other varieties The North Swedish horse, the Percheron, the Polish draft, the Pinto draft, the shire, the Rhenish-German cold blood, the Russian heavy draft, the Schleswig Holstein, the South German cold blood, the Soviet heavy draft, the spotted heavy draft, the Vladimir heavy draft, and the Suffolk horse are all still used as alternatives to modern machines.
In order to make this essay more understandable for those who are unfamiliar with the notion of horsepower, we must first take a brief look at the meaning of the term “horsepower” itself.
The reason for this might be due to the widespread use of the horsepower measurement unit, or it could be because of its simplicity.
## What is horsepower?
To put it another way, a horsepower is a unit of measurement for the power of motors and engines. For example, one horsepower unit is equivalent to the amount of power required to draw a weight of 500 pounds over a distance of 1 foot in one second or to elevate 33,000 pounds over a distance of 1 foot in one minute. The term “horsepower” was coined in the 18th century when farmers contrasted the effectiveness of this foreign contraption to that of the horse, which at the time was the only source of assistance.
The notion of this power unit is based on the assumption that a healthy horse can produce a certain amount of work in a minute.
## The history of the unit ” horsepower”.
Prior to the development of the automobile and the invention of machines, the majority of the labor on roads and in industries was done by people or horses. Because of this, James Watt (the inventor of the steam engine) required a means to express the power of the engine that was about to be introduced to the public. In order to make people comprehend the strength of his revolutionary engine, he employed horses, who were at the time the only available source of assistance. Because the engine was powerful enough to handle the work of numerous horses at the same time, he devised a simple device that can indicate how many horses are about to be put to rest.
He created a unit that could quickly and readily answer their inquiries in order to put their disturbed minds at ease.
A further result of this comparison was the creation of the unit of “horsepower” to measure power.
## Horsepower means the power of one horse- how did the misconception arise?
What do you receive when you hear the phrase “Horsepower” for the first time if you have never heard it before? “The strength of a horse,” you will almost certainly respond. When the “horsepower” was formally acknowledged as a unit of measurement for power, no one was aware of the history that led to its creation; instead, we simply assumed that the power of one horse was equivalent to the unit of measurement “horsepower.”
## Types of horsepower unit
Horsepower is classified in the same way that definitions are classified. The two most often encountered are as follows:
• Mechanical horsepower, also known as Imperial horsepower
• Metric horsepower
### Mechanical or Imperial horsepower
hp stands for mechanical horsepower, and it is a unit of power defined as the amount of force required to raise an object 550 pounds/250 kg by one foot in one minute. It is symbolized by the sign hp.
### Metric horsepower
Metric horsepower is the same as mechanical horsepower or imperial horsepower in terms of measurement. Due to the fact that it is computed and displayed differently, it was split apart. As a result, the metric horsepower is approximately 735.5 watts, but the imperial horsepower is 745.7 watts There are several distinct meanings for the horsepower unit. While looking for an exact response, you may have came across numerous different definitions, which may have left you perplexed as to which one was correct.
The following are the reasons: In order to determine the definition of horsepower, you must consider geographical variances as well as the sort of machinery or object that will be measured.
One unit of metric horsepower is equivalent to 735.5 watts, while one unit of electric motor horsepower is equal to 746 watts.
The view of the horsepower unit for the steam boiler is not even near to the views of the above-mentioned examples, as one unit is equal to 9,810 watts in the steam boiler. This is one of the reasons why we become confused when it comes to the many meanings of the horsepower.
## Horsepower formula
Because horsepower is just the rate at which work is completed at a given period, the following are some popular horsepower formulas: Horsepower equals work multiplied by time. (This method is not typically used to compute horsepower since it does not provide as exact information as you want, but it does provide a thorough explanation of the philosophy behind this power measurement unit.) Horsepower is calculated as torque*RPM/5252 (In this formula, RPM is the speed of the engine, T is torque, and 5252 represents the radians per second.)
What exactly is hp? HP is an abbreviation for the horsepower, which is a power measurement unit. Can you tell me how many horsepower a horse has? Horses have a maximum power output of either 14.9 or 15 horsepower. And this computation is based on the horsepower of all horse breeds, with a particular emphasis on workhorse kinds. What exactly does the term “horsepower” mean? Horsepower is commonly understood to signify “the power of a horse,” but in actuality, it is only a term for the power measurement unit, which is referred to as a Watt in this case.
Because a healthy horse may produce up to fifteen horsepower at a time, this statement (one horsepower is equivalent to the power of one horse) is inaccurate and misleading.
Is it correct to say that horsepower is equivalent to horse power?
In reality, it is used to assess the amount of output a horse can produce while executing a very difficult activity.
## Final Words
A horse has 15 horsepower, which is equivalent to one horsepower. For the purpose of this estimate, the horses that worked in industries throughout the 18th century have been taken into consideration. Horsepower is a well recognized unit of output measurement that is used to quantify not just the output of machinery and vehicles, but also the output of horses. This mistake about the meaning of horsepower being the power of a horse is simply that: a misunderstanding. Horsepower does not equate to horsepower.
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# Geographic coordinate systems
A geographic coordinate system uses a three-dimensional spherical surface to determine locations on the earth. Any location on earth can be referenced by a point with longitude and latitude coordinates.
For example, Figure 1 shows a geographic coordinate system where a location is represented by the coordinates longitude 80 degree East and latitude 55 degree North.
The lines that run east and west each have a constant latitude value and are called parallels. They are equidistant and parallel to one another, and form concentric circles around the earth. The equator is the largest circle and divides the earth in half. It is equal in distance from each of the poles, and the value of this latitude line is zero. Locations north of the equator have positive latitudes that range from 0 to +90 degrees, while locations south of the equator have negative latitudes that range from 0 to -90 degrees.
Figure 2 illustrates latitude lines.
The lines that run north and south each have a constant longitude value and are called meridians. They form circles of the same size around the earth, and intersect at the poles. The prime meridian is the line of longitude that defines the origin (zero degrees) for longitude coordinates. One of the most commonly used prime meridian locations is the line that passes through Greenwich, England. However, other longitude lines, such as those that pass through Bern, Bogota, and Paris, have also been used as the prime meridian. Locations east of the prime meridian up to its antipodal meridian (the continuation of the prime meridian on the other side of the globe) have positive longitudes ranging from 0 to +180 degrees. Locations west of the prime meridian have negative longitudes ranging from 0 to -180 degrees.
Figure 3 illustrates longitude lines.
The latitude and longitude lines can cover the globe to form a grid, called a graticule. The point of origin of the graticule is (0,0), where the equator and the prime meridian intersect. The equator is the only place on the graticule where the linear distance corresponding to one degree latitude is approximately equal the distance corresponding to one degree longitude. Because the longitude lines converge at the poles, the distance between two meridians is different at every parallel. Therefore, as you move closer to the poles, the distance corresponding to one degree latitude will be much greater than that corresponding to one degree longitude.
It is also difficult to determine the lengths of the latitude lines using the graticule. The latitude lines are concentric circles that become smaller near the poles. They form a single point at the poles where the meridians begin. At the equator, one degree of longitude is approximately 111.321 kilometers, while at 60 degrees of latitude, one degree of longitude is only 55.802 km (this approximation is based on the Clarke 1866 spheroid). Therefore, because there is no uniform length of degrees of latitude and longitude, the distance between points cannot be measured accurately by using angular units of measure.
Figure 4 shows the different dimensions between locations on the graticule.
A coordinate system can be defined by either a sphere or a spheroid approximation of the earth's shape. Because the earth is not perfectly round, a spheroid can help maintain accuracy for a map, depending on the location on the earth. A spheroid is an ellipsoid, that is based on an ellipse, whereas a sphere is based on a circle.
The shape of the ellipse is determined by two radii. The longer radius is called the semimajor axis, and the shorter radius is called the semiminor axis. An ellipsoid is a three-dimensional shape formed by rotating an ellipse around one of its axes.
Figure 5 shows the sphere and spheroid approximations of the earth and the major and minor axes of an ellipse.
A datum is a set of values that defines the position of the spheroid relative to the center of the earth. The datum provides a frame of reference for measuring locations and defines the origin and orientation of latitude and longitude lines. Some datums are global and intend to provide good average accuracy around the world. A local datum aligns its spheroid to closely fit the earth's surface in a particular area. Therefore, the coordinate system's measurements are not be accurate if they are used with an area other than the one that they were designed.
Figure 6 shows how different datums align with the earth's surface. The local datum, NAD27, more closely aligns with Earth's surface than the Earth-centered datum, WGS84, at this particular location.
Whenever you change the datum, the geographic coordinate system is altered and the coordinate values will change. For example, the coordinates in DMS of a control point in Redlands, California using the North American Datum of 1983 (NAD 1983) are: -117 12 57.75961 34 01 43.77884. The coordinates of the same point on the North American Datum of 1927 (NAD 1927) are: -117 12 54.61539 34 01 43.72995.
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# 18599 (number)
18,599 (eighteen thousand five hundred ninety-nine) is an odd five-digits composite number following 18598 and preceding 18600. In scientific notation, it is written as 1.8599 × 104. The sum of its digits is 32. It has a total of 2 prime factors and 4 positive divisors. There are 15,936 positive integers (up to 18599) that are relatively prime to 18599.
## Basic properties
• Is Prime? No
• Number parity Odd
• Number length 5
• Sum of Digits 32
• Digital Root 5
## Name
Short name 18 thousand 599 eighteen thousand five hundred ninety-nine
## Notation
Scientific notation 1.8599 × 104 18.599 × 103
## Prime Factorization of 18599
Prime Factorization 7 × 2657
Composite number
Distinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 2 Total number of prime factors rad(n) 18599 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 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 18,599 is 7 × 2657. Since it has a total of 2 prime factors, 18,599 is a composite number.
## Divisors of 18599
1, 7, 2657, 18599
4 divisors
Even divisors 0 4 2 2
Total Divisors Sum of Divisors Aliquot Sum τ(n) 4 Total number of the positive divisors of n σ(n) 21264 Sum of all the positive divisors of n s(n) 2665 Sum of the proper positive divisors of n A(n) 5316 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 136.378 Returns the nth root of the product of n divisors H(n) 3.49868 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 18,599 can be divided by 4 positive divisors (out of which 0 are even, and 4 are odd). The sum of these divisors (counting 18,599) is 21,264, the average is 5,316.
## Other Arithmetic Functions (n = 18599)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 15936 Total number of positive integers not greater than n that are coprime to n λ(n) 7968 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 2125 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 15,936 positive integers (less than 18,599) that are coprime with 18,599. And there are approximately 2,125 prime numbers less than or equal to 18,599.
## Divisibility of 18599
m n mod m 2 3 4 5 6 7 8 9 1 2 3 4 5 0 7 5
The number 18,599 is divisible by 7.
## Classification of 18599
• Arithmetic
• Semiprime
• Deficient
• Polite
• Square Free
### Other numbers
• LucasCarmichael
## Base conversion (18599)
Base System Value
2 Binary 100100010100111
3 Ternary 221111212
4 Quaternary 10202213
5 Quinary 1043344
6 Senary 222035
8 Octal 44247
10 Decimal 18599
12 Duodecimal a91b
20 Vigesimal 269j
36 Base36 ecn
## Basic calculations (n = 18599)
### Multiplication
n×y
n×2 37198 55797 74396 92995
### Division
n÷y
n÷2 9299.5 6199.67 4649.75 3719.8
### Exponentiation
ny
n2 345922801 6433818175799 119662584251685601 2225604404497100492999
### Nth Root
y√n
2√n 136.378 26.495 11.6781 7.14328
## 18599 as geometric shapes
### Circle
Diameter 37198 116861 1.08675e+09
### Sphere
Volume 2.69499e+13 4.34699e+09 116861
### Square
Length = n
Perimeter 74396 3.45923e+08 26303
### Cube
Length = n
Surface area 2.07554e+09 6.43382e+12 32214.4
### Equilateral Triangle
Length = n
Perimeter 55797 1.49789e+08 16107.2
### Triangular Pyramid
Length = n
Surface area 5.99156e+08 7.58233e+11 15186
## Cryptographic Hash Functions
md5 7f841c9b91fcd2891b7ffbc612b80934 3fb1efff04bebc850461b41e4f597a770db9f926 5a0abe2b91a49e0d8f22314539af1a42bb33a268136175c8bcede46a4c0f4395 4a825a5358726cb0e4b6ae094fd4d518de4a503dd1a8de50a51d81c144c039a0190b490690f6ca5cd2b1b9bf942f7caea7bb39cb640daf167600d02f7aba2e6e 0c78bd2c44811aa4a86b9960c86639eb0746c831
| 1,450 | 4,118 |
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# Solve this
Asked by venkat 25th December 2017, 8:56 PM
Since Q is projected after 1 second, P will reach the maximum height and meet Q when it is returning back.
Let maximum height reached by P is H.
H = (u×u) /(2×g), where u is initial projection velocity 30 m/s
H = (30×30) / (2*10) = 45 m.
after reaching the maximumheight let P travels back for t seconds and meet Q.
Let the distance travelled by P in that t seconds is h and it is given by
h = (1/2)g×t2 = (1/2)×10×t2 = 5×t2 .........................(1)
Then distance travelled by Q when it meets P is 45-h and the time taken is (2+t)s. This distance is given by
45-h = 30×(2+t) - (1/2)×10×(2+t)2
after substituting for h from eqn.(1), we will get
45 - 5×t2 = 30×(2+t)-5×(2+t)2
solving the above eqn for t, we get t = 0.5 s.
Hence P travelled 3.5 s before meetin Q
Answered by Expert 27th December 2017, 3:13 PM
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https://www.varsitytutors.com/precalculus-help/convert-rectangular-coordinates-to-polar-coordinates-and-vice-versa?page=3
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# Precalculus : Convert Rectangular Coordinates To Polar Coordinates and vice versa
## Example Questions
1 3 Next →
### Example Question #75 : Polar Coordinates
Which polar-coordinate point is not the same as the rectangular point ?
Explanation:
Plotting this point creates a triangle in quadrant I:
Using our knowledge of Special Right Triangles, we can conclude that the angle is and the radius/hypotenuse of this triangle is . Our polar coordinates are therefore , so we can eliminate that as a choice since we know it works.
Looking at the unit circle [or just the relevant parts] can give us a sense of what happens when the angles and/or the radii are negative:
Now we can easily see that the angle would correspond with our angle of , so works.
We can see that if our radius is negative we'd want to start off at the angle , so the point works.
As we can see from looking at this excerpt from the unit circle, another way of writing the angle would be to write , so the point works.
The only one that does not work would be because that would place us in quadrant II rather than I like we want.
### Example Question #76 : Polar Coordinates
Which of the following is a set of polar coordinates for the point with the rectangular coordinates
Explanation:
The relation between polar coordinates and rectangular coordinates is given by and .
You can plug in each of the choices for and and see which pair gives the rectangular coordinate .
The answer turns out to be .
Alternatively, you can find by the equation
, thus
.
As for finding , you can use the equation
, and since
.
Thus, the polar coordinate is
### Example Question #77 : Polar Coordinates
Convert the point to polar form
Explanation:
First, find r using pythagorean theorem,
Then we can find theta by doing the inverse tangent of y over x:
Since this point is in quadrant II, add 180 degrees to get
### Example Question #78 : Polar Coordinates
Convert the following rectangular coordinates to polar coordinates:
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https://howto.org/how-big-is-one-meter-90484/
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## What is about 1 meter long?
A meter is a standard metric unit equal to about 3 feet 3 inches. … Guitars, baseball bats, and yard sticks are examples of objects that are about one meter long. Meters are also used to measure distances in races, such as running and swimming.
## Which is bigger 1 foot or 1 meter?
A meter is approximately equal to 3.28084 feet. … Remember, there are 12 inches in 1 foot.
## What is a meter of length?
metre (m), also spelled meter, in measurement, fundamental unit of length in the metric system and in the International Systems of Units (SI). It is equal to approximately 39.37 inches in the British Imperial and United States Customary systems.
## What is the difference between a meter and a foot?
Meter to Feet Conversion
For converting meter to feet firstly we should know the difference between their lengths. That is one meter is equal to 3.28 feet and one foot is equal to 12 inches as per rule. So, to convert meter to feet just simply multiply the number of meter to the value of feet per meter. … = 19.91 ft.
## Is a meter smaller than a foot?
Feet are smaller than meters, so there should be more feet than meters. The conversion is 1 ft = 0.3048 meters.
## Is a meter bigger than a yard?
Answer: The difference between meter and yard is that the meter is a SI unit of length and a yard is a unit of length. Also, 1 meter is about 1.09 yards.
## How big is a metre squared?
A square meter is a measurement of area. One square meter is the equivalent of the area of a square that is one meter in length on each side. The perimeter of such a square (the total distance around it) would be four meters.
## How big is a domestic water meter?
PD meters are generally very accurate at the low-to-moderate flow rates typical of residential and small commercial users and commonly range in size from 5/8″ to 2″.
## What is bigger than a meter?
Units larger than a meter have Greek prefixes: Deka- means 10; a dekameter is 10 meters. Hecto- means 100; a hectometer is 100 meters. Kilo- means 1,000; a kilometer is 1,000 meters.
## How many meters is a football field?
FIFA recommendations for field dimensions in professional football are 105 metres in length and 68 metres in width. Clubs are encouraged where possible to mark their fields in accordance with this standard.
## Is a meter bigger than a kilometer?
Kilometers are 1,000 times larger than meters. The meter is the base unit for measuring length or distance in the metric system.
## What is the smallest meter?
micrometer
A micrometer, also called a micron, is one thousand times smaller than millimeter. It is equal to 1/1,000,000th (or one millionth of meter). Things on this scale usually can’t be seen with your eyes.
## Is a meter bigger than a mile?
A mile is a unit of length or distance measurement that is equal to 5,280 feet. … Compared with the metric system, a mile is about 1,609 meters. Its abbreviation is m. A kilometer is a unit of length or distance measurement that is equal to 1,000 meters.
## Which is smaller nanometer or micrometer?
Nanometer A nanometer is 1000 times smaller than a micrometer. 1 micrometer (μm) = 1000 nanometers.
## Are nanometers bigger than meters?
A nanometer is a one-billionth of a meter, and used to measure things that are very, very small. … By definition a nanometer is one-billionth of a meter. A meter is about 39 inches long. A billion is a thousand times bigger than a million, as a number you write it out as 1,000,000,000.
## What part of a meter is a micrometer?
The micrometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American spelling), also commonly known as a micron, is an SI derived unit of length equalling 1×106 metre (SI standard prefix “micro-” = 106); that is, one millionth of a metre (or one …
## What is smaller than a meter?
A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters). A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).
## How small is a nano meter?
Just how small is “nano?” In the International System of Units, the prefix “nano” means one-billionth, or 109; therefore one nanometer is one-billionth of a meter.
## How much smaller is a micrometer than a meter?
Therefore, one micrometer is 1/1,000,000 of a meter.
## How many micrometers make a meter?
There are 1,000,000 micrometers in a meter, which is why we use this value in the formula above. Meters and micrometers are both units used to measure length.
## Is 1 nm smaller than 10nm?
The standard measure of length in science is in meters (m). One nanometer (1 nm) is equal to 109 m or 0.000000001 m.
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# Finding which functions are bounded by $O(n^2)$
I am asked to select the functions that are bounded by the Big-Oh function O(n^2): $$f(n) \in O(n^2)$$.
1. $$f(n) = \sum_{i=1}^{n} n$$
2. $$f(n) = \sum_{i=1}^{n} i$$
3. $$f(n) = n + n^2$$
4. $$f(n) = 1$$
I choose the answers 1, 2, and 4:
1. Simplifies to $$n \cdot n = n^2$$, which is bounded by $$O(n^2)$$
2. Simplifies to $$\frac{n(n + 1)}{2} = n^2/2 + 1/2$$, which is bounded by $$O(n^2)$$
3. $$n^2 + n$$ is obviously not bounded by $$O(n^2)$$
4. $$1$$ is obviously bounded by $$O(n^2)$$
However, it was alerted to me that one of my answers is incorrect. That seems strange, considering that after expanding all the expressions mathematically, I believe that I have chosen whether they are bounded by $$n^2$$ correctly.
• Are you familiar with the definition of big O? Jan 17 at 15:59
We have $$n^2 + n = O(n^2)$$. Indeed, if $$n \geq 1$$ then $$n^2 \geq n$$ and so $$n^2 + n \leq 2n^2.$$
I know, that answer is done, is correct and is accepted, but let me add, that "function is bounded by $$O(n^2)$$" is not completely correct sentence and as such can lead to errors: $$O(n^2)$$ is set, so we can speak about boundedness by some member from it, but exact is to say, that function is bounded by some $$\boldsymbol n^{ \boldsymbol 2}$$ factor, starting from some number up to infinity.
| 463 | 1,345 |
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https://fansided.com/2017/09/20/kawhi-leonard-mvp-prediction-2017-18-season/
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# Nylon Calculus: Using math to predict the 2017-18 NBA MVP race
The 2016-17 NBA MVP race was one of the most exciting in recent memory, and possibly ever. Almost all of the major challengers for the award last year (Russell Westbrook, James Harden, Kawhi Leonard, LeBron James) are in position to compete again this season.
So, how can we use the history of the MVP race to predict the 2017-18 season? Who will be the 2017-18 NBA MVP?
## Building a model
To start off with, I built a database of every winner of the award going back to the 1979-80 season, the first with an NBA 3-point line. Then, using MVP award shares as a target variable, I performed a linear regression of a few key variables that I found to be predictive of future results.
The final equation for MVPScore is as follows:
Where the constants are:
This equation misses on players who win MVP for narrative driven reasons more than on-court reasons, like Derrick Rose in 2011 or Allen Iverson in 2001. Both of those players were very good the year they won MVP, but may not have carried their teams in the traditional sense compared to other MVPs.
This methodology has been able to correctly predict 25 of the last 38 MVPs, including the 2016-17 MVP Russell Westbrook.
A 66.0 percent hit rate is not perfect but does a good job at identifying at the very least who the right candidates are. Of the last 38 MVPs, 34 have received one of the top three MVPScores in the season they won. In fact, only two of the last 38 MVPs did not rate as the top-5 most likely winners by this metric and both times it was the same player: Steve Nash, who came in eighth by this metric for his 2004-05 MVP and ninth for his 2005-06 MVP.
Knowing that the eventual MVP was always a top-10 player by MVPScore, and 95.0 percent of the time a top-5 player, let us look at the players for the 2017-18 NBA season who project to receive the best score.
## The 2017-18 NBA MVP is…
With a projected MVPScore of 37.7 percent, the 2017-18 MVP favorite in my model is currently Kawhi Leonard. Leonard projects to have per game averages of 23.7 points, 6.2 rebounds, 3.7 assists and 1.8 steals while being the best player on a 50+ win team.
Here is the entire projected top 10 for the 2017-18 NBA MVP Race:
The biggest surprise is Jimmy Butler coming in at sixth. The MVPScore metric sees Butler as the most important player on a Timberwolves teams that is expected to win about 48-52 games. Butler projects to per game averages of 19.2 points, 5.6 rebounds, 4.2 assists and 1.6 steals. While more casual fans may view Karl-Anthony Towns as Butler’s equal, Butler rated out as one of the 10 most impactful players last season while Towns did not.
Players of note who just missed the cut for the top-10: Rudy Gobert, Kemba Walker, Kyle Lowry, Chris Paul, Damian Lillard and Gordon Hayward.
Next: How accurately can we predict NBA Playoff berths?
The 2017-18 NBA MVP race is shaping up to be a really great one once again with no clear favorite. Narrative will matter a lot. It hopefully is able to live up to the high bar the 2016-17 MVP set. The season is just under a month away. Soon we will see who steps up and leads their team.
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### Financial Algebra - OCPS Teacher Server
```5
5-1
5-2
5-3
5-4
5-5
Slide 1
AUTOMOBILE
OWNERSHIP
Graph Frequency Distributions
Automobile Insurance
Linear Automobile Depreciation
Financial Algebra
5
5-6
5-7
5-8
5-9
Slide 2
AUTOMOBILE
OWNERSHIP
Historical and Exponential Depreciation
Driving Data
Driving Safety Data
Accident Investigation Data
Financial Algebra
5-3
GRAPH FREQUENCY
DISTRIBUTIONS
OBJECTIVES
Create a frequency distribution from a
set of data.
Use box-and-whisker plots and stemand-leaf plots to display information.
Slide 3
Financial Algebra
Key Terms (from 5-2 & 5-3)
from 5-2
mean
outlier
ascending order
median
range
quartiles
lower quartile
upper quartile
Slide 4
5-2 cont.
interquartile range (IQR)
mode
from 5-3
frequency distribution
stem-and-leaf plot
box-and-whisker plot
Financial Algebra
Why are graphs used so frequently
in mathematics, and in daily life?
Can graphs be used to mislead people?
Slide 5
Financial Algebra
Example 1
Jerry wants to purchase a car stereo. He found 33 ads for
the stereo he wants and arranged the prices in ascending
order:
\$540 \$550 \$550 \$550 \$550 \$600 \$600 \$600 \$675 \$700 \$700 \$700
\$700 \$700 \$700 \$700 \$750 \$775 \$775 \$800 \$870 \$900 \$900 \$990
\$990 \$990 \$990 \$990 \$990 \$1,000 \$1,200 \$1,200 \$1,200
He is analyzing the prices, but having trouble because
there are so many numbers. How can he organize his
Slide 6
Financial Algebra
Example 1 (cont.)
Create a frequency distribution.
Add the frequencies to be sure
no numbers are left out.
Slide 7
Price
540
550
600
675
700
750
775
800
870
900
990
1000
1200
Total
Frequency
1
4
3
1
7
1
2
1
1
2
6
1
3
33
Financial Algebra
Use the frequency distribution from
Example 1 to find the number of car
stereos selling for less than \$800.
Slide 8
Price
540
550
600
675
700
750
775
800
870
900
990
1000
1200
Total
Frequency
1
4
3
1
7
1
2
1
1
2
6
1
3
33
Financial Algebra
Example 2
Find the mean of the car
stereos prices from Example 1.
Create a 3rd column to show
product of 1st 2 columns.
Find total of 3rd column,
divide by total prices
Slide 9
Price
540
550
600
675
700
750
775
800
870
900
990
1000
1200
Total
Frequency
1
4
3
1
7
1
2
1
1
2
6
1
3
33
Total
540
2,200
1,800
675
4,900
750
1,550
800
870
1,800
5,940
1,000
3,600
26,425
Financial Algebra
Jerry, from Example 1, decides
he is not interested in any of the
car stereos priced below \$650
because they are in poor
condition and need too much
work. Find the mean of the data
set that remains after those
prices are removed.
Slide 10
Price
540
550
600
675
700
750
775
800
870
900
990
1000
1200
Total
Frequency
1
4
3
1
7
1
2
1
1
2
6
1
3
33
Total
540
2,200
1,800
675
4,900
750
1,550
800
870
1,800
5,940
1,000
3,600
26,425
Financial Algebra
Histogram
A histogram is a graph of frequencies.
Examples: Making Histograms
Put frequencies on vertical axis.
Bars should touch.
Frequency
8
7
6
5
4
3
2
Price
540
550
600
675
700
750
775
800
870
900
990
1000
1200
Total
Frequency
1
4
3
1
7
1
2
1
1
2
6
1
3
33
1
0
540
Slide 11
550
600
675
700
750
775
800
870
900
990
1000 1200
Financial Algebra
Histogram (cont.)
Group numbers into ranges to
make data more meaningful
Price Frequency
500
5
600
4
12
700
10
10
800
2
8
900
8
1000
1
1100
0
1200
3
Frequency
6
4
2
0
500
Slide 12
600
700
800
900
1000 1100 1200
Financial Algebra
EXAMPLE 3
Rod was doing Internet research on the number of
gasoline price changes per year in gas stations in his
county. He found the following graph, called a stemand-leaf plot. What are the mean and the median of
this distribution?
Note the key
Slide 13
Financial Algebra
EXAMPLE 3 (cont.)
The mean: Add the data and
divide by the frequency (the
number of leaves).
1,188 ÷ 30 39.6
The median: The frequency is
30. Since it is even, find the
mean of the 15th and 16th
positions.
15th =39; 16th = 39
median = 39
Slide 14
Financial Algebra
Find the range and the upper and lower quartiles.
Range: highest – lowest
72 – 11 = 61
Slide 15
Financial Algebra
Quartiles
divide the data into 4 equal groups.
Q2, the median, creates 2 groups
Q2 = 39
Q1 is the lower quartile.
Find the median of the data below Q2.
There are 15 numbers in lower group. Q1 is in the 8th position, 23.
Q3 is the upper quartile.
Find the median of the data above Q2.
There are 15 numbers in the upper group. Q3 is in the 23rd
position, 55.
Slide 16
Financial Algebra
EXAMPLE 4
Rod, from Example 3, found another graph called a boxand-whisker plot, or boxplot.
•Plot the minimum, 3 quartiles and maximum on a
number line.
•Draw a box using Q1 and Q3 at either end.
•Draw a line through Q2, the median of all the data.
Find the interquartile range (IQR), Q3 – Q1
•55 – 23 = 32
Slide 17
Financial Algebra
Based on the box-and-whisker plot from Example 4,
what percent of the gas stations had 55 or fewer price
changes?
Slide 18
Financial Algebra
EXAMPLE 5
The following box-and-whisker plot gives the purchase
prices of the cars of 114 seniors at West High School.
Are any of the car prices outliers?
Slide 19
Financial Algebra
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# How do you find the test statistic for a population proportion?
How do you find the test statistic for a population proportion? The basic procedure is: State the null hypothesis H0 and the alternative hypothesis HA. Set the level of significance . Calculate the test statistic:
## How do you find the test statistic for a population proportion?
The basic procedure is:
1. State the null hypothesis H0 and the alternative hypothesis HA.
2. Set the level of significance .
3. Calculate the test statistic: z = p ^ − p o p 0 ( 1 − p 0 ) n.
4. Calculate the p-value.
5. Make a decision. Check whether to reject the null hypothesis by comparing p-value to .
### What is the 95% confidence interval for the population proportion?
The 95% confidence interval for the true binomial population proportion is ( p′ – EBP, p′ + EBP) = (0.810, 0.874).
#### What test statistic is used for constructing a confidence interval on the population proportion?
To use the standard error, we replace the unknown parameter p with the statistic p̂. The result is the following formula for a confidence interval for a population proportion: p̂ +/- z* (p̂(1 – p̂)/n)0.5.
What is the test statistic for confidence interval?
So, if your significance level is 0.05, the corresponding confidence level is 95%. If the P value is less than your significance (alpha) level, the hypothesis test is statistically significant. If the confidence interval does not contain the null hypothesis value, the results are statistically significant.
What is the test statistic for proportion?
Statistics – One Proportion Z Test The test statistic is a z-score (z) defined by the following equation. z=(p−P)σ where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and σ is the standard deviation of the sampling distribution.
## How do you find P-value from proportion?
Since we have a two-tailed test, the P-value is the probability that the z-score is less than -1.75 or greater than 1.75. We use the Normal Distribution Calculator to find P(z < -1.75) = 0.04, and P(z > 1.75) = 0.04. Thus, the P-value = 0.04 + 0.04 = 0.08.
### What is the formula for the confidence interval for the population proportion?
To calculate the confidence interval, you must find p′, q′, andEBP. p′ = 0.842 is the sample proportion; this is the point estimate of the population proportion. Since CL = 0.95, then α = 1 – CL = 1 – 0.95 = 0.05 (α) = 0.025.
#### What is the formula for z test proportion?
How is the confidence interval for a population proportion determined?
Here the value of z* is determined by our level of confidence C. For the standard normal distribution, exactly C percent of the standard normal distribution is between -z* and z*. Common values for z* include 1.645 for 90% confidence and 1.96 for 95% confidence.
How to do a hypothesis test for a population proportion?
A Hypothesis Test for a Population Proportion 1. Intro 3. Using Confidence Intervals to Test Hypotheses Our main goal is in finding the probability of a difference between a sample mean p̂ and the claimed value of the population proportion, p0.
## What is the confidence level for proportion statology?
The following table shows the z-value that corresponds to popular confidence level choices: Confidence Level z-value 0.90 1.645 0.95 1.96 0.99 2.58
### What is the formula for the sample proportion statistic?
Recall from Linking Probability to Statistical Inference that the formula for the z -score of a sample proportion is as follows: For this example, we calculate: This z -score is called the test statistic. It tells us the sample proportion of 0.83 is about 2.12 standard errors above the population proportion given in the null hypothesis.
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# How a Rubik's Cube Is Made
The Rubik's Cube is a puzzle that consists of a handheld cube whose edges are slightly less than 3 inches. The cube seems to be composed of 27 smaller brightly colored cubelets so each face is a 3-by-3 array. Each face of the cube can be turned independently. The puzzle is solved when each face of the cube is a solid color.
## History
Hungarian architecture professor Enro Rubik invented the best selling toy in the world (current approximation is a half billion sold) in 1974. He invented it to help his students visualize three dimensional rotations. He sold it to Ideal Toys in 1980. The world will celebrate the Rubik's Cube 30th anniversary in 2010.
## Significance
The chief cultural significance of the Rubik's Cube is as a proof of intelligence. It has appeared in dozens of movies as a device for quickly demonstrating a character's intelligence. Besides the problem of solving the puzzle, there is another interesting problem with the cube--the mechanical problem. How is a Rubik's cube made and how do all those pieces work together?
## Construction
Mechanically, the Rubik's cube is partly an illusion. There appear to be 27 identical cubelets, but there are actually only 26 cubelets (no center cubelet) and they are not all identical--there are three different types: six center cubelets with one face each, 12 edge cubelets with two faces each, and eight corner cubelets with three faces each. To disassemble the cube, rotate one face 45 degrees and pry off the exposed corner. Once you remove one of the cublets, the rest follow easily.
## Function
The center cubelets are the framework upon which the rest of the cubelets rotate. The center cublets are the center of each face of the Rubik's cube connected in a fixed three-dimensional cross. The edge cublets are beveled so they can rotate around the center cublets and the corner cublets are cut so that they ride between the edge cublets when you rotate the faces.
## Considerations
The simplest way to see how a Rubik's cube is made is to disassemble it. If you take a reasonable amount of care it snaps back together as good as new. One thing you should be aware of is that if you take apart the cube and don't put each cubelet back in exactly the proper place you might have created a cube that is impossible to solve. To avoid this problem, only disassemble a solved cube--then it is easy to see how to reassemble it.
## Expert Insight
The current world record (as of April 2010) for single time on a Rubik's Cube in an international competition is 7.08 seconds. The world record average solve (average of the middle three of five trials) is currently 9.21 seconds. One of the most amazing events in these competitions is the blindfold solve: the contestant is handed a randomized cube to examine as long as he wants, then blindfolded before doing any rotations. The total time--examination plus blindfolded rotation--is recorded. The current world's record is 32.27 seconds.
## References
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I am given a set of coefficients such that the affine combination $$\{x_1, x_2, ..., x_n \} \notin conv(x_1, x_2, ..., x_n)$$. How do I prove that under such given conditions the Jensen's inequality direction is reversed? That is
$$f(\sum_{i=0}^n \lambda_ix_i) \geq \sum_{i=0}^n \lambda_if(x_i)$$
I know this result and I have verified this but I can't prove the statement.
• I think you want to say that $\lambda_i$ are such that $\sum_{i} \lambda_i=1$ and at least one of them is not in $[0,1]$. Feb 10, 2023 at 6:48
• Yes, that is true. Feb 10, 2023 at 6:54
This is false, let $$f:\mathbb R^2\to\mathbb R$$ be defined as $$f(a,b)=a^2+b^2$$, let $$x_1=[0,0]^T$$, $$x_2=[1,0]^T$$ and $$x_3=[0,1]^T$$ with $$\lambda_1=-\varepsilon$$, $$\lambda_2=\lambda_3=\frac{1+\varepsilon}{2}$$. For any $$\varepsilon >0$$, this is not in the convex hull of $$x_1$$, $$x_2$$ and $$x_3$$. We can compute explicitly
\begin{align*} f\left( \sum_{i} \lambda_i x_i \right)&=\frac{(1+\varepsilon)^2}{2}\\ \sum_{i=1}^n \lambda_i f(x_i) &= 1+\varepsilon \end{align*}
observe that $$\frac{(1+\varepsilon)^2}{2}-1-\varepsilon=\frac{\varepsilon^2-1}{2}$$, therefore for $$\varepsilon\in ]0,1[$$, $$f\left( \sum_{i} \lambda_i x_i \right)<\sum_{i=1}^n \lambda_i f(x_i)$$.
Observe however that your result is true for $$n=2$$, indeed if $$x=\lambda x_1+(1-\lambda)x_2$$ with $$\lambda < 0$$, then $$x_2=\frac{1}{1-\lambda} x - \frac{\lambda}{1-\lambda} x_1$$ where both coefficients are positive and sum to one, therefore by Jensen inequality we get \begin{align*} (1-\lambda)f(x_2)&\leq (1-\lambda)\frac{1}{1-\lambda} f(x) - (1-\lambda)\frac{\lambda}{1-\lambda}\\ &=f(x) - \lambda f(x_1) \end{align*} which proves your result. In the example above try drawing the points and the non-convex combination to see why we can break the condition, in particular it is clear when $$\varepsilon \to 0$$.
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# Program to find length of shortest sublist with maximum frequent element with same frequency in Python
Suppose we have a list of numbers called nums. If the frequency of a most frequent number in nums is k. We have to find the length of a shortest sublist such that the frequency of its most frequent item is also k.
So, if the input is like nums = [10, 20, 30, 40, 30, 10], then the output will be 3, because here the most frequent numbers are 10 and 30 , here k = 2. If we select the sublist [30, 40, 30] this is the shortest sublist where 30 is present and its frequency is also 2.
To solve this, we will follow these steps −
• L := size of nums
• rnums := reverse of nums
• d := a map containing frequencies of each elements present in nums
• mx := maximum of list of all values of d
• vs := a list of k for each k in d if d[k] is same as mx
• mn := L
• for each v in vs, do
• mn := minimum of mn and ((L - (index of v in rnums) - (index of v in nums))
• return mn
## Example
Let us see the following implementation to get better understanding −
from collections import Counter
def solve(nums):
L = len(nums)
rnums = nums[::-1]
d = Counter(nums)
mx = max(d.values())
vs = [k for k in d if d[k] == mx]
mn = L
for v in vs:
mn = min(mn, (L - rnums.index(v)) - nums.index(v))
return mn
nums = [10, 20, 30, 40, 30, 10]
print(solve(nums))
## Input
[10, 20, 30, 40, 30, 10]
## Output
3
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# Histogram Area Problem Using C++
Filed Under: C++
In today’s article, we’ll learn to solve the histogram area problem using C++. This is very popular related to technical interviews. There are high chances for your recruiter to ask about this problem during a technical coding interview.
Today we’ll find the solution to this problem using the stack-based approach. It’s going to be interesting, so without wasting any time, let’s get started.
## What Is A Histogram?
Before we move to the problem statement, let’s quickly go through a histogram. Let’s see how does it look like and what it means.
Using a histogram, we represent the frequency of a variable and observe how frequently this variable falls into a particular bin.
## Problem Statement for Histogram Area Problem in C++
Given a histogram, find the maximum area that you can get by making a rectangle in this histogram. Take the width of each bar as 1 unit.
## Concept for Histogram Area Problem
There are many rectangles possible. Out of all these rectangles, our task is to find the one with the maximum area. It is a classical data structure and algorithm problem, so there are multiple ways to solve this problem. In this article, we’ll go with the stack-based approach. Let’s go through the pseudocode and get the intuition behind the algorithm.
• Create a stack of integers. This stack will contain the indices of the elements of the array
• Now for each bar of the histogram, we will do the following
• Push the bar into the stack if it’s of higher value than the stack top.
• Otherwise, pop all the bars of greater height than the current bar.
• Based on the above two operations, governing formulas for the area will be:
• if(stack.empty())
• area = arr[top] * i
• else
• area = arr[top] * (i – s.top() – 1)
• Here, i represent the rightmost lower element
• And s.top() represent the previous top element(left)
## Algorithm for Histogram Area Problem in C++
```// function to calculate the maximum area
int maximum_area(vector <int> arr)
{
int n = arr.size();
int max_area = 0;
stack <int> s;
// start iterating over the elements
for(int i = 0; i < n; i++)
{
// check if the stack is empty or not
while(!s.empty())
{
// Case: Current bar is greater than or
// equal to the previous bar(push)
if(arr[s.top()] <= arr[i])
break;
// Case: Current bar is smaller than the
// previous bar(pop)
int j = s.top();
s.pop();
// Area calculation formulas
if(!s.empty())
max_area = max(max_area, arr[j] * (i - s.top() - 1));
else
max_area = max(max_area, arr[j] * i);
}
s.push(i);
}
// if the stack is still not empty,
// calculate the area again
while(!s.empty())
{
int j = s.top();
s.pop();
if(!s.empty())
max_area = max(max_area, arr[j] * (n - s.top() - 1));
else
max_area = max(max_area, arr[j] * n);
}
return max_area;
}
```
## Histogram Area Problem in C++
```#include <iostream>
#include <stack>
#include <vector>
using namespace std;
int maximum_area(vector <int> arr)
{
int n = arr.size();
int max_area = 0;
stack <int> s;
for(int i = 0; i < n; i++)
{
while(!s.empty())
{
if(arr[s.top()] <= arr[i])
break;
int j = s.top();
s.pop();
if(!s.empty())
max_area = max(max_area, arr[j] * (i - s.top() - 1));
else
max_area = max(max_area, arr[j] * i);
}
s.push(i);
}
while(!s.empty())
{
int j = s.top();
s.pop();
if(!s.empty())
max_area = max(max_area, arr[j] * (n - s.top() - 1));
else
max_area = max(max_area, arr[j] * n);
}
return max_area;
}
int main()
{
vector <int> histogram;
cout << "Enter the height of each bar of the histogram, press -1 to stop" << endl;
while(true)
{
int ele;
cin >> ele;
if(ele == -1)
break;
histogram.push_back(ele);
}
cout << "The maximum area under the current histogram is: " << maximum_area(histogram) << endl;
return 0;
}
```
## Conclusion
In this article, we learned to solve the maximum histogram area problem. Being a classical problem, it is very important for interview preparation and conceptual understanding of the subject. We used the stack-based approach to find the solution to this problem. If you notice the time complexity of the algorithm, you’ll find it to be linearly proportional to the size of the vector, i.e. O(N). This is one of the most efficient approaches to solving this problem. That’s all for today, thanks for reading.
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# Air Pressure, Forces, and Motion Meteorology 101 Dr. Robert M MacKay.
## Presentation on theme: "Air Pressure, Forces, and Motion Meteorology 101 Dr. Robert M MacKay."— Presentation transcript:
Air Pressure, Forces, and Motion Meteorology 101 Dr. Robert M MacKay
Pressure Temperature Volume The Gas Law P=C*density*Temp P=2.87 T
Gas Laws
Constant P as T increases V Increases/decreases Constant V as T increases P Increases/decreases Constant T as V increases P Increases/decreases Constant V,T as M increases P Increases/decreases
Gas Laws Constant P as T increases V Increases/decreases Constant V as T increases P Increases/decreases Constant T as V increases P Increases/decreases Constant V,T as M increases P Increases/decreases
Mercury Barometer Pressure Measurement 1013 mb = 1013 hPa Millibar hectoPascal 1013 mb 29.92 in Hg 76 cm Hg 760 mm Hg 760 Torr 14.7 psi
Station Pressure + Elevation(meter)/10 = Sea Level Pressure
Newtons Laws of Motion Newtons 3 laws of motion 1. Law of inertia 2. Net Force = mass x acceleration ( F = M A ) 3. Action Reaction
1st Law (Law of Inertia) Every object continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. acceleration = 0.0 unless the objected is acted on by an unbalanced force
1st Law Inertia (The intrinsic tendency of an object to rest changes in motion) Mass is a measure of an objects inertia Mass is also a measure of the amount of an objects matter content. (i.e. protons, neutrons, and electrons)
Newtons 2nd Law Net Force = Mass x Acceleration F = M A
Newtons Law of Action Reaction (3rd Law) You can not touch without being touched For every action force there is and equal and oppositely directed reaction force
Forces that influence the wind 1. Pressure Gradient Force 2. Coriolis Force 4. Friction On average gravity nearly balances the vertical Pressure gradient (hydrostatic balance)
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Simple & Compound Interest
# Simple & Compound Interest - F at end of year 1 is...
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Simple Interest is a historical concept whereby interest is calculated only on the original principal borrowed. Compound Interest , in widespread use today, bases the interest calculation on the original principal borrowed plus any accrued interest from prior periods.
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Simple Interest Interest earned on original principal only I = i * P * N F = P + I F = P + (i * P * N) or F = P(1 + iN) Example: \$100 @ 10% per year for 2 years F = \$100(1 + 10% * 2) = \$120 \$100 @ 10% per year for 20 years F = \$100(1 + 10% * 20) = \$300
Compound Interest Example: \$100 @ 10% per year compounded annually for 2 years F @ end Year 1 = P (1 + i) = \$100 * (1 + 10%) = \$110
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Unformatted text preview: F at end of year 1 is P at beginning of Year 2 F @ end Year 2 = P (1 + i) * (1 + i) = \$110 (1 + 10%) or Combining Terms: F = P (1 + i) 2 F @ end Year 2 = \$100 (1 + 10%) 2 = \$121.00 What is the balance in 20 years? Yr 1 Yr 2 … Yr 20 F @ end of year 20 = \$100(1+10%)(1+10%)…(1+10%) F @ end of Year 20 = \$100 (1 + 10%) 20 = \$672.75 Compound Interest – Cont’d For any number of Periods N: F = P (1 + i) N [(1 + i) N ] is known as the “Compound Amount Factor” Solving for P we obtain: P = F (1 + i)-N [(1 + i)-N ] is known as the “Discount Factor” or “Present Worth Factor”...
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# Posed in regional mathematics Olympiad 2005 [duplicate]
This question is an exact duplicate of:
Let a, b, c be three positive real numbers such that $a+b+c = 1$.
Let $\Delta= \min( a^{3} + a^{2}bc, b^{3}+ab^{2}c, c^{3}+abc^{2} )$.
Prove that the roots of the equation $x^{2} + x + 4 \Delta = 0$ are real. The last line is equivalent to $\Delta \leq \frac{1}{16}$. So I tried to prove by contradiction. I assumed all of them are $> \frac{1}{16}$and tried to draw a contradiction with the fact $a+ b+ c=1$ but failed
## marked as duplicate by user10354138, Xander Henderson, max_zorn, Lord Shark the Unknown, NosratiNov 3 '18 at 7:12
This question was marked as an exact duplicate of an existing question.
• You should post your work and approach as well. – Arpan Mar 18 '15 at 15:19
• Hint: The conclusion is equivalent to saying $\Delta<\frac{1}{16}$. – Thomas Andrews Mar 18 '15 at 15:20
• @Thomas Andrews Yes, I have figured that out. – Aniket Bhattacharyea Mar 18 '15 at 15:22
• If any value is $\leq 1/4$, you can show $\Delta<\frac{1}{16}$. So you can assume that $\frac{1}{4}< a,b,c<\frac{1}{2}$. – Thomas Andrews Mar 18 '15 at 15:24
• @ThomasAndrews Is it okay to assume that? – Arpan Mar 18 '15 at 15:26
First, we see that the roots of the polynimium are real iff:
$$1-16\Delta \ge 0 \Leftrightarrow \Delta \le \frac{1}{16}$$
Assume without loss of gennerality that $a \le b \le c$. Then:
$$\Delta = a^3 + a^2bc = a^2(a+bc)$$
$bc$ is biggest when $b=c=\frac{1-a}{2}$, so:
$$\Delta\le a^2\left(a+\left(\frac{1-a}{2}\right)^2\right)=\frac{a^4+2a^3+a^2}{4}$$
Notice that this expression grows monotonically with $a$, but $a\le \frac13$, so it's maximum is at $a=\frac13$. But then:
$$\Delta\le\frac{\frac{1}{81}+\frac{2}{27}+\frac{1}{9}}{4}=\frac{4}{81} < \frac{1}{16}$$
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# I need help with cubic splines containing pre existing conditions
3 views (last 30 days)
Leo Fischer on 19 Nov 2023
Edited: John D'Errico on 20 Nov 2023
My spline passes through (0,0), (1,1),and (2,2), but also S0’(0)=S1’(2)=1, I don’t know how to implement this last condition, my coefs don’t fit this criteria, so I know they’re wrong
Torsten on 19 Nov 2023
Edited: Torsten on 19 Nov 2023
s(x) = x
Show us your linear system that you use to determine the spline coefficients.
Leo Fischer on 20 Nov 2023
So, my Sn = {S0 = a0(x-x0)^3 + b0(x-x0)^2 + c0(x-x0) + d0
D0 = S0(x0) = y0 B0 = s0’’(x0)/2 And then I just go around the clamped properties to find out the other values, was that what you were asking?
John D'Errico on 19 Nov 2023
Edited: John D'Errico on 19 Nov 2023
Easy enough. And ... also somewhat useless, because this is not how you will have been asked to solve the problem. But it might give you an idea of what you missed. And that is what I want you to do, to think about the problem, and what you did.
a = sym('a',[1,4])
b = sym('b',[1,4]);
syms x real
y1(x) = dot(x.^(3:-1:0),a);
y2(x) = dot(x.^(3:-1:0),b);
dy1 = diff(y1,x);
dy2 = diff(y2,x);
ddy1 = diff(dy1);
ddy2 = diff(dy2);
absol = solve(y1(0) == 0,y1(1) == 1,y2(1) == 1, y2(2) == y1(2),dy1(0) == 1,dy2(2) == 1,dy1(1) == dy2(1),ddy1(1) == ddy2(1),[a,b])
absol = struct with fields:
a1: 0 a2: 0 a3: 1 a4: 0 b1: 0 b2: 0 b3: 1 b4: 0
subs(y1,absol)
ans(x) =
x
subs(y2,absol)
ans(x) =
x
The cubic spline is now easily seen to be that which @Torsten suggested. Note that I have used an absolute form for the polynomial segments, not a relative one, as tools like spline will do.
What did you do wrong? Take a careful look at the constraints I posed in the call to solve. Now think about what is missing or incorrect in the code you wrote.
Leo Fischer on 20 Nov 2023
Basically, my spline had to follow those three points I mentioned, but when it came to that derivative part, my c0 = 0,5, when it had to be equal to 1, but then, when I used the same code for the other stuff I had to do, it worked perfectly. I guess it might have been something about the order I was inputting my code, I’ll try to revise some more before updating you
John D'Errico on 20 Nov 2023
Edited: John D'Errico on 20 Nov 2023
All I can suggest is to be more careful about how you implemented the various conditions. Look very carefully at what you wrote, since you show no code at all for us to see. Having written splines code that have been used an uncountable number of times, I can say that it will work, but that you need to do it right.
If I had to guess, it is that you computed a derivative incorrectly, so that it works on some specia lcase, but it fails otherwise.
Matt J on 19 Nov 2023
Edited: Matt J on 19 Nov 2023
e=0:2;
xq=linspace(0,2,20);
yq=spline(e,[1,e,1],xq);
plot(e,e,'o',xq,yq,'.'); legend('Control Points','Interpolated','Location','southeast')
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# The Reciprocal of a Number Online Quiz
Following quiz provides Multiple Choice Questions (MCQs) related to The Reciprocal of a Number. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using Show Answer button. You can use Next Quiz button to check new set of questions in the quiz.
Q 1 - Find the reciprocal of $\frac{4}{9}$
### Explanation
Step 1:
To find the reciprocal of $\frac{4}{9}$, its numerator and denominator are flipped and the reciprocal = $\frac{9}{4}$
Step 2:
So, the reciprocal of $\frac{4}{9}$ is $\frac{9}{4}$.
Q 2 - Find the reciprocal of $\frac{3}{8}$
### Explanation
Step 1:
To find the reciprocal of $\frac{3}{8}$, its numerator and denominator are flipped and the reciprocal = $\frac{8}{3}$
Step 2:
So, the reciprocal of $\frac{3}{8}$ is $\frac{8}{3}$.
Q 3 - Find the reciprocal of $\frac{2}{5}$
### Explanation
Step 1:
To find the reciprocal of $\frac{2}{5}$, its numerator and denominator are flipped and the reciprocal = $\frac{5}{2}$
Step 2:
So, the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.
Q 4 - Find the reciprocal of $\frac{5}{7}$
### Explanation
Step 1:
To find the reciprocal of $\frac{5}{7}$, its numerator and denominator are flipped and the reciprocal = $\frac{7}{5}$
Step 2:
So, the reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.
### Explanation
Step 1:
First 15 is written as a fraction $\frac{15}{1}$
To find the reciprocal of $\frac{15}{1}$, its numerator and denominator are flipped and the reciprocal = $\frac{1}{15}$
Step 2:
So, the reciprocal of 15 is $\frac{1}{15}$.
Q 6 - Find the reciprocal of $\frac{6}{11}$
### Explanation
Step 1:
To find the reciprocal of $\frac{6}{11}$, its numerator and denominator are flipped and the reciprocal = $\frac{11}{6}$
Step 2:
So, the reciprocal of $\frac{6}{11}$ is $\frac{11}{6}$
Q 7 - Find the reciprocal of $\frac{8}{9}$
### Explanation
Step 1:
To find the reciprocal of $\frac{8}{9}$, its numerator and denominator are flipped and the reciprocal = $\frac{9}{8}$
Step 2:
So, the reciprocal of $\frac{8}{9}$ is $\frac{9}{8}$.
Q 8 - Find the reciprocal of $\frac{5}{13}$
### Explanation
Step 1:
To find the reciprocal of $\frac{5}{13}$, its numerator and denominator are flipped and the reciprocal = $\frac{13}{5}$
Step 2:
So, the reciprocal of $\frac{5}{13}$ is $\frac{13}{5}$.
### Explanation
Step 1:
First 19 is written as a fraction $\frac{19}{1}$
To find the reciprocal of $\frac{19}{1}$, its numerator and denominator are flipped and the reciprocal = $\frac{1}{19}$
Step 2:
So, the reciprocal of 19 is $\frac{1}{19}$.
Q 10 - Find the reciprocal of $\frac{2}{7}$
### Explanation
Step 1:
To find the reciprocal of $\frac{2}{7}$, its numerator and denominator are flipped and the reciprocal = $\frac{7}{2}$
Step 2:
So, the reciprocal of $\frac{2}{7}$ is $\frac{7}{2}$.
reciprocal_of_number.htm
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# How much wire to fence in 1.25 acres square?
I have 1.25 acres...its nearly perfectly square. I want to put up welded wire fence. Can you give me an idea of how much wire is needed? and show me the calculations and explain them? I believe what I am looking for is the linear feet but not positive. Any help is appreciated thanks!.
I had to look up the square footage for an acre! One acre is equal to 43,560 square feet. So, 1.25 acres is equal to (43560 sq ft / acre))(1.25 acres) = 54450 sq feet.
So that's the area in terms of square feet. The area of a square is A = s^2, where s is the side. So you want to FIRST find the length of a side of your square. That's where square roots come in! You have (54450 ft^2) = s^2, which means s = SQRT(54450 ft^2).
You actually take the square root of BOTH 54450 AND the "square feet". The square root of "square feet" is just "feet", a linear foot. So you get:
s = SQRT(54450 ft^2)
s = 233.34 ft
You have to do it on a calculator - google will do it for you, or Wolfram Alpha can do it too. There are algorithms to do square roots by hand, but I think they thankfully stopped teaching that back in the 50s.
Now you have ONE side of your yard. Your square has FOUR sides. So you want to find the perimeter, which is equal to s + s + s + s = 4s, since all sides are the same. In this case, you get:
P = 4s = 933.38 ft
So you need about 933 feet, or 311 yards, of wire to surround your field once. If you want to string, say, four pieces of wire around the fence, you'd need four times as much.
However, you said that your field is "almost square". There's a fun theorem that says that if you have two fields with the same area, and one is a square, and the other is a rectangle, the square will ALWAYS have the smallest perimeter. Think about this in an extreme case...if you squashed your 54450 square feet into a 1 foot by 54450 foot rectangular field, you'd need 1 + 1 + 54450 + 54450 feet of wire!
So, in practical terms, you'll need MORE than 311 yards of wire to surround your field once. How much more depends on how "non-square" it is.
Jul 5 at 11:39
1 acre = 43560 ft^2 = s^2
s = 208.7103256... ≈ 208.71 ft
P = 4s = 4(208.71) ≈ 834.84 ft
Approximately 834.84 ft of wire would be needed.
Jul 5 at 15:25
1 ac = 43560 sqft.
1.25 ac = 54450 sqft
approx. side of a square that measures 1.25 acres: √54450 ≈ 233.35 feet
How much fence ? 4 * 233.35 ≈ 933.4
to be safe...about 950 linear feet of fencing...
go for it.
Jul 5 at 19:34
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# Math Nerds, Help!
I’m trying to understand the difference between the following operators (not sure if that’s the right term in this case) and when each would be used. Hopefully the math geeks and semantic pedants here can be of help.
What is the difference between:
I know this has to do with sets, but I’m not entirely clear how to visualize the differences between “is a member of” and “Contains” or “Subset of” and Super set of".
This has a lot of them: http://www.rapidtables.com/math/symbols/Set_Symbols.htm
I’m not sure what the “backwards” element-of operator is.
Edit: Nevermind, it’s just for listing things backwards. So, x ∈ A is the same as A ∋ x.
The / denotes negation. So x ∈ {x, y, z}, but x ∉ {y, z}.
With regards to membership and containment, it’s a matter of distinguishing between objects and sets. x and {x} are not the same thing, and so we would say that x ∈ {x, y, z} and {x} ⊆ {x, y, z}.
The relation between ⊂ and ⊆ is pretty similar to the relation between < and ≤: the first means “is a strict subset of”, while the second one means “is a strict subset of or is equal to”. So we could say that {x} ⊆ {x, y, z}, and {x, y, z} ⊆ {x, y, z}; but it isn’t true that {x, y, z} ⊂ {x, y, z}.
Note that the “backwards” operators (∋, ∌, ⊃, ⊅) are used far less commonly than the others.
In other words, only a set can be a subset of another set. So you’d only put a set to the left of a ⊂ or ⊄ or ⊆ symbol. On the other hand, a set usually isn’t an element of another set (∈, ∉), though it can be if that set is itself a set of sets.
If the you want a really basic explanation of this set stuff, try this.
For those who like Venn diagrams:
http://www.purplemath.com/modules/venndiag2.htm
Just a warning: I’ve seen ⊂ used this way, but I’ve also seen ⊂ used in exactly the same way as ⊆. To be better understood, instead of using ⊂, you might want to use ⊆ with a slash through just the underline part (don’t know how to make that symbol for display, though).
Unfortunately, ⊂ is almost always used to mean included in or equal to. It would certainly make sense to use it to mean contained in and not equal to, but mathematicians are not always as logical as the ought to be.
Ok, sorry for the delayed follow up, I got sidetracked over the weekend, but hopefully some of you are still paying attention.
If you have a set of {x, y, z}.
Is {x,y} ∈ {x,y,z} evaluated as true?
Is {x,y} ⊂ {x,y,z} evaluated as true?
Is {x,b} ∈ {x,y,z} evaluated as true?
Is {x,b} ⊂ {x,y,z} evaluated as true?
Is {x,b} ∉ {x,y,z} evaluated as true?
Is {x,b} ⊄ {x,y,z} evaluated as true?
If I read the comment above right, you’d never have the ∈ or ∉ operators with a set on the left side only a single value, or is that mistaken?
It’s on the right conceptual track, but it’s not quite true. The way it works is this:
∈: Set of thingamajigs on the right, a particular thingamajig on the left. It says the item on the left is one of the members of the set on the right.
⊂: On both left and right, there is a set of thingamajigs. It says every member in the set on the left is also a member in the set on the right.
So on the right of ∈, and the left and right of ⊂, you must always have a set.
You don’t have to have a set on the left of ∈, but you can, if what’s on the right of ∈ is a set whose members are themselves sets. (For example, {x, y} ∈ {{a, b, c}, {x, y}, {x, z}}).
No. There are only three true facts ending in " ∈ {x,y,z}". Those three true facts are: x ∈ {x,y,z}, y ∈ {x,y,z}, and z ∈ {x,y,z}.
Yes. The two things in {x, y} are also both in {x, y, z}.
No. There are only three true facts ending in " ∈ {x,y,z}". Those three true facts are: x ∈ {x,y,z}, y ∈ {x,y,z}, and z ∈ {x,y,z}.
No. It’s not the case that both of the things in {x, b} are also in {x, y, z}. Specifically, b is not in {x, y, z}.
Yes. This is just another way of saying “It’s NOT the case that ‘{x, b} ∈ {x,y,z}’ is true”.
Yes. This is just another way of saying “It’s NOT the case that ‘{x, b} ⊂ {x,y,z}’ is true”.
In my case, we will never have a Set of Sets so the statement that a ∈ will always have a single value on the left is apt.
Is there an operator that would evaluate {x,b} as true when compared to {x,y,z}? Intersection or something?
Oh, crap, that was wrong… nevermind.
Well, there are many operators that would (such as, as you say, the “Do these have any element in common?” operator), but most of them have not been considered so ubiquitously useful as to be typically written with a single symbol rather than words (or multiple symbols).
What exactly is your case? Perhaps we can give better advice with more details.
We have some software that uses these operators to compare the values of multi-select pick lists. It was originally implemented incorrectly and I need to define what the new solution should look like and I’m trying to capture all possibilities and make sure I understand the principles right.
One scenario is the user inputs a value, either a single or multiple selection, and that needs to be compared against a different set of inputs (some or all of a different multi-select list). This is pretty straightforward I think. The first value is either going to be a element of or a subset of or not a element of or not a subset of the second set.
The other scenario is that the user picks from a set of items, either one or many, and I need to decide when they’ve selected the right combination of choices. This one is a little tougher for me to get my head around.
Here’s an example:
Suppose we have a selection of choices of pizza toppings in a set.
{Pepperoni, Sausage, Onion, Basil, Peppers, Mushrooms, Olives}
Anytime a combination of these choices contains Pepperoni I need to set Pepperoni to true. Any time Pepperoni, Sausage and Olives is selected I need to set Pepperoni, Olives and Sausage to true. For some reason this is harder for me to grasp and I’m not sure why.
There are people who don’t like Venn diagrams?
I don’t completely understand the example you posted, but I want to comment on this:
In this case, {x, b} is not a subset of {x, y, z}. In more formal notation, you’d write {x, b} ⊄ {x, y, z}.
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# Magnitude of average force
1. Mar 1, 2009
### Crichar1
A woman with a mass of 44 kg runs at a speed of 8 m/s and jumps onto a giant 30 kg skateboard initially at rest. What is the combined speed of the woman and the skateboard?
44 kg / 8 m/s=5.5
30 kg / 8m/s=3.75
5.5+3.75=9.25m/s
2. Mar 1, 2009
### The Bob
What do you know about mechanics that will help? Any equations?
The Bob
3. Mar 1, 2009
### Freyster98
Think momentum.
4. Mar 1, 2009
### Crichar1
i tried the linear momentum equation..mass * velocity and the answer was still incorrect
5. Mar 1, 2009
### The Bob
But what did you do to get this incorrect answer?
The Bob
6. Mar 1, 2009
### Crichar1
mass x velocity
44kg x 8m/s= 352
30kg x 8m/s=240
352-240=112m/s
didn't seem right but i thought i would try it anyways
7. Mar 1, 2009
### The Bob
Right, now I see what you're doing. Have you heard of something called 'conservation of momentum'? Basically, the momentum before a 'change' must be equal to the combined momentum afterwards. You've, therefore, wrongly assumed that the skateboard has velocity of 8ms-1.
So... try imagining the woman and the skateboard as one mass after they have 'combined'.
The Bob
8. Mar 1, 2009
### Freyster98
Conservation of momentum. The total momentum before she jumps on the board=total momentum after she jumps on the board.
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# Work done by friction by wheels
Hello all.
I have a doubt about the work done by friction force on a wheels , in two diferent situations:
a) In Morin's mechanics book, chapter 5 (page 146) he considers the situation in which a car is braking without skidding. He claims that the friction from the ground on the tires causes the car to slow down. But then, he says that this force is not doing work on the car, because the force acts over zero distance. I do not really undertand why there is no work here.
b) Imagine the same situation, but now, the car slows down while skidding. In this situation, the wikipedia says that there is work done by friction (see https://en.wikipedia.org/wiki/Work_(physics), section "Moving in a straight line (skid to a stop)". I guess that friction does work only if the wheels of the car are skidding, but it does no work if the wheels are just rolling. I tink it is related to the point of application of the force, but I can't see the point obout it
Can anybody explain me why?
Best regards
PeroK
Homework Helper
Gold Member
2020 Award
Hello all.
I have a doubt about the work done by friction force on a wheels , in two diferent situations:
a) In Morin's mechanics book, chapter 5 (page 146) he considers the situation in which a car is braking without skidding. He claims that the friction from the ground on the tires causes the car to slow down. But then, he says that this force is not doing work on the car, because the force acts over zero distance. I do not really undertand why there is no work here.
b) Imagine the same situation, but now, the car slows down while skidding. In this situation, the wikipedia says that there is work done by friction (see https://en.wikipedia.org/wiki/Work_(physics), section "Moving in a straight line (skid to a stop)". I guess that friction does work only if the wheels of the car are skidding, but it does no work if the wheels are just rolling. I tink it is related to the point of application of the force, but I can't see the point obout it
Can anybody explain me why?
Best regards
First, take a simple example. If you push a heavy object, but can't get it moving because of static friction, then no work is done. But, if you get it moving then work is done by kinetic friction.
In case b), you have an example of kinetic friction at work. The tyres will heat up etc.
In case a) the work is being done in the braking mechanism. The brakes will get hot. But, braking does rely on static friction on the ground. If the car were moving through the air, then applying the brakes would do nothing. To explain this, you can see that the tyre is not actually moving along the ground - it's rolling. One of the things about rolling (without slipping) is that the part of the tyre/wheel that touches the ground is instantaneously at rest. It grips using static friction, rather than kinetic friction. This is what allows rolling to be so efficient: each small area of the wheel touches the ground without slipping, grips momentarily using static friction, then moves off the ground. And the next small area of the wheel takes over. So, there is effectively no heat or energy loss to friction: either in accelerating, braking or moving with constant speed.
Thanks a lot.
Trata helps clarify.
Sorry, but after reading Kleppner - Kolenkow (2nd ediction) example 7.16, I have another doubt about work done by friction.
In this example, a uniform drum of radius b, mass M, weight W = Mg, and moment of inertia I = Mb2/2 is on a plane of angle β. The drum starts from rest and rolls without slipping, and we have to find the speed V of its center of mass after it has descended a height h.
In the solution, they calculate the total work done by all the forces while the certer of mass moves along the plane. For them, the forces doing work are the weight W (that is acting at the center of mass), and the friction f between the drum and the plane as well. So they consider that work done by friction is not zero, but -fL (being L the distance covered) (!!).
After that, they calculate the work done by the total torque on the drum. The only force providing a torque is friction (because weight is acting on the center of mass, and normal force is paralell to position vector). So this torque is responsible for the rolling of the drum, and accounts for the change of rotational kinetic energy of the drum. In adition, they get that work done by torque is fL, so it equals the work done by friction.
Finally, they conclude that work done by friction (-fL) is decreasing the the center of mass kinetic energy in exactly the same amount that torque exerted by friction (fL) is increasing the rotational energy. So here friction simply transforms mechanical energy from one mode to another.
Now, if it is true, the friction must be doing work, but previously you told me that friction is not doing work. I am really confused. What is really happening here??
PeroK
Homework Helper
Gold Member
2020 Award
Sorry, but after reading Kleppner - Kolenkow (2nd ediction) example 7.16, I have another doubt about work done by friction.
In this example, a uniform drum of radius b, mass M, weight W = Mg, and moment of inertia I = Mb2/2 is on a plane of angle β. The drum starts from rest and rolls without slipping, and we have to find the speed V of its center of mass after it has descended a height h.
In the solution, they calculate the total work done by all the forces while the certer of mass moves along the plane. For them, the forces doing work are the weight W (that is acting at the center of mass), and the friction f between the drum and the plane as well. So they consider that work done by friction is not zero, but -fL (being L the distance covered) (!!).
After that, they calculate the work done by the total torque on the drum. The only force providing a torque is friction (because weight is acting on the center of mass, and normal force is paralell to position vector). So this torque is responsible for the rolling of the drum, and accounts for the change of rotational kinetic energy of the drum. In adition, they get that work done by torque is fL, so it equals the work done by friction.
Finally, they conclude that work done by friction (-fL) is decreasing the the center of mass kinetic energy in exactly the same amount that torque exerted by friction (fL) is increasing the rotational energy. So here friction simply transforms mechanical energy from one mode to another.
Now, if it is true, the friction must be doing work, but previously you told me that friction is not doing work. I am really confused. What is really happening here??
K & K sum up that example very well at the end by saying that in rolling without slipping friction transforms linear to rotational energy with no energy being dissipated as heat. Although, you could also say that friction transforms gravitational PE to rotational KE.
"Doing work" is putting energy in or taking energy out of a system. Transforming energy from one form of KE to another isn't "work".
In fact, if you look at the definition of work in K & K: secion 5.3.3. it gives ##W_{ba} = K_b - K_a##. In this case, all the change in KE is due to gravity. Friction, by definition, does no work in this example.
A.T.
Finally, they conclude that work done by friction (-fL) is decreasing the the center of mass kinetic energy in exactly the same amount that torque exerted by friction (fL) is increasing the rotational energy. So here friction simply transforms mechanical energy from one mode to another.
Now, if it is true, the friction must be doing work, but previously you told me that friction is not doing work. I am really confused. What is really happening here??
Just as they say: The linear and rotational work done by friction cancel each other, so friction is doing no work in total.
But then, or Morin is wrong, or Kleppner and Kolenkow are wrong. Let me explain:
1) As I mentioned in the first contribution to this thread, in Morin's mechanics book, chapter 5 (page 146) he considers the situation in which a car is braking without skidding. He claims that the friction from the ground on the tires causes the car to slow down. But then, he says that this force is not doing work on the car, because the force acts over zero distance. So, for Morin, friction is nor doing work.
2) On the other hand, in Kleppner and Kolenkow mechanics book (2nd ediction) example 7.16, there is a drum rolling without slipping on a plane. The situation is the same than in the example in Morin's book. Nevertheless, they do not claim that work is zero, and compute the work that is being done by friction as the integral of f (the friction force) between x1 and x2, obtaining that this work equals to -fL (where L is the distance travelled between x1 and x2). So, for Kleppner and Kolenkow, work done by friction is not zero.
Evidently, one of them must be wrong, because the reasoning they follow is opposite, indepentdently that the torque due to the friction force is doing work or not.
What do you think about it?
PS: By the way, if friction does no work, how can Morin claim that the friction from the ground on the tires causes the car to slow down? By the W-E theorem, if friction does no work, the friction would not contributte to the loss of kinetic energy of the car...
A.T.
Evidently, one of them must be wrong,
They both agree that friction is doing zero work. They just decompose that work into linear and rational parts differently, by choosing a different reference point.
I see...
So in Morin, the reference point is the contact point between the wheel and the road. But in Kleppner- Kolenkow, the referece point is the center of mass of the drum. Am I right?
Nevertheless, when we change the reference point to the center of mass, is the friction force is still acting over zero distance, or not? This issue is not still clear to me...
Thank you so much.
PeroK
Homework Helper
Gold Member
2020 Award
2) On the other hand, in Kleppner and Kolenkow mechanics book (2nd ediction) example 7.16, there is a drum rolling without slipping on a plane. The situation is the same than in the example in Morin's book. Nevertheless, they do not claim that work is zero, and compute the work that is being done by friction as the integral of f (the friction force) between x1 and x2, obtaining that this work equals to -fL (where L is the distance travelled between x1 and x2). So, for Kleppner and Kolenkow, work done by friction is not zero.
PS: By the way, if friction does no work, how can Morin claim that the friction from the ground on the tires causes the car to slow down? By the W-E theorem, if friction does no work, the friction would not contributte to the loss of kinetic energy of the car...
You're missing a subtlety in K&K's solution. They integrate ##f## with respect to the angle around the drum. From the drum's reference frame ##f## works it's way round the circumference and does work equal to ##fb\theta##. And that's what causes the rotational motion.
Then, they take the force down the slope as ##F-f## where ##f## is doing negative work equal to the positive work it does above. Overalll, therefore, the work done is:
##(Fl - fl) + fl##
You can now interpret this two ways:
a) Friction is doing no overall work.
b) Friction is transforming linear KE into rotational KE by doing both positive and negative work at the same time.
The problem is perhaps clearer if you look at the overall energy. ##E_{initial} = GPE = E_{final} = LKE + RKE##. With no energy lost to friction/heat. So that's clear. If you now reread K&K's final note, I think this is what they are trying to explain: friction in this case was not a dissipative force. They used it (rather cleverly) as a transformative force.
PeroK
Homework Helper
Gold Member
2020 Award
I see...
So in Morin, the reference point is the contact point between the wheel and the road. But in Kleppner- Kolenkow, the referece point is the center of mass of the drum. Am I right?
Nevertheless, when we change the reference point to the center of mass, is the friction force is still acting over zero distance, or not? This issue is not still clear to me...
Thank you so much.
The final point, which perhaps you may get from my post above, is that from the drum's perspective friction works its way round the circumference as a motive force.
Mmmmm.
I think now I understand it all.
The key point is that Morin chooses as his reference point the point of contact between the wheel and the ground (which is instantaneously at rest); so for him, friction does no "linear" work nor "rotational" work at all. On the other hand, Kleppner and Kolenkow choose as their reference point the center of mass of the rolling drum (which, in the center of mass reference frame is at rest); so for them, friction does both "linear" and "rotational" work, which happen to be identical (but with opposite signs), and cancel each other, so the total work done by frction is zero.
I find this argumentation plausible, but I need to work it out to check if it is right.
Thanks A.T, and PeroK for your clever indications. If I had any other doubts, I would post them here as well.
Regards.
OK, I have tried to solve Kleppner - Kolenkow example taking the point of contact between the drum and the ground as the origin of the reference frame.
That implies the new moment of inertia of the drum is I=ICM+Mb2=(3/2)Mb2. Now the problem is easier, because the W-E theorem for the rotational motion equation is enough to get the answer. Now the torque is only due to the weight, and not to the friction, so we get:
∫τ dθ=(1/2)Iω2
bWsenβ θ=(1/2)(3/2)Mb2ω2
, so:
V=√(4gh/3)
which is the same answer that Kleppner - Kolenkow obtain by using the W-E ecuations both for the lineal and rotation motion.
Now, I have a doubt. Altough it is not necesary for the problem, how should I use the W-E theorem for the traslational motion with this new reference point (the point of contact between the drum and the ground)??
∫F dr=T2-T1
Now the only force doing work is weight, because friction acts over zero distance, and normal force is perpendicular to the displacement. But, with this reference point, what is the kinetic energy?? I mean, what is the speed I shoud consider? The linear speed of the center of mass? The linear speed of that point? Both the rotational and the linear speed(in this case, what linear speed and why?)? Maybe the linear speed is zero (because this point is instantaneously at rest)?...
Regards.
Doc Al
Mentor
PS: By the way, if friction does no work, how can Morin claim that the friction from the ground on the tires causes the car to slow down? By the W-E theorem, if friction does no work, the friction would not contributte to the loss of kinetic energy of the car...
One must be careful about interpreting the W-E theorem. Despite the name, it's not really a statement about "real" work, but of pseudo-work (sometimes called "center of mass" work). It is an application of Newton's 2nd law:
$$F_{net}\Delta x_{cm}=\Delta (\frac{1}{2}m v_{cm}^2)$$
Note that you are multiplying the net force times the displacement of the center of mass, which allows you to calculate the resulting change in the kinetic energy of the center of mass. In Morin's example, the friction provides the net force and you can use the theorem to calculate the change in KE of the car. The friction causes the car to slow, but it does no work on the car. (The real work would involve friction times the displacement of the point of application--which is zero, since there is no slipping.)
The same analysis, of course, applies to an accelerating car. Friction causes the car to accelerate, but does no (real) work on the car (assuming no slipping). Nonetheless, the W-E theorem applies as before. The energy comes from within the car, not from the road. (But that external force from the road is certainly needed to create the motion of the car.)
The key point is that Morin chooses as his reference point the point of contact between the wheel and the ground (which is instantaneously at rest); so for him, friction does no "linear" work nor "rotational" work at all. On the other hand, Kleppner and Kolenkow choose as their reference point the center of mass
Here there's nothing like choosing a reference , its all about choosing a system - centre of mass or
- Whole body
First, take a simple example. If you push a heavy object, but can't get it moving because of static friction, then no work is done. But, if you get it moving then work is done by kinetic friction.
In case b), you have an example of kinetic friction at work. The tyres will heat up etc.
In case a) the work is being done in the braking mechanism. The brakes will get hot. But, braking does rely on static friction on the ground. If the car were moving through the air, then applying the brakes would do nothing. To explain this, you can see that the tyre is not actually moving along the ground - it's rolling. One of the things about rolling (without slipping) is that the part of the tyre/wheel that touches the ground is instantaneously at rest. It grips using static friction, rather than kinetic friction. This is what allows rolling to be so efficient: each small area of the wheel touches the ground without slipping, grips momentarily using static friction, then moves off the ground. And the next small area of the wheel takes over. So, there is effectively no heat or energy loss to friction: either in accelerating, braking or moving with constant speed.
All this is explaining that change in internal energy of the system is manifesting in form of energy , cause there's friction inspiring that !!
I will try to work it out again keeping this in mind.
What I still do not understand is the fact that, if friction is nor doing work, why it is sais that friction from the ground on the tires causes the car to slow down...
In my textbooks, nobody mention pseudo-work, they just tlak about work. So by W-E theorem, if frction does no work, it should not cause the car to slow down...
Thanks.
I will try to work it out again keeping this in mind.
What I still do not understand is the fact that, if friction is nor doing work, why it is sais that friction from the ground on the tires causes the car to slow down...
In my textbooks, nobody mention pseudo-work, they just tlak about work. So by W-E theorem, if frction does no work, it should not cause the car to slow down...
Thanks.
Look,
For a complicated system(where motion of all the parts of the system is not same), we can apply work energy theorem in 2 ways
1] WE theorem for COM, i.e you treate the whole body as a point particle, and find work done by all the forces on COM i.e integral of force times dx(COM)
So you will equate all this to change in KE of COM (here you will not include rotational KE of the body as, now your body is a point particle)
2]Apply WE theorem for whole body , here rotational KE will be considered and work done by those forces which are not causing displacement at the point of contact will be zero.
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# Differential Equations for Engineers
### Description
This course is all about differential equations. Both basic theory and applications are taught. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations.
The course contains 56 short lecture videos, with a few problems to solve after each lecture. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz.
http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf
Watch the promotional video:
https://youtu.be/eSty7oo09ZI
### What you will learn
First-Order Differential Equations
A differential equation is an equation for a function with one or more of its derivatives. We introduce differential equations and classify them. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Then we learn analytical methods for solving separable and linear first-order odes. An explanation of the theory is followed by illustrative solutions of some simple odes. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.
Homogeneous Linear Differential Equations
We generalize the Euler numerical method to a second-order ode. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. We make use of an exponential ansatz, and transform the constant-coefficient ode to a quadratic equation called the characteristic equation of the ode. The characteristic equation may have real or complex roots and we learn solution methods for the different cases.
Inhomogeneous Linear Differential Equations
We now add an inhomogeneous term to the constant-coefficient ode. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum.
The Laplace Transform and Series Solution Methods
We present two new analytical solution methods for solving linear odes. The first is the Laplace transform method, which is used to solve the constant-coefficient ode with a discontinuous or impulsive inhomogeneous term. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. We also discuss the series solution of a linear ode. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses.
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# Homework Help: Experiment to determine the focal distance and magnification of a converging lens
1. Jul 13, 2008
### kthouz
1. The problem statement, all variables and given/known data
My task was to determine the focal distance and the magnification of a converging lens. I've done everything as stated in the procedure. The problem that i have is that when i am calculating the magnification in two differents ways i.e M1=b/a and M2=h'/h where a=distance form object to mirror, b=distance from image to mirror, h=height of the objet and h' height of the image, those two magnifications are very divergent. for example in 5 trials if i choose two : M1= 2 while M2=0.95
M1=1.57 while M2=0.85
...
So how can i explain it? Can you give me a hint or just some error analysis methods (link if possible)? Cause i've never get a course about how to analyse errors.
2. Relevant equations
3. The attempt at a solution
2. Jul 13, 2008
### Redbelly98
Staff Emeritus
That's strange. Just what are the 4 distances involved in your experiments? (Image distance, object distance, image height, object height)
Possible errors are either inaccuracies in the measurements, or that the object or image distance is comparable to the lens thickness (the lens should be much thinner than either distance for the calculations to work out simply).
3. Jul 13, 2008
### kthouz
But do you know any mathematical theory used in error analysis so i can try to interpret my results? Please give me a refence!
4. Jul 13, 2008
### Redbelly98
Staff Emeritus
The simplest theory works as follows:
When you are adding or subtracting two numbers, the error of the result is the sum of the errors.
When multiplying or dividing two numbers, the percentage error of the result is the sum of the percentage errors.
There are more complicated ways of calculating errors, but if you're new to error analysis I would use this simple method.
Example 1, adding or subtracting numbers:
$$(5.0 \ \pm \ 0.3) - (2.0 \ \pm \ 0.1) = 3.0 \ \pm \ (0.3 + 0.1) = 3.0 \ \pm \ 0.4$$
Check by adding or subtracting errors from terms, in order to get the maximum result possible:
(5.0 + 0.3) - (2.0 - 0.1)
= 5.3 - 1.9
= 3.4
This "maximal error" calculation gives an answer that is 0.4 higher than the original calculation of 3.0, so the calculated error of 0.4 is correct.
Example 2, multiplying or dividing:
$$\frac{6.0 \pm 0.5}{2.0 \pm 0.1} = 3.0 \ \pm \ ???$$
To find the error, add the percentage or fractional errors:
$$\frac{0.5}{6.0}\ 100 \ + \ \frac{0.1}{2.0} \ 100$$
= 8% + 5% = 13%
So the error is 13% of 3.0:
0.13 x 3.0 = 0.39 or roughly 0.4
The final answer is
$$3.0 \pm 0.4$$
Check the result by adding or subtracting errors to produce the maximum result possible:
$$\frac{6.0 + 0.5}{2.0 - 0.1} = \frac{6.5}{1.9} = 3.42...$$
This result is 0.4 higher than 3.0 (to the nearest 0.1), so the calculated error of 0.4 is correct.
Hope that helps.
5. Jul 13, 2008
### kthouz
Thank you very much!
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# Thread: Evalute an integral with a weird measure.
1. ## Evalute an integral with a weird measure.
Let $\mu$ be a measure on the set of natural numbers.
And define $\mu ( \{ n, n+1, ... \} ) = \frac {n}{2^n}$ for $n = \{ 1, 2, 3, ... \}$
Find $\int x d \mu (x)$
My solution so far:
To be honest I'm a bit lost in this one. So should I have something like $\sum ^ \infty _{x=1}\frac {x}{2^x }$?
2. ## Re: Evalute an integral with a weird measure.
Let $\mu$ be a measure on the set of natural numbers.
And define $\mu ( \{ n, n+1, ... \} ) = \frac {n}{2^n}$ for $n \in \mathbb {N}$
Taking $n=0$ we get that $\mu(\mathbb N)=0$:maybe there is a typo.
3. ## Re: Evalute an integral with a weird measure.
Sorry, n should be equal to 1, 2, 3, ... I changed it in the original post.
4. ## Re: Evalute an integral with a weird measure.
Ok. Hint: write the integral as $\sum_{n=1}^{+\infty}\int_{\{n\}}xd\mu(x)$ (use monotone convergence theorem) and you are almost done. Compute the measure of $\{n\}$ for $n\geq 1$.
5. ## Re: Evalute an integral with a weird measure.
So would I have $\sum ^ \infty _{n=1} x \frac {n}{2^n}$?
6. ## Re: Evalute an integral with a weird measure.
So would I have $\sum ^ \infty _{n=1} x \frac {n}{2^n}$?
There shouldn't be any $x$ in the sum, since you sum over $n$. We have $\mu(\{n\})=\mu(\{n,n+1,\ldots\})-\mu(\{n+1,\ldots\})=\frac{n}{2^n}-\frac{n+1}{2^{n+1}} = \frac{2n-n-1}{2^{n+1}}$.
We have $\int_{\{n\}}xd\mu(x)=n\mu(\{n\})$, since on $\{n\}$ we must have $x=n$.
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# Clarifications about the definition of algebraic systems and algebraic structures
I am learning about Groups from a discrete mathematics textbook for a computer science course by Grimaldi.
An algebraic system is defined as a set along with operations on elements of the set. An algebraic structure is defined as a set, operations on elements of the set and relations between elements of the set which leads to a structure on the elements of a set.
The author then goes on to talk about groups as being algebraic structures(so does wikipedia). I am failing to see what structure a group imposes on a set it is defined on. This is making me think about a group an algebraic system instead of an algebraic structure.
What should I see in a group that will help me relate it the definition of an algebraic structure? What should I understand by "structure on the elements of a set"? Explanation with examples would help a lot.
EDIT - Exact definitions as given in the textbook
An algebraic system is a system consisting of a nonempty set $A$ and one or more n-ary operations on the set $A$. It is denoted by $\langle A, f_1, f_2, ... \rangle$.
An algebraic structure is an algebraic system, $\langle A, f_1, f_2, ... , R_1, R_2, ...\rangle$, wherein addition to operations $f_i$, the relations $R_i$ are defined on A. This leads to a structure on the elements of A.
-
Your definitions are rather vague. Can you give exact quotations? – Zhen Lin Dec 19 '12 at 5:13
@ZhenLin, I've added the exact definitions. – Abhijith Dec 19 '12 at 5:45
Hmmm. I think by ‘algebraic structure’ most people mean a set with operations but no relations. – Zhen Lin Dec 19 '12 at 7:33
Following the definitions, an algebraic system is for example $\langle \Bbb Z, +\rangle$ and an algebraic structure is $\langle \Bbb Z,+,\leq\rangle$ – leo Dec 19 '12 at 20:34
Maybe the easiest example is the integers. As a set, $\mathbb{Z}$ is just a collection of elements which have nothing to do with each other - $0$ is no more special than $79$, $10$ and $-10$ have no particular relationship. But once we consider the integers as a group under the binary operation 'addition' (which I think we're all familiar with) we are able to see that $10+(-10)=0$ and $z + 0 = z$ for every $z\in \mathbb{Z}$.
You probably mean addition modulo $n$ in your definition of $\mathbb Z_n$. Also, all groups can be realized as groups of permutations, not just finite groups. – Santiago Canez Dec 19 '12 at 5:01
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# Tagged Questions
Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.
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Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
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Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $X+1$, ...
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I face the problem of finding how many non-trivial ring or group homomorphisms there are from $\mathbb{Z}_m$ to $\mathbb{Z}_n$, where $m<n$. Is there any general formula? At the moment, I want to ...
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# Physics in the Lab I
Doing experiments in a laboratory allows us to understand how the laws of physics work. We develop a knowledge of the quantity that we are measuring and how it depends on other factors. Experimental work involves making measurements and looking for patterns in the measurements. There are two concepts in experimental work that students have difficulty with. These are uncertainty (error) analysis and plotting data. As the NSW HSC syllabus no longer includes uncertainties in practical work many students arrive at first year university labs not fully prepared for the requirements of experimental work in both calculator skills and uncertainty analysis. The international examining boards, Cambridge International Examinations and the International Baccalaureate, still include questions with uncertainties in their examination papers. To help students learn uncertainty analysis a tutorial problem set is provided below.
1. A student drops a ball from the same height and measures the time of fall. Their measurements are 1.75s, 1.85s, 1.60s, 1.70 s and 1.71 s. Determine the average time of fall.
2. A student measures the dimensions of a desk top. The average value of the length was found to be 2.524 ± 0.004 m and the average value of the width was found to be 0.622 ± 0.004 m. Determine the perimeter and area of the desk top.
3. A student releases a ball from rest and measures the time it takes to fall to the ground. The average time was found to be 1.32 ± 0.08 s. Given that the height of release is 8.61 ± 0.05 m, determine the acceleration due to gravity.
4. A student measures the mass of a block as 117.56 ± 1.24 g. The volume of the block was measured as 22.67 ± 0.36 cm3. Determine the density of the block.
5. In a laboratory experiment a student measures the time of 10 oscillations of a simple pendulum of length 3.25 ± 0.03 m. Their time was 37.21 ± 0.86 s. Use this data to determine the acceleration due to gravity.
6. A dynamics trolley is moving along a smooth laboratory bench at a speed of 0.26 ± 0.03 m/s. The trolley accelerates at 0.84 ± 0.04 m/s2 for 6.53 ± 0.08 s. Determine the distance travelled by the trolley.
7. Determine the volume of a right circular cylinder of radius 3.215 ± 0.025 m and height 7.512 ± 0.025 m.
8. Determine the volume of a sphere of radius 3.219 ± 0.038 m.
9. The density of plutonium is 19.8 ± 0.4 gcm-3. Determine the radius in centimetres of a sphere of plutonium of mass 15.0 ± 0.5 kg.
10. When the radius r of the bob of a simple pendulum of length L is included in the calculation the period T of the small oscillations of the pendulum is given by the equation
T = 2𝜋√(L/g + 2r2/(5gL))
If L = 2.00 ± 0.02 m, g = 9.81 ± 0.03 ms-2 and r = 10.0 ± 1.0 cm, determine the value of T.
11. When the angle of oscillation of a simple pendulum is not small, the approximate period of the oscillations is given by the equation
T = 2𝜋√(L/g) (1 + 𝜽2/16)
where L is the length of the string, g is the acceleration due to gravity and 𝜽 is the angle of release of the string from the vertical measured in radians. Taking L = 6.57 ± 0.05 m, g = 9.81 ± 0.04 and 𝜽 = 22 ± 3 , determine the value of T.
12. Determine the area of a triangle of sides 1.236 ± 0.015 m, 3.256 ± 0.023 m and 2.887 ± 0.023 m.
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# Simplify: ${{\log }_{9}}729+{{\log }_{9}}81+{{\log }_{9}}9$
Last updated date: 24th Jul 2024
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Hint: To solve these kinds of problems, we need to know about the basic formulas related to logarithm. Firstly, we assume the given expression as ‘E’. This problem is solved step by step using related formulas and identities and thus obtaining the answer. Also, it is very important for us to know the properties of logarithm and which we are used in this problem:
${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$
${{\log }_{a}}1=0$
${{\log }_{a}}{{a}^{r}}=r$
${{\log }_{a}}a=1$
Logarithm is the exponent that indicates the power to which a base number is raised to produce a given number Or A logarithmic is defined as the power to which number which must be raised to get some values. Logarithmic functions and exponential functions are inverses to each other. We should be clear when the rule is applied to the power, then the exponent rule is used.
Example: The logarithm of 100 to the base 10 is 2.
As given in the question,
$\Rightarrow {{\log }_{9}}729+{{\log }_{9}}81+{{\log }_{9}}9$
As we know that, $729={{9}^{3}}$and $81={{9}^{2}}$, we get the above expression as
$\Rightarrow {{\log }_{9}}\left( {{9}^{3}} \right)+{{\log }_{9}}\left( {{9}^{2}} \right)+{{\log }_{9}}9$
According to logarithmic property, ${{\log }_{a}}{{x}^{n}}=n{{\log }_{a}}x$ , we get
$\Rightarrow 3{{\log }_{9}}9+2{{\log }_{9}}9+{{\log }_{9}}9$
For further evaluation,
We are using the logarithm identity ${{\log }_{a}}a=1$, we get
$\Rightarrow 3+2+1$
$=6$.
Therefore, ${{\log }_{9}}729+{{\log }_{9}}81+{{\log }_{9}}9=6$.
Note: One must remember the log values of arguments from 1 to 10 with the base value of the log equal to 10. More generally, exponentiation allow any positive real number as base to be raised to any real power, always producing a positive result, so ${{\log }_{b}}x$ for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y.
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## Key Concepts
• Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus $P\left(r,\theta \right)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
• A conic is the set of all points $e=\frac{PF}{PD}$, where eccentricity $e$ is a positive real number. Each conic may be written in terms of its polar equation.
• The polar equations of conics can be graphed.
• Conics can be defined in terms of a focus, a directrix, and eccentricity.
• We can use the identities $r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{ }\cos \text{ }\theta$, and $y=r\text{ }\sin \text{ }\theta$ to convert the equation for a conic from polar to rectangular form.
## Glossary
eccentricity
the ratio of the distances from a point $P$ on the graph to the focus $F$ and to the directrix $D$ represented by $e=\frac{PF}{PD}$, where $e$ is a positive real number
polar equation
an equation of a curve in polar coordinates $r$ and $\theta$
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1. ## Implicit differentiation
Can someone spot my mistake, if there is one? Evaluate y' at the point (-1,2) if xe^y - 6y = 2x - 10 - e^2. Here's what I did: Taking the derivative of both sides, I got e^y + xe^y(y') - 6yy' = 2. Solving for y': y' = (2-e^y)/(xe^y - 6y). Putting in the values x=-1 and y = 2, I got: y' at that point = (2-e^2)/(-e^2-12). The computer homework system I'm using says this is incorrect. Did I make a mistake somewhere? Thanks for speedy help.
2. $\displaystyle \text{Evaluate } y' \text{ at the point } (-1,2) \text{ if } xe^y - 6y = 2x - 10 - e^2$.
[math] xe^y - 6y = 2x - 10 - e^2 [/tex]
Isolate constant terms:
$\displaystyle -2x + xe^y - 6y = -10 - e^2$
Differentiate both sides:
$\displaystyle -2+[(1*e^y)+(x*e^y)y'] -6y' = 0$
$\displaystyle -2 + e^y + (xe^y)y' -6y' = 0$
$\displaystyle (xe^y)y' -6y' = 2 - e^y$
$\displaystyle (xe^y - 6) y' = 2 - e^y$
y' = $\displaystyle \frac{2-e^y}{x e^y - 6} = -1 \frac{e^y-2}{x e^y - 6}$
You made a differentiation error on the 6y term. If f(y) = 6y,
then f'(y) = 6*(y'). If this were not implicit differentiation: if f(y)=6y,
f'(y) =6.
3. Thanks - I think I was needlessly employing the chain rule there.
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Digital task cards are highly engaging and require no prep and no paper! All you need is a device, a free Google account and access to the internet. Students type their answers directly on the slides in the provided text boxes. An optional printable recording sheet is included if you’d like your students to turn in a paper copy. A bonus self-grading Google Forms™ assessment is also included.
These task cards are aligned to the Common Core State Standards: 3.NF.1, 3.NF.3, 4.NF.1, 4.NF.2. The following skills are covered:
Slides 1-4: Identify fractions of the whole
Slides 5-8: Identify fractions of a group
Slides 9-16: Compare two fractions using >, <, =
Slides 17-24: Order 3 fractions from least to greatest/greatest to least
Slides 25-28: Tell whether 2 fractions are equivalent
Slides 29-32: Find the missing numerator or denominator to make the fractions equivalent
***This product is part of a DISCOUNTED TASK CARD MEGA BUNDLE***
Be sure to check out my other digital math task cards for grades 3 & 4 below, organized by domain:
Operations & Algebraic Thinking:
Word Problems (Sports Themed)
Word Problems (School Themed)
Properties of Multiplication
Factors & Multiples
Continuing Number Patterns & Skip Counting
Ways to Show Multiplication
Number & Operations in Base 10:
Place Value: Rounding, Comparing, Ordering, Number Forms
Comparing & Ordering Multi-digit Numbers
Rounding
Multiplying by 1 and 2-digit Numbers
Long Division
Number & Operations: Fractions:
Fractions on a Number Line
Simplifying Fractions
Multiplying Fractions by Whole Numbers
Mixed Numbers & Improper Fractions
Decimals: Identifying, Comparing, Ordering
Measurement & Data:
Measurement Conversions
Measuring Angles with a Protractor
Measuring to the Half & Quarter Inch (With Line Plots)
Line Plots
Telling Time to the Minute
Elapsed Time
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Geometry: Lines, Angles, Polygons, Area/Perimeter
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## Algebra 1
$w=24$
$\frac{1}{3}w+3=\frac{2}{3}w-5\longrightarrow$ subtract $\frac{1}{3}w$ from each side; add 5 to each side $\frac{1}{3}w+3-\frac{1}{3}w+5=\frac{2}{3}w-5-\frac{1}{3}w+5\longrightarrow$ combine like terms $8=\frac{1}{3}w\longrightarrow$ multiply each side by 3 $8\times3=\frac{1}{3}w\times3\longrightarrow$ multiply $24=w$
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A point moves on the parabola y2 = 4ax. Its distance from the focus is minimum for the following value(s) of x. : Kaysons Education
# A Point Moves On The Parabola y2 = 4ax. Its Distance From The Focus Is Minimum For The Following Value(s) Of x.
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## Question
### Solution
Correct option is
#### SIMILAR QUESTIONS
Q1
The locus of a point whose some of the distance from the origin and the line x = 2 is 4 units, is
Q2
The length of the subnormal to the parabola y2 = 4ax at any point is equal to
Q3
The slope of the normal at the point (at2, 2at) of parabola y2 = 4ax is
Q4
Equation of locus of a point whose distance from point (a, 0) is equal to its distance from y-axis is
Q5
Through the vertex O of parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is
Q6
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix
Q7
The equation of common tangent to the curves y2 = 8x and xy = –1 is
Q8
From the point (–1, 2) tangent lines are drawn to the parabola y2 = 4x, then the equation of chord of contact is
Q9
For the above problem, the area of triangle formed by chord of contact and the tangents is given by
Q10
The line x – y + 2 = 0 touches the parabola y2 = 8x at the point
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## Sanjay buys some articles for Rs. 1,80,000. He sells 2/5th of it at a loss of 12%. If he wants to earn overall profit of 18% on selling all the articles, then at what profit % he should sell the remaining articles?
A) 48% B) 73 % C) 42 % D) 60 % Solution Lets assume Total 5 articles (sold at 100) Then 5 x100 —–500 2 x 100 —–200 But sold at 12 % loss 200x 12/100 = 176 3—-300 As per question total is % profit 500x 18/100 = 590 But already 2 sold at … Read more
## Mohit Calculates his profit% on the selling Puice where as soma calculates on her cost price. They find that the difference of their profit is rs 195. If the selling Price is same for both and mohit earned 25 % profit where a Soma earned 30% Profit Find Cost Price of mohit
A) 7804 B) 6335 C) 6405 D) 7605 Solution Mohit = 25% = 1/4 (1= profit; 4= sp) Cp =3. Sp = 4 profit=1 Soma 30% 3/10 (3 profit ; 10 cp) Cp =10 sp = 13. Profit = 3 CP. SP Mohit. 3. 4. (13) Soma. 10. 13. (4) As per question sp to … Read more
## A shopkeeper sells his article at 16*2/3% profit on S.P. What is his actual profit?
A) 20% B) 21% C) 25 % D) 19 % Solution
## An egg seller sells his eggs only in the packs of 3 eggs, 6 eggs, 9 eggs, 12 eggs etc., but the rate is not necessarily uniform. One day Raju purchased at the rate of 3 eggs for a rupee and the next hour he purchased equal number of eggs at the rate of 6 eggs for a rupee. Next day he sold all the eggs at the rate of 9 eggs for Rs.2. What is his percentage profit or loss?
A) loss 10 % B) 11.11% loss C) 20 % loss D) 16 % loss Solution
## A man purchased some eggs at 3 for Rs. 5 and sold them at 5 for Rs.12. Thus he gained Rs. 143 in all. the number of egges he bought is
A) 210 B) 200 C) 195 D) 190 Solution
## 0.023g of sodium metal is reacted with 100cm 3 of water. The pH of the resulting solution is?
A) 10 B) 11 C) 9 D) 12 Solution
## Excess of carbon dioxide is passed through 50 mL, of 0.5 M calcium hydroxide solution. After the completion of the reaction, the solution was evaporated to dryness. The solid calcium carbonate was completely neutralised with 0.1 N hydrochloric acid. The volume of hydrochloric acid required is (Atomic mass of calcium = 40)
A) 300 cm3 B) 500 cm3 C) 290cm3 D) 450 cm3 Solution
## SIMPLE NOTE ON GUPTA DYNASTY
The Gupta dynasty was a ruling dynasty of ancient India that flourished from the 3rd to 6th century CE. The dynasty was founded by Sri Gupta, who was succeeded by his son Ghatotkacha Gupta. The most famous king of the Gupta dynasty was Chandragupta I, who ruled from 320 to 335 CE. Under the rule … Read more
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## Algebra 1
You have the values:3.7, 0.2, 4.3, 4, 4.3 To find the mean you have to add all the values together and divide by 5:$\frac{3.7+0.2+4.3+4+4.3}{5}$=3.3 Arrange the numbers from least to greatest (0.2, 3.7, 4, 4.3, 4.3) to find the median. The middle number is 4. The range is the largest value minus the smallest so 4.3-0.2=4.1 The mode is the number that repeats the most which is 4.3 because it occurs twice. So the mean=3.3, Range=4.1,Median=4,Mode=4.3
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# Math
posted by on .
Joe's electric made 16 customer calls last week and 22 calls this week. How many calls must be made next week in order to maintain an average of at least 21 calls for the three week period?
My answer was at least 19 calls must be made to maintain an average of 21 calls is this correct?
• Math - ,
3 * 21 = 63 calls in 3 weeks
16 + 22 = 38 calls made in 2 weeks
63 - 38 = 25 calls next week
Check it out: 16 + 22 + 25 = 63
63/3 = 21
• Math - ,
thank you ms. sue
• Math - ,
You're welcome.
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# BASE Function
Converts a number into the supplied base
## What is the BASE Function?
The BASE function is available under Mathematical and Trigonometric functions. The function returns a text representation of the calculated value and will convert a number into the supplied base (radix). BASE was introduced in Excel 2013 and is unavailable in earlier versions.
### Formula
The BASE function uses the following arguments:
1. Number (required argument) – It is the number that we wish to convert. The number should be an integer and greater than or equal to 0 and less than 2^53.
2. Radix (required argument) – The base radix is what we want to convert the number into. It must be an integer greater than or equal to 2 and less than or equal to 36. Radix is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system, the radix is 10, as it uses ten digits from 0 to 9.
3. Min_length (optional argument) – It is the minimum length of the string returned. If specified, it must be an integer greater than or equal to 0.
### How to use the BASE Function in Excel?
To understand the uses of the BASE function, let’s consider a few examples:
#### Example 1
Suppose we want to convert the number 10 to base 2:
The BASE function will convert the decimal number 10 to base 2 (binary system) and give us the following result:
On a similar basis, we can give different radix to get the desired output. Few examples are shown below:
#### Example 2
Now, let’s see how this function behaves when we specify a minimum_length argument. Suppose we specify the number 12, with radix 2 and minimum length 10:
In the example, the function will convert the decimal number 12 to base 2 (binary), with a minimum length of 10. The result would be 0000001100, which is 1100 with 6 leading zeros to make the string 10 characters long.
### Few notes about the BASE Function
1. #VALUE! error – Occurs when the number given is a non-numeric value.
2. #NUM! error – When any of the argument that is number, radix, or min_length is outside the minimum or maximum constraints. So, the error will occur:
• When the given number argument is < 0 or is ≥ 2^53.
• When the given radix argument is < 2 or > 36.
• When the [min_length] argument is supplied and is < 0 or ≥ 256.
3. #NAME! error – Occurs when the formula contains an unrecognized value in any of the arguments.
4. If we enter a non-integer number, the argument is truncated to an integer.
5. The maximum value of the min_length argument is 255.
6. If we enter the min_length argument, leading zeros are added to the result if the result would otherwise be shorter than the minimum length specified. For example, BASE(32,2) returns 100000, but BASE(32,2,8) returns 00100000.
7. The function can also be used by specifying a cell reference. If the cell referred to is empty, it will take the number as zero.
For example, if we give the formula =BASE(A3,2,10), wherein A3 is the cell referred to and is empty. In such scenario, the BASE function will return the following result as A3 is empty.
Thanks for reading CFI’s guide to important Excel functions! By taking the time to learn and master these functions, you’ll significantly speed up your financial analysis. To learn more, check out these additional resources:
• Excel Functions for Finance
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# Number 1267
### Properties of number 1267
Cross Sum:
Factorization:
7 * 181
Divisors:
1, 7, 181, 1267
Count of divisors:
Sum of divisors:
Prime number?
No
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 2 (Binary):
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
4f3
Base 32:
17j
sin(1267)
-0.80647188074013
cos(1267)
-0.59127244614938
tan(1267)
1.3639598563948
ln(1267)
7.1444071803211
lg(1267)
3.1027766148834
sqrt(1267)
35.594943461115
Square(1267)
### Number Look Up
Look Up
1267 (one thousand two hundred sixty-seven) is a amazing figure. The cross sum of 1267 is 16. If you factorisate the figure 1267 you will get these result 7 * 181. The number 1267 has 4 divisors ( 1, 7, 181, 1267 ) whith a sum of 1456. The number 1267 is not a prime number. The number 1267 is not a fibonacci number. The figure 1267 is not a Bell Number. The number 1267 is not a Catalan Number. The convertion of 1267 to base 2 (Binary) is 10011110011. The convertion of 1267 to base 3 (Ternary) is 1201221. The convertion of 1267 to base 4 (Quaternary) is 103303. The convertion of 1267 to base 5 (Quintal) is 20032. The convertion of 1267 to base 8 (Octal) is 2363. The convertion of 1267 to base 16 (Hexadecimal) is 4f3. The convertion of 1267 to base 32 is 17j. The sine of 1267 is -0.80647188074013. The cosine of 1267 is -0.59127244614938. The tangent of 1267 is 1.3639598563948. The root of 1267 is 35.594943461115.
If you square 1267 you will get the following result 1605289. The natural logarithm of 1267 is 7.1444071803211 and the decimal logarithm is 3.1027766148834. You should now know that 1267 is very special figure!
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# Learn Number 8
We will learn about number 8.
How to teach number 8 to your child?
Ask the child to pick up eight objects from a collection of items such as books, crayons, pencils and show it to you.
I. Circle the number 8
II. Count and write:
III. (i) Draw eight circles.
(ii) Draw eight triangles.
(iii) Draw eight squares.
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
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What is the difference in radii of two concentric circles given an angle and length of a triangle that is inscribed in the annulus?
In relation to this geometric construction:
where D is the center of both circles, if the inner radius (x = length of line segments DA and DE), the angle φ = ∠CAB, and the length Δg of line segment AB are given, what is the length Δx of line segment AC?
-
@muad: The inner radius is non-zero, but if it were, then Δx—the length that I seek—would be equal to the given value Δg. I cannot assign a value to the inner radius, Δg, or φ because these are all given quantities. – Daniel Trebbien Sep 26 '10 at 15:19
Consider the triangle DAB. You know two sides and an angle. The length of the third side is $x + \Delta x$.
Notice that we have, $\angle DAB=\pi-\phi$, $AD=x$, $AB=\Delta g$ & $BD=x+\Delta x$
Now, applying Cosine rule in $\Delta ABD$ as follows $$\cos \angle DAB=\frac{(AD)^2+(AB)^2-(BD)^2}{2(AD)(AB)}$$ $$\cos (\pi-\phi)=\frac{(x)^2+(\Delta g)^2-(x+\Delta x)^2}{2(x)(\Delta g)}$$ $$\implies (x+\Delta x)^2=x^2+(\Delta g)^2+2x(\Delta g)\cos\phi$$ $$\implies (x+\Delta x)=\pm \sqrt{x^2+(\Delta g)^2+2x(\Delta g)\cos\phi}$$ $$\implies \Delta x=\pm \sqrt{x^2+(\Delta g)^2+2x(\Delta g)\cos\phi}-x$$ But $\Delta x>0$, Hence, we get $$\color{blue}{\Delta x=\sqrt{x^2+(\Delta g)^2+2x(\Delta g)\cos\phi}-x}$$
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# 13732
## 13,732 is an even composite number composed of two prime numbers multiplied together.
What does the number 13732 look like?
This visualization shows the relationship between its 2 prime factors (large circles) and 6 divisors.
13732 is an even composite number. It is composed of two distinct prime numbers multiplied together. It has a total of six divisors.
## Prime factorization of 13732:
### 22 × 3433
(2 × 2 × 3433)
See below for interesting mathematical facts about the number 13732 from the Numbermatics database.
### Names of 13732
• Cardinal: 13732 can be written as Thirteen thousand, seven hundred thirty-two.
### Scientific notation
• Scientific notation: 1.3732 × 104
### Factors of 13732
• Number of distinct prime factors ω(n): 2
• Total number of prime factors Ω(n): 3
• Sum of prime factors: 3435
### Divisors of 13732
• Number of divisors d(n): 6
• Complete list of divisors:
• Sum of all divisors σ(n): 24038
• Sum of proper divisors (its aliquot sum) s(n): 10306
• 13732 is a deficient number, because the sum of its proper divisors (10306) is less than itself. Its deficiency is 3426
### Bases of 13732
• Binary: 110101101001002
• Base-36: ALG
### Squares and roots of 13732
• 13732 squared (137322) is 188567824
• 13732 cubed (137323) is 2589413359168
• The square root of 13732 is 117.1836166023
• The cube root of 13732 is 23.9466407133
### Scales and comparisons
How big is 13732?
• 13,732 seconds is equal to 3 hours, 48 minutes, 52 seconds.
• To count from 1 to 13,732 would take you about three hours.
This is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)
• A cube with a volume of 13732 cubic inches would be around 2 feet tall.
### Recreational maths with 13732
• 13732 backwards is 23731
• The number of decimal digits it has is: 5
• The sum of 13732's digits is 16
• More coming soon!
#### Copy this link to share with anyone:
MLA style:
"Number 13732 - Facts about the integer". Numbermatics.com. 2023. Web. 5 December 2023.
APA style:
Numbermatics. (2023). Number 13732 - Facts about the integer. Retrieved 5 December 2023, from https://numbermatics.com/n/13732/
Chicago style:
Numbermatics. 2023. "Number 13732 - Facts about the integer". https://numbermatics.com/n/13732/
The information we have on file for 13732 includes mathematical data and numerical statistics calculated using standard algorithms and methods. We are adding more all the time. If there are any features you would like to see, please contact us. Information provided for educational use, intellectual curiosity and fun!
Keywords: Divisors of 13732, math, Factors of 13732, curriculum, school, college, exams, university, Prime factorization of 13732, STEM, science, technology, engineering, physics, economics, calculator, thirteen thousand, seven hundred thirty-two.
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baltazar (kmb2869) – HW1 – ditmire – (58216) 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fly is crawling on an adjacent wall at a point having coordinates (3 . 2 m , 2 . 4 m), what is the distance of the fly from the corner of the room? Correct answer: 4 m. Explanation: Let : Δ x = 3 . 2 m and Δ y = 2 . 4 m . Using the Pythagorean Theorem, d = radicalBig x ) 2 + (Δ y ) 2 = radicalBig (3 . 2 m) 2 + (2 . 4 m) 2 = 4 m . 002 10.0 points When an object falls through air, there is a drag force ( with dimension M · L / T 2 ) that depends on the product of the surface area of the object and the square of its velocity; i.e. , F air = C A v 2 , where C is a constant. What is the dimension for constant C ? 1. [ C ] = M T · L 2 2. [ C ] = T · L M 3. [ C ] = T M 4. [ C ] = M L 3 correct 5. [ C ] = M T 2 · L 2 6. [ C ] = M T 7. [ C ] = T 2 · L M 8. [ C ] = T 2 · L 2 M 9. [ C ] = M L 2 10. [ C ] = T · L 2 M Explanation: [ F ] = M · L / T 2 , [ A ] = L 2 , and [ v ] = L / T , so [ C ] = [ F ] [ A ] [ v ] 2 = M · L / T 2 L 2 · (L / T) 2 = M · L T 2 · T 2 L 4 = M L 3 . 003 10.0 points Consider a cube of soft, spongy material. Which piece below has the larger density? 1. cutting out a piece of the cube that has one-eighth the volume 2. Unable to determine 3. compressing the cube until it has one- eighth the volume correct 4. Densities are the same. Explanation:
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baltazar (kmb2869) – HW1 – ditmire – (58216) 2 ρ 1 = m V Compressing the cube results in a denser ma- terial. Compared to the piece cut out, the compressed piece has density ρ 2 = m 1 8 V = 8 m V = 8 ρ 1 . 004 (part 1 of 3) 10.0 points There are roughly 10 59 neutrons and protons in an average star and about 10 11 stars in a typical galaxy. Galaxies tend to form in clus- ters of (on the average) about 10 3 galaxies, and there are about 10 9 clusters in the known part of the Universe. Approximately how many neutrons and protons “#” are there in the known Universe? 1. # 10 43 2. # 10 87 3. None of these 4. # 10 52 5. # 10 47 6. # 10 82 correct Explanation: Let : N n = 10 59 , N s = 10 11 , N g = 10 3 and N c = 10 9 . The number of particles in the observable Universe equals the product of the numbers of particles in each astrophysical unit N nU = N n N s N g N c = ( 10 59 ) ( 10 11 ) ( 10 3 ) ( 10 9 ) = 10 82 neutrons and protons . 005 (part 2 of 3) 10.0 points Suppose all this matter were compressed into a sphere of nuclear matter such that each nuclear particle occupied a volume of 1 . 401 × 10 45 m 3 (which is approximately the “volume” of a neutron or proton). What would be the radius of this sphere of nuclear matter? 1. R 10 12 m correct 2. R 10 14 m 3. R 10 23 m 4. R 10 35 m 5. R 10 25 m 6. None of these Explanation: Let : V p = 1 . 401 × 10 45 m 3 The volume of the sphere would equal the product of the number of protons and neu- trons in the observable Universe and the vol- ume of such particles; i.e. , N nU V p = 4 3 π r 3 r = parenleftbigg 3 N nU V p 4 π parenrightbigg 1 3 = bracketleftBigg 3 ( 10 82 ) ( 1 . 401 × 10 45 ) 4 π bracketrightBigg 1 3
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My Ferris Wheel
(Part 1)
By Javiera Fernandez
Description of the wheel
The wheel has a radius of 5 meters. The center
of the wheel is 8 meters from the ground and 12 meters from the wall. The wheel spins 0.6284 meters per second. It takes 50 seconds for a full spin (the wheel has a 31.42 meters
circumference). People get on the ride at the
lowest point, 3 meters away from the ground. I will be depicting the distance from the ground.
The Wheel
Characteristics
The size:
Diameter: 10 meters
Location of center:
8 meters to the ground 12 meters to the wall
More Characteristics
Rotation Speed
0.6284 meters per second
50 seconds per spin
Where do people get on the Ferris wheel
At the point closer to the ground (minimum value)
How many spins
Everyone gets 3 spins
The rules
Sine function:
Y=5 sin pi/25 (x-12.5) +8
Cosine function:
Y= -5 cos pi/25x +8
Validating the rules
I will validate the rules by checking a point
that we already know should be there: (25,13)
Sine function Y= 5 sin pi/25 (x-12.5)+8 Y= 5 sin pi/25 (25-12.5)+8 Y= 13
Cosine function Y= -5 cos pi/25 x+8 Y= -5 cos pi/25 (25)+8 Y= 13
So now we know that both our rules work
The Graph
The Graph
Max: 13 (diameter + the
ground)
distance from the end of the wheel to the ground) Min: 3 (the distance from the end of the wheel to the
The Graph
L.O.O.= Max+Min/2 = 8
spin
Period= Total length of one
The Graph
the total amount of time
the ride lasts = 50 x 3 =
150 seconds
Total length of the graph=
My work(validating the graph)
We are going to validate the graph by using
the formula to find a point and then locate it on the graph: What is the y value when x is
37.5
Sine function:
Y=5 sin pi/25 (x-12.5) +8
Y= 5 sin pi/25 (37.5-12.5) + 8
Y= 8
My work(validating the graph)
We can see that in the graph we have the
point (37.5,8)
(37.5,8)
Problem #2
(Part 2)
By Javiera Fernandez
The Problem
40 cm
The information we are given
Cycle begins at the bottom of the cylinder,
goes to the top of the cylinder and then returns to the bottom to complete the cycle
The piston does 20 cycles every second
The cylinder is 40 cm tall
What we are looking for
The rule for the situation
Define the variables
Graph of the situation (label axes)
Solution to the problems:
- How far from the bottom of the cylinder will the piston be at 3.08 seconds?
- Find two different times at which the piston is 20 cm from the bottom of the cylinder
Table of Values
Table of values explanation
I got the table of value by knowing that there
is 20 movements to the min, max and min in one second. I multiplied 20 by 3 to have the
time divided exactly to the points where the
function reaches a max or min. So we know have the amount of time by sixtieth of a
second. Every 3/60 of a second represent a
full cycle. I divided the time between each min
and max to know where the point would be in
the l.o.o.
My work (the variables)
A= Amplitude
*Max-Min/2 40-0/2 = 20
Period: The length of one full cycle = 3/60 of a
second
B= 2pi/period 2pi/(3/60) = 125.66
H: The distance from the y axis to where the cos,
sin, -cos or sin start. In this case I will use cos (umbrella) and it starts right at the y axis. H=0
K= l.o.o. As we can see in the table of values:
L.o.o. = 20 *Max + Min/2 40+0/2 = 20
My work(The equation)
From the variables stated in the prior slide we
can form our equation Y=-a cos b(x-h)+k Y= -20 cos 125.66 x + 20
My work (Validating rule)
To make sure our rule is right we are going to
test a value we already know and see if the rule gives us the right answer: (1.5/60, 40) Y= -20 cos 125.66 x + 20 Y= -20 cos 125.66 (1.5/60) + 20
Y= 40
So we now know that our rule is fine
My work (The graph)
/60
/60
/60
/60
/60
/60
/60
My work(validating the graph)
We are going to validate the graph by revising
to see that we have the right max, min, l.o.o.
and period
Max: 40
Period: 3/60 seconds
/60
/60
/60
/60
/60
/60
l.o.o.: 20
/60
Min: 0
Solutions to the problems
How far from the bottom of the cylinder will
the piston be at 3.08 seconds? Y= -20 cos 125.66 x + 20 Y= -20 cos 125.66 (3.08) + 20 Y= -20 cos 387.0328 + 20
Y= -20 x -.81567 + 20
Y= 16.31 + 20
Y= 36.31 centimetres from the ground
Solutions to the problems
Find two different times at which the piston is
20 cm from the bottom of the cylinder Here we can refer back to our table of values
Solutions to the problems (continued)
We can see in the table of values that it is at 20
centimeters from the ground at .75/60 and 2.25/60
Just to make sure this answer is right we are going to
check it using our formula
.75/60
Y= -20 cos 125.66 x + 20
Y= -20 cos 125.66 (.75/60)+20
Y= 20
2.25/60
Y= -20 cos 125.66 x + 20
Y= -20 cos 125.66 (2.25/60)+20
Y= 20
Conclusion
How far from the bottom of the cylinder will
the piston be at 3.08 seconds? 36.31 centimetres from the ground
Find two different times at which the piston is 20 cm from the bottom of the cylinder
When y= .75/60 and 2.25/60
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# college algebra
posted by .
write the function whose grapg is the graph of y=(x+5)^2, but is reflected about the y-axis.
would it look like this: y=(x+5)-2??
• college algebra -
Nope. Replace x by -x:
y = (-x+5)^2
y = (x-5)^2
The vertex moves from x = -5 to x = +5
• college algebra -
Thnaks Steve!!
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Cody
# Problem 44359. 5th Time's a Charm
Solution 1335583
Submitted on 11 Nov 2017 by Svyatoslav Golousov
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
x = 1; y1 = fifth_times_a_charm(x); assert(~isequal(y1,x)) y2 = fifth_times_a_charm(x); assert(~isequal(y2,x)) assert(abs(x-y2)<abs(x-y1)) y3 = fifth_times_a_charm(x); assert(~isequal(y3,x)) assert(abs(x-y3)<abs(x-y2)) y4 = fifth_times_a_charm(x); assert(~isequal(y4,x)) assert(abs(x-y4)<abs(x-y3)) y5 = fifth_times_a_charm(x); assert(isequal(y5,x))
y = 5 y = 4 y = 3 y = 2 y = 1
2 Pass
x = -1; y1 = fifth_times_a_charm(x); assert(~isequal(y1,x)) y2 = fifth_times_a_charm(x); assert(~isequal(y2,x)) assert(abs(x-y2)<abs(x-y1)) y3 = fifth_times_a_charm(x); assert(~isequal(y3,x)) assert(abs(x-y3)<abs(x-y2)) y4 = fifth_times_a_charm(x); assert(~isequal(y4,x)) assert(abs(x-y4)<abs(x-y3)) y5 = fifth_times_a_charm(x); assert(isequal(y5,x))
y = 3 y = 2 y = 1 y = 0 y = -1
3 Pass
x = 42; y1 = fifth_times_a_charm(x); assert(~isequal(y1,x)) y2 = fifth_times_a_charm(x); assert(~isequal(y2,x)) assert(abs(x-y2)<abs(x-y1)) y3 = fifth_times_a_charm(x); assert(~isequal(y3,x)) assert(abs(x-y3)<abs(x-y2)) y4 = fifth_times_a_charm(x); assert(~isequal(y4,x)) assert(abs(x-y4)<abs(x-y3)) y5 = fifth_times_a_charm(x); assert(isequal(y5,x))
y = 46 y = 45 y = 44 y = 43 y = 42
4 Pass
x = i; y1 = fifth_times_a_charm(x); assert(~isequal(y1,x)) y2 = fifth_times_a_charm(x); assert(~isequal(y2,x)) assert(abs(x-y2)<abs(x-y1)) y3 = fifth_times_a_charm(x); assert(~isequal(y3,x)) assert(abs(x-y3)<abs(x-y2)) y4 = fifth_times_a_charm(x); assert(~isequal(y4,x)) assert(abs(x-y4)<abs(x-y3)) y5 = fifth_times_a_charm(x); assert(isequal(y5,x))
y = 4.0000 + 1.0000i y = 3.0000 + 1.0000i y = 2.0000 + 1.0000i y = 1.0000 + 1.0000i y = 0.0000 + 1.0000i
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# geometric sequence. Determine the numbers a, b if 3 ,a ,b , 24 is a geometric sequence.
The consecutive terms of a geometric sequence have a common ratio.
We have the sequence given by 3 ,a ,b , 24
24/b = b/a = a/3
=> 24a = b^2 or a = b^2/24
substitute in b/a = a/3
=> 24b/b^2 = b^2/3*24
=> 24/b = b^2/24*3
=> b^3 = 24*24*3
=> b = 3^3*8*8
=> b = 3*4
=> b = 12
a = 144/24 = 6
The value of a = 6 and b = 12
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## Calculus: The Derivative
Since I am finally TAing for a calculus class it seemed like a good time to write about this topic. Unfortunately I still can’t make nice pictures and a good visualization is extremely useful in intuitively understanding calculus.
Before taking calculus students are taught the equations which represent lines on a graph, i.e. $y = mx+b$. The letter “m” represents the important information, the slope of the line. The slope is just how quickly the line rises compared to how far across it goes. A line with slope 2, would go up 2 units for every 1 unit it moved to the right. A line with slope $\frac{5}{13}$ would move up 5 units for every 13 units it moves to the right. Hence the familiar idea of “rise over run.”
Higher slopes indicate steeper lines. This is because the line is rising much faster than it is moving to the right. Negative slope just means going to the down and to the right. A line with slope -5 will move down 5 units and to the right one unit.
After learning how to graph, students will typically learn a bunch of other seemingly random stuff, but slope is actually the most important idea. One of the random things is the mysterious function called the “difference quotient.” The difference quotient involves a starting function, f(x), and doing what seems like an arbitrary mash up of things.
Given a function f(x), the difference quotient of f(x) is the function, $\frac{f(x+h)-f(x)}{h}$. It looks like gibberish to the untrained eye but it is secretly the slope, and here’s why.
First, $f(x+h)$ means to replace x with “x+h” in the function f(x). For example, if the function was $f(x) = \sin(x)$ then $f(x+h) = sin(x+h)$. Replacing the input with “x+h” is supposed to represent plugging in slightly more than x into the function. More precisely, plugging in “h more than x”.
For example if $f(x) = x^2 + 5$ then $f(1) = 6$ and $f(1+h) = (1+h)^2 + 5 = 1 + 2h + h^2 + 5 = 6 + 2h + h^2$. The idea here is that adding just a little more to 1, namely h, affects the function much differently than simply adding h to the end result, 6. In fact we can see here that by adding h into the input of the function, we get in general $f(x+h) = x^2 + 2xh + h^2 + 5$. So again, adding h into the input, can have a drastic affect on the output.
Now that we understand $f(x+h)$ we can look closely at $f(x+h) - f(x)$. Again, starting with an example makes it easier to see.
Let $f(x) = x^2 + 5$ and lets use an actual x and an actual h. How about, $x = 5$ and $h = 1$. Then we get:
$f(x+h) -f(x)= f(5 + 1) -f(5)$ and stop there. We are just subtracting two outputs, aka two y-values. This should remind you of calculating the slope between two points, $\frac{f(x_1)-f(x_2)}{x_1-x_2}$. Here the first point is given, and the second point comes from adding a little bit more to the first point, (h more).
Right now we have $f(x+h)-f(x)$ and to get a slope equation we need to divide by the two inputs subtracted, just like with the slope between two points we subtract the two x-values in the denominator. Well what are the two inputs here? “x+h” and “x”. Therefore, the slope between the two points $(x,f(x))$ and $(x+h,f(x+h))$ is given by
$\frac{f(x+h)-f(x)}{x+h-x} = \frac{f(x+h)-f(x)}{h}$ This is the difference quotient! So this mysterious equation was secretly the slope equation written in a special way all along. It will become clear later why we write it in this way, but for now just know that it is the slope between the points, $(x,f(x))$ and $(x+h,f(x+h))$.
Students don’t typically dissect this formula in a good way so they just forget about it. Then much later they learn about the idea of limits, which really should be shown right after slope. Limits can take a lot of rigorous explaining which I won’t go through. Instead I will talk about them in a very informal way.
Think of a function, represented by a graph, as “moving” as you plug in values. As you plug in $x= 1,2,3,4...$ what is happening to the graph? For something simple like a line, $f(x) = 2x+1$ you can easily visualize as the input x moves left and right, that the line follows what it should. The idea of seeing what happens to the function as the input x moves around is what limits are all about.
We represent this with the following notation, $\lim_{x \rightarrow c} f(x)$. This means, “the limit of f(x) as x approaches c”. We are looking at what the function does, as the inputs approach some value c. Note that we don’t care what actually happens when plugging c into f(x), but rather we care about what the function is doing as the input gets close to the value c. The typical example is a piece-wise function, i.e. a function that consists of putting other functions together. We can have functions approach some limit, L, as the inputs approach something, c, but the actual value of the function at c, f(c), is not equal to L. This can be hard to grasp and leads to the idea of continuity, but that is a topic for another time. So instead of worrying about that, just think of limits of functions as what happens to outputs as the input goes toward a certain number.
So, they teach you this thing called the difference quotient, and then they later teach you this idea of limits. The logical step is to put the two topics together in a smart way.
What happens to $\frac{f(x+h)-f(x)}{h}$ when we move h around? Consider x as just some number, and we are not moving it around. It is immovable. Think of x as a fixed point and we add the value h to it. Well, we already know that the difference quotient will give us the slope between the two points $(x,f(x))$ and $(x+h,f(x+h))$. What happens when we make h get smaller and smaller? This means we are adding less and less to the fixed point x. This corresponds to the slope of two points, where the second point is getting closer to the stationary one. Now imagine we keep making h smaller and smaller, in fact we want to make it as small as possible. We can’t make it be exactly zero, or else we would be dividing by zero. However, we can take the limit of the function as h goes toward zero. This would be written as:
$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$
When h goes to zero, we can think of the second point, $(x+h,f(x+h))$ as actually right on top of the point $(x,f(x))$. So if we could actually take this limit to when h becomes zero, we would get the slope of two points which are actually the same point. Thus we magically have the slope of a line going through a single point instead of two points! Note I said “if we could actually take this limit.” This is because the limit does not have to exist in the first place, but when it does we have this neat idea of a slope of a line going through a single point.
This slope is tied to the original function we started with. So it’s not exactly just a slope of a line going through some random point, but instead is the slope of a line related to f(x). We have a special way of saying what this thing is. It is the slope of the line tangent to the graph, at the point x. A tangent line is a line that barely touches the graph once (a topic for another day).
In practice what happens is that you are given some function. Then you get the difference quotient and if it is a “good” function you will be able to manipulate it in a way that gets rid of the h in the denominator. Assuming you do mathematically correct manipulations (multiply by 1 or add zero) the end result will have the same limit that the non manipulated equation had. Then because it no longer has the risk of zero being the denominator, you will be able to plug in $h = 0$ directly to find the exact limit as h approaches zero.
After doing this entire process we finally have what is known as the derivative of a function!
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6740
6,740 (six thousand seven hundred forty) is an even four-digits composite number following 6739 and preceding 6741. In scientific notation, it is written as 6.74 × 103. The sum of its digits is 17. It has a total of 4 prime factors and 12 positive divisors. There are 2,688 positive integers (up to 6740) that are relatively prime to 6740.
Basic properties
• Is Prime? No
• Number parity Even
• Number length 4
• Sum of Digits 17
• Digital Root 8
Name
Short name 6 thousand 740 six thousand seven hundred forty
Notation
Scientific notation 6.74 × 103 6.74 × 103
Prime Factorization of 6740
Prime Factorization 22 × 5 × 337
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) 3370 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 6,740 is 22 × 5 × 337. Since it has a total of 4 prime factors, 6,740 is a composite number.
Divisors of 6740
1, 2, 4, 5, 10, 20, 337, 674, 1348, 1685, 3370, 6740
12 divisors
Even divisors 8 4 4 0
Total Divisors Sum of Divisors Aliquot Sum τ(n) 12 Total number of the positive divisors of n σ(n) 14196 Sum of all the positive divisors of n s(n) 7456 Sum of the proper positive divisors of n A(n) 1183 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 82.0975 Returns the nth root of the product of n divisors H(n) 5.69738 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 6,740 can be divided by 12 positive divisors (out of which 8 are even, and 4 are odd). The sum of these divisors (counting 6,740) is 14,196, the average is 1,183.
Other Arithmetic Functions (n = 6740)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 2688 Total number of positive integers not greater than n that are coprime to n λ(n) 336 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 872 Total number of primes less than or equal to n r2(n) 16 The number of ways n can be represented as the sum of 2 squares
There are 2,688 positive integers (less than 6,740) that are coprime with 6,740. And there are approximately 872 prime numbers less than or equal to 6,740.
Divisibility of 6740
m n mod m 2 3 4 5 6 7 8 9 0 2 0 0 2 6 4 8
The number 6,740 is divisible by 2, 4 and 5.
• Arithmetic
• Abundant
• Polite
Base conversion (6740)
Base System Value
2 Binary 1101001010100
3 Ternary 100020122
4 Quaternary 1221110
5 Quinary 203430
6 Senary 51112
8 Octal 15124
10 Decimal 6740
12 Duodecimal 3a98
20 Vigesimal gh0
36 Base36 578
Basic calculations (n = 6740)
Multiplication
n×i
n×2 13480 20220 26960 33700
Division
ni
n⁄2 3370 2246.67 1685 1348
Exponentiation
ni
n2 45427600 306182024000 2063666841760000 13909114513462400000
Nth Root
i√n
2√n 82.0975 18.8895 9.06077 5.83085
6740 as geometric shapes
Circle
Diameter 13480 42348.7 1.42715e+08
Sphere
Volume 1.28253e+12 5.7086e+08 42348.7
Square
Length = n
Perimeter 26960 4.54276e+07 9531.8
Cube
Length = n
Surface area 2.72566e+08 3.06182e+11 11674
Equilateral Triangle
Length = n
Perimeter 20220 1.96707e+07 5837.01
Triangular Pyramid
Length = n
Surface area 7.86829e+07 3.60839e+10 5503.19
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# How can I create 2D projections from a 3D object?
130 views (last 30 days)
Eric on 10 May 2016
Commented: Seung Jae Lee on 5 Apr 2023
I know I can use "radon" for creating 1D projections from 2D, but how can I get 2D projections from 3D objects (such as from a 3D file like an STL file).
Mike Garrity on 10 May 2016
One approach is the technique I showed in answers to this question, and this question.
The basic idea is that you take keep the faces of your geometry, but project the vertices onto the plane. That does mean that you've got multiple conincident polys, so it's not good for things like area computation. But if you just want the visual result, it works well and it's simple.
##### 2 CommentsShow 1 older commentHide 1 older comment
Naveen Pathak on 7 Jun 2021
Hi Mike,
I have a simliar problem. Please, could you help me to plot the following:
Thank you.
Y.S. on 29 Mar 2023
2 methods I found that I want to place here for future reference:
[1] fast, but only works for convex shapes. Assuming you have a struct (fvIn) with vertices & faces
fvIn.vertices(:,3)=0; % squash all Z coords
verts = fvIn.vertices;
faces = fvIn.faces;
a = verts(faces(:, 2), :) - verts(faces(:, 1), :); % compute area of all triangles
b = verts(faces(:, 3), :) - verts(faces(:, 1), :);
c = cross(a, b, 2);
area2 = 1/2 * sum(sqrt(sum(c.^2, 2))); % Calculate total area, but this gives double the area because the shape is squashed
Ap = area2/2; %
[2] slow, but works for all shapes
Loop over all triangles, project them on the Z=0 plane, create a polyshape and combine(union) it with the other projected triangles
P = fvIn.vertices(fvIn.faces(1,:),1:2);
psSurfTot = polyshape(P);
for N = 2:size(fvIn.faces,1)
P = fvIn.vertices(fvIn.faces(N,:),1:2);
psSurfTMP = polyshape(P);
psSurfTot = union(psSurfTot,psSurfTMP);
end
% calculate projected area
Ap = area(psSurfTot);
I am still looking for a fast method that works for all shapes, but havent found any
##### 2 CommentsShow 1 older commentHide 1 older comment
Seung Jae Lee on 5 Apr 2023
Hi, Naveen. How could you solve this problem? Can you please share your findings?
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## Friend of our site
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# How To Calculate Index Of Agreement In Excel
By Zach Arnold | December 10, 2020
Thank you for solving the problem. I learned a lot from reading your contributions and it is an excellent page. Congratulations For adjusting the rarest errors, we bill root Mean Square Error with Data Tool. By scouring errors before calculating their average, and then taking the square root of the average, we reach a measure of error size that gives more weight to large but rare errors than average. We can also compare Root Mean Square Error and Mean Absolute Error to determine if the prognosis contains significant but rare errors. The larger the difference between Root Mean Square Error and Mean Absolute Error, the more inconsistent the error size. 2) If I can calculate separate kappas for each subcategory, how can I calculate Kappa Cohens for the category (which contains z.B 3 subcategory)? Do I have to sum up the kappa values and divide them by the number of subcategory? Based on the initial measurement values x 0 “displaystyle x_{0}”, the final values observed or measured x `displaystyle x_`m` and the final calculated values x c `displaystyle x_`c`, there are several adaptable quality statistics that can be calculated. The definition of some of the most common uses is given below. The average distortion error is generally not used as a measure of pattern error, as high individual errors in the forecast can also produce a low MBE. The average distortion error is primarily used to estimate the average distortion of the model and to determine whether measurements are needed to correct the model distortion. The average Bias Error (MBE) records the average distortion in the forecast and is calculated as follows: I have a large sheet and I want to calculate the presence of each student for a certain period of time. For example, present and 0-absence, I want to summarize every student who is present from date10 to date80. What do you mean? Date: date1, date2, date3, date4 …
Date99, date100 student1, 1.1,0,1 … 1.1 students2, 1.0,1.0 … 0.1 SDR is a measure of dynamic correspondence. Smaller values suggest better consistency. RMSE, ME, STD are related to the following formula: Oi refers to observational data and Pi displays forecast or modeling data. We applied To Mean Absolute Error, Root Mean Square Error, Index of Agreement and Nash Sutcliffe Efficiency to verify the accuracy of CMIP5 output models, i.e. maximum and minimum temperatures and precipitation. For a better understanding, see Die Ullah et al. (2018). These criteria were also used by Salehnia et al. (2017) to assess the performance of AgMERRA precipitation data. Alex, Q1 – Yes, Cohen`s Kappa is used for this purpose with Q2-grade data — it really depends on what you expect from the Cohens Kappa aggregation.
You can use the average kappas to represent the average kappa. You can use the minimum of kappas to represent the most pessimistic chord, etc. Charles I. has a lot of data in which I try to calculate weeks of supply. I look for a formula that goes to a specified cell (which contains my construction for a particular category), then multiplies the sales of the last few weeks by this build. If it`s not equal to 0, I want it to do the same thing and move it to the nearest cell and make the same formula. Then continue until my result is zero, and finally, count all the cells that made this formula, which gives me my weeks of supply. I have a table with line 1, which dates from July to the end of October, and line 2 indicates whether a person is working (1) or not (0).
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# Help to understand the basis for a dual space
I've been introduced to the concept of dual space in linear algebra. I can understand perfectly that the dual space of the space $V$ is a space $V^*$ made of all possible linear maps from $V$ to $\mathbb{R}$. So, for example, let $V$ be $\mathbb{R^3}$, then we have, as elements of the dual space, for example:
$$F(x,y,z) = x + y\\F(x,y,z) = -x + z\\F(x,y,z) = 3z\\F(x,y,z) = 3x + 4y + 5z\\ \cdots$$
What I don't understand, it's why these things called functionals, span the dual space. I've seen a proof but I didn't understand. I know (at least have an intuition) that the dual space can be represented as the space of all possible linear combinations of $x,y,z$ like:
$$F(x,y,z) = ax + by + cz$$
but how to prove that this is suficient to generate the entire space? And why that rule that maps to $0$ and $1$ form a basis to this space?
Sorry by all these questions, but this concept seemed a lot strange for me, and I can't understand why dual spaces and finding its basis are so important.
• Perhaps it's a typo, but the dual space for a vector space $V$ over $\mathbb{R}$ should be all linear maps $V \to \mathbb{R}$. – Maanroof Jan 29 '15 at 21:58
• @Maanroof thanks, fixed – Guerlando OCs Jan 29 '15 at 22:02
• Does it also help your understanding? Assuming we are talking finite-dimensional vector spaces, am I correct you want to know why a dual basis is a basis for $V^*$? – Maanroof Jan 29 '15 at 22:05
• @Maanroof yes, I want to know why those linear maps defined that way (that $0$ and $1$ thing) spans the dual space. – Guerlando OCs Jan 29 '15 at 22:07
• Well, you need to show two things then: linearly independence and the fact that they span the whole space. For the first, fix a basis $\mathcal{B} = e_1,e_2,...,e_n$ of our space $V$ over $k$, and let $e^1,e^2,....,e^n$ be the associated dual basis. Take an linear combination $\sum_{i=1}^n \lambda_i e^i$ equal to zero and deduce that all $\lambda_i$'s must be zero. As I recall this is fairly straightforward (using the $\delta_i^j$'s). For the second point, use that an arbitrary $\varphi \in V^*$ is linear, and consider what it does on $\mathcal{B}$. – Maanroof Jan 29 '15 at 22:15
Any linear map $$fu$$ from $$V=\mathbf R^3$$ to $$\mathbf R$$ is determined by its values on the vectors of a base $$\mathcal B =(e_1, e_2, e_3)$$. For if $$v=\lambda e_1+\mu e_2+\nu e_3$$, then $$f(v)=\lambda f(e_1)+\mu f(e_2)+\nu f(e_3)$$.
Now if $$f(e_1)=\alpha_1$$, $$f(e_2)=\alpha_2$$, $$f(e_3)=\alpha_3$$ and if $$e_1^*, e_2^*,e_3^*$$ is the dual basis of $$\mathcal B$$, it's easy to check that $$f=\alpha_1 e_1^*+\alpha_2 e_2^*+\alpha_3 e_3^*$$ since both sides take the same value for $$\,e_1,e_2,e_3$$.
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# How to do the calculations for a Wheatstone Bridge
songoku
I know at balanced condition, the current flowing through galvanometer = 0 and R1 . Rx = R2 . R3
What I want to ask is how to analyze the circuit until we can conclude the current will be zero. I have learned about Kirchhoff's 1st and 2nd law and also about potential divider. I am guessing there is something to do with potential divider. By changing the value of R2, we can get VD = VB so no current flowing through galvanometer but how to do the calculation / analysis? Can we do it by using Kirchhoff's 1st and 2nd law and potential divider or we need something else beyond high school scope?
Thanks
Last edited by a moderator:
All you need is Ohm's law. Take the voltage at point C to be 0, and the voltage at point A to be V, which is the voltage supplied by the battery. Some current I3 flows through R3. So the voltage at point B is VB=V - I3*R3. The current I3 all flows through Rx, since no current flows through the galvanometer. So the current at point B is also VB=I3*Rx. Eliminating I3 from these two equations gives that VB=V*Rx/(R3+Rx). Similarly on the other side, VD=V*R2/(R1+R2). When VB=VD, you can solve for Rx=R2*R3/R1.
songoku
All you need is Ohm's law. Take the voltage at point C to be 0, and the voltage at point A to be V, which is the voltage supplied by the battery. Some current I3 flows through R3. So the voltage at point B is VB=V - I3*R3. The current I3 all flows through Rx, since no current flows through the galvanometer. So the current at point B is also VB=I3*Rx. Eliminating I3 from these two equations gives that VB=V*Rx/(R3+Rx). Similarly on the other side, VD=V*R2/(R1+R2). When VB=VD, you can solve for Rx=R2*R3/R1.
Sorry I think I don't explain myself clearly. I mean how we can get VB = VD, not how to get Rx . R1 = R2 . R3
Thanks
Gold Member
For the out of balance case, there are a couple of choices. In one you assume that you have a very high impedance voltmeter (much larger than the other resistors), then you just calculated the voltages with two independent voltage dividers, as above.
For the more general case you can put an impedance in place of the galvanometer. You can then solve this circuit by labeling all of the branch currents (6 of them) and write a set of quite simple equations for the currents from KVL and KCL. It is a 6x6 matrix, but the elements are simple.
Sorry I think I don't explain myself clearly. I mean how we can get VB = VD, not how to get Rx . R1 = R2 . R3
Thanks
That's how the bridge works. You adjust the variable resistor R2 until the galvanometer reads 0. Maybe I don't understand your question.
2022 Award
Sorry I think I don't explain myself clearly. I mean how we can get VB = VD, not how to get Rx . R1 = R2 . R3
Thanks
If ##V_B\neq V_D## then there is a potential difference between ##B## and ##D## and current flows one way or another through the galvanometer. We balance the bridge so that no current flows, so we know ##V_B=V_D##.
Homework Helper
Gold Member
I suspect OP wants to start from the general case, that is replace the galvanometer with a resistor, find the current in that resistor using Kirchhoff rules and then see under what conditions that current is zero.
nasu, sophiecentaur and vanhees71
Gold Member
I mean how we can get VB = VD,
You may be looking at this in the 'wrong order'. The point about the condition VB = VD is that you have to adjust the resistors to achieve it. The values of the Rs will then fit the equation Rx . R1 = R2 . R3
You could say that VB=VD is an imposed condition.
songoku
That's how the bridge works. You adjust the variable resistor R2 until the galvanometer reads 0. Maybe I don't understand your question.
If ##V_B\neq V_D## then there is a potential difference between ##B## and ##D## and current flows one way or another through the galvanometer. We balance the bridge so that no current flows, so we know ##V_B=V_D##.
I am sorry. My question is just like what kuruman said (although the idea of changing galvanometer with a resistor did not cross my mind)
I suspect OP wants to start from the general case, that is replace the galvanometer with a resistor, find the current in that resistor using Kirchhoff rules and then see under what conditions that current is zero.
For the out of balance case, there are a couple of choices. In one you assume that you have a very high impedance voltmeter (much larger than the other resistors), then you just calculated the voltages with two independent voltage dividers, as above.
For the more general case you can put an impedance in place of the galvanometer. You can then solve this circuit by labeling all of the branch currents (6 of them) and write a set of quite simple equations for the currents from KVL and KCL. It is a 6x6 matrix, but the elements are simple.
Let say I change galvanometer to R5 and Rx to R4. I am thinking of stating I5 (current passing through R5) in terms of R1 to R5 then find what the condition is for I5 = 0
Can that be done by using KCL and KVL? I think I can write down three equations of KVL by using three different loops and maybe three equations using KCL (something like I1 = I2 + I5 , I4 = I3 + I5). By using algebra, can I state I5 in terms of R1 to R5 using this method? I have to deal with a lot of variables and it seems impossible to do that.
You may be looking at this in the 'wrong order'. The point about the condition VB = VD is that you have to adjust the resistors to achieve it. The values of the Rs will then fit the equation Rx . R1 = R2 . R3
You could say that VB=VD is an imposed condition.
I understand now it is an imposed condition. For balanced wheatstone bridge, it means current through galvanometer = 0 so it implies that VB = VD. Now I am looking for something more general about wheatstone bridge.
I apologize to everyone for not stating my question clearly.
Thanks
Gold Member
I have to deal with a lot of variables and it seems impossible to do that.
Not impossible. Maybe a little tedious. If you've solved systems of linear equations it will be familiar. Linear algebra is your friend here, but it's not necessary. You'll get one equation for every loop and one equation for every node; 7 equations, but only 6 unknown currents.
I understand now it is an imposed condition. For balanced wheatstone bridge, it means current through galvanometer = 0 so it implies that VB = VD.
@songoku , I think there is still a point that you are missing. An ideal voltmeter (galvanometer) has infinite resistance, so the current through the galvanometer is always zero, even if VB ≠ VD.
Gold Member
I have to deal with a lot of variables and it seems impossible to do that.
Now I am looking for something more general about wheatstone bridge.
If you can understand how do do the calculation for two or three resistors then there is no shame in reaching for a circuit analysis application do do it for more (hundreds, even). It's just a tool.
vanhees71
songoku
Not impossible. Maybe a little tedious. If you've solved systems of linear equations it will be familiar. Linear algebra is your friend here, but it's not necessary. You'll get one equation for every loop and one equation for every node; 7 equations, but only 6 unknown currents.
Ok I will try it again. I have not gotten 7 equations, only 5 equations so I am missing something
@songoku , I think there is still a point that you are missing. An ideal voltmeter (galvanometer) has infinite resistance, so the current through the galvanometer is always zero, even if VB ≠ VD.
I thought galvanometer is ammeter, for measuring current so ideal galvanometer will have zero resistance. So galvanometer is actually voltmeter?
If you can understand how do do the calculation for two or three resistors then there is no shame in reaching for a circuit analysis application do do it for more (hundreds, even). It's just a tool.
At first I wonder whether I need knowledge beyond what I know right now to do what I ask in this thread but it seems I just need to use KCL and KVL and the problem is my algebra skill is lacking.
Thanks
I thought galvanometer is ammeter, for measuring current so ideal galvanometer will have zero resistance. So galvanometer is actually voltmeter?
You are correct that a galvanometer is an ammeter and ideally has zero resistance. Ignore my comment. I though a Wheatstone bridge typically had a voltmeter in the middle.
Homework Helper
Gold Member
Ok I will try it again. I have not gotten 7 equations, only 5 equations so I am missing something
You are not missing anything. Starting with the diagram shown below, write 3 Kirchhoff voltage and two node current equations which will give you a system of 5 equations and 5 unknowns, the currents. I don't count the total current as unknown since ##I=I_1+I_2=I_3+I_4.##
The two current equations that I used are ##I_1+I_2=I_3+I_4## and ##I_1=I_3+I_{\text{AB}}##.
For the 3 voltage loops equations, be sure none of them is the sum of the other two. Good luck with solving the system.
Disclaimer
The system is one of those things that one has to do at least once in one's life. I have already done it many years ago, so I used Mathematica to get the solution. However, the answer for ##I_{\text{AB}}## makes eminent sense in retrospect.
DaveE
songoku
For the 3 voltage loops equations, be sure none of them is the sum of the other two.
This is where my mistake is. I will try doing it again
Thank you very much for all the help and explanation phyzguy, DaveE, Ibix, kuruman, sophiecentaur
sophiecentaur
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## Nick2019 3 years ago I need help!!!
1. Nick2019
2. cherio12
okay well the theorem is $z^2=x^2+y^2$ and you know everything has to be an integer (whole number)
3. cherio12
you can just try different values. For instance for the first one i might try z=51 and x=24. so 51^2=24^2+y^2. Solving i get y=45. So this works
4. Nick2019
can you try 19?
5. Nick2019
@cherio12 can you try #19 plz
6. cherio12
you just have to play with the numbers. I first tried $z^2=25^2+20^$ but i got z= 32.0156..so this didn't work next i tried $25^2=20^2+y^2$ y=15 (this is an integer) so it checks out
7. cherio12
does this make sense?
8. Nick2019
thats the answer in the book so i should be able to learn off your 2 answers
9. Nick2019
wait so on 18# i got 2601=576+y2 then do i get y by itself? @cherio12
10. cherio12
yes. you should go 2601-576=y^2. then take the squareroot of both sides to just get y
11. Nick2019
ok how do you know were to put the numbers on the equation?
12. cherio12
what do you mean?
13. Nick2019
wait oh i see you said move them around and it has to be a whole number
14. cherio12
yup, its always (leg)^2+(leg)^2=(hypotenuse)^2
15. Nick2019
thanks man! helped alot
16. cherio12
its commonly written as x^2+y^2=z^2
17. cherio12
no problem =D
18. Nick2019
k
19. Nick2019
wait so on # 20 i got 91.82592222ect
20. cherio12
okay so that means you picked the wrong combo of hypotenuse and legs. try another combo (by the way the hypotenuse will always be the largest of the 3 numbers)
21. Nick2019
so set it up as x2= 96 + 28?
22. cherio12
i would try $96^2=x^2+28^2$
23. Nick2019
do get x by its self?
24. cherio12
yes
25. cherio12
|dw:1359577137858:dw|
26. Nick2019
but i still will get that 91.etc right?
27. cherio12
i was just showing you the triangle to help show what i meant by x, y and z
28. Nick2019
i know im saying when i get x by its self and take 9216 and square it i get that 91.00202 number
29. cherio12
o you mean for the 96, 28 problem?
30. Nick2019
yes
31. cherio12
$96^2=x^2+28^2$ $9216=x^2+784$ $9216-784=x^2$ $x=\sqrt{8432}$ x=91.8259 so this way didn't work.
32. cherio12
i would then try $z^2=96^2+28^2$
33. Nick2019
but you would still get 91.8259 right?
34. cherio12
no i got 100
35. cherio12
you add 96^2 and 28^2 in this example instead of subtracting them (like we tried before)
36. Nick2019
ohhh
37. Nick2019
now i see thakns agian lol
38. Nick2019
hey can you help @cherio12 with # 23
39. cherio12
sure
40. cherio12
okay so i first tried z^2=72^2+75^2 But i got z=103.966 so that's not right then i tried 75^2=x^2+72^2
41. Nick2019
ok i did the first but didnt try that one what did you get?
42. cherio12
21
43. Nick2019
can you show how you did that?
44. cherio12
75^2=x^2+72^2 solve for x
45. Nick2019
ok thanks i got 21
46. cherio12
no problem
47. Nick2019
hey @cherio12 for #24 you you do the same?
48. cherio12
yes, your two legs are 6 and 3
49. Nick2019
so would it be 6(2)=3(2)+x2
50. Nick2019
@cherio12
51. cherio12
by(2) do you mean you are squaring it?
52. Nick2019
no 6 to the power 2 like 6*6
53. cherio12
that means squaring, but yes that is correct
54. Nick2019
wait that doesnt work
55. Nick2019
i tried both ways and got not a whole number
56. cherio12
I don't believe you are supossed to get an integer for this problem
57. Nick2019
so would 5.19 work?
58. Nick2019
@cherio12
59. cherio12
i got 7.937
60. cherio12
6^2+3^2=x^2
61. Nick2019
ya that was my second way
62. Nick2019
so would you set up # 25 and # 26 the same way?
63. cherio12
yes for 25, but for 26 you will use a ratio
64. cherio12
i gotta go eat..be back later
65. Nick2019
ok when you get back can you explain ratio?
66. Nick2019
@cherio12 you back?
67. cherio12
yes
68. cherio12
okay well for starters find the other leg on the 5,3 triangle
69. Nick2019
wait on the # 25 its just 11 and x so is it 121 then sqare root answer is 11?
70. cherio12
no, there are two 11 sides
71. Nick2019
oh i see cuase of that line thingy
72. Nick2019
73. cherio12
no for 25, it is 11^1+11^2=x^2
74. Nick2019
so its 15.49
75. cherio12
yes
76. cherio12
does that make sense why?
77. Nick2019
yes i just didnt try that way and remember there not whole numbers
78. Nick2019
so how do you do #26 is it 2 problems in 1
79. Nick2019
@cherio12
80. cherio12
find the other leg in the 5,3 triangle first
81. Nick2019
5.83
82. Nick2019
wait no its 4
83. cherio12
yes 4
84. Nick2019
now do the other triangel?
85. cherio12
now we can use like triangle ratios 4/5=7/x
86. Nick2019
so 4 sqare + 5 square = 7sqare?
87. cherio12
no, no square
88. cherio12
just ratio, just division
89. Nick2019
so 0.8=7
90. cherio12
4/5=7/x .8=7/x .8x=7
91. Nick2019
oh ok wow being sick for one day of notes can rweally heart you
92. cherio12
haha, no problem
93. Nick2019
wait # 28 its 2x and 2x+4 and the length of the hypot is 4x-4
94. Nick2019
@cherio12
95. cherio12
yes, just apply the Pythagorean theorem
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# Footing to support 20,000 pounds water
Question:
Hi ,how deep does concrete have to be for a 16 by 16 floor to hold 20,000 pounds of water for fish on half of it ?
Thanks for visiting all-concrete-cement.com. Assuming the 16 by 16 mention is 16 feet x 16 feet, with area of 256 square feet ( 16x16 = 256 square feet).
Assuming the 20,000 pounds are being distributed at the entire area of 256 square feet, the pressure exerted on the concrete will be 20,000/256 ( 20,000 divide by 256 square feet) which is equal to 78.12 pounds per square foot.
In other words, each square foot of the slab or footing needs to be able to withstand 78 pounds which is a small load for concrete.
In most cases, concrete of this type of application at minimum needs to be 4000 psi(pounds per square inch) mix. Converting 4000 psi to pound per foot i.e multiply 4000 by 144 (144 square inch in a square foot).
The answer is 576,000 pounds per square foot compared to 78.12 psf which is very small load to be supported by concrete slab or footing.
A 4 inch thick slab will be more than enough to be able to carry the weight but structure of this type usually have 10 to 12 inches thick slab or footing and it has two layers of rebars, one at top and one at bottom.
Since the structure is to sustain water, epoxy coated rebars are required. Should water finds its way to the reinforcements, corrosion will severly cut the life of the structure.
Since water sideway movement will cause the footings and walls to go from compression to tension, it is very common to see this type of concrete application have two layers of reinforcements at walls and slab.
Hope this helps
Good luck
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# ENGI 1313 Mechanics I - PowerPoint PPT Presentation
1 / 20
ENGI 1313 Mechanics I. Lecture 07:Vector Dot Product. Chapter 2 Objectives. to review concepts from linear algebra to sum forces, determine force resultants and resolve force components for 2D vectors using Parallelogram Law to express force and position in Cartesian vector form
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## ENGI 1313 Mechanics I
Lecture 07:Vector Dot Product
### Chapter 2 Objectives
• to review concepts from linear algebra
• to sum forces, determine force resultants and resolve force components for 2D vectors using Parallelogram Law
• to express force and position in Cartesian vector form
• to examine the concept of dot product
### Lecture 07 Objectives
• to examine the concept of dot product
### Overview of Dot Product
• Definition
• Laws of Operations
• Commutative law
• Scalar Multiplication
• Distributive law
### Overview of Dot Product (cont.)
• Dot Product of Cartesian Vectors
Go to zero
### Application of Dot Product
• Angle between two vectors
• Cables forces and the pole?
• and ?
Component magnitudes
If A||has + sense then same direction as u
^
### Application of Dot Product (cont.)
• Component magnitudeof A on a parallel or collinear linewith line aa
• Recall
Component A||
### Application of Dot Product (cont.)
• The vector A|| canbe determined by:
Vector A||
Application of Dot Product for Component A||
Multiply by Unit Vector ûto obtain Vector A||
### Application of Dot Product (cont.)
• For force vector F at Point A: What is the component magnitudeparallel (|F1|) to the pipe (OA)?
### Application of Dot Product (cont.)
• For force vector F at Point A: what is the component magnitudeperpendicular (F2) to the pipe (OA)?
• Method 1
• Method 2
### Comprehension Quiz 7-01
• The dot product of two vectors results in a _________ quantity.
• A) scalar
• B) vector
• C) complex number
• D) unit vector
A
### Example Problem 7-01
• For the Cartesian force vector, find the angle between the force vector and the pole, and the magnitude of the projection of the force along the pole OA
A
### Example Problem 7-01 (cont.)
• Position vector rOA
• Magnitude of |rOA|
• Magnitude of |F|
A
### Example Problem 7-01 (cont.)
• Find the angle between rOA and F
A
### Example Problem 7-01 (cont.)
• Find magnitude of the projection of the force F along the pole OA
### Comprehension Quiz 7-02
• If the dot product of two non-zero vectors is 0, then the two vectors must be ______ to each other.
• A) parallel (pointing in the same direction)
• B) parallel (pointing in the opposite direction)
• C) perpendicular
• D) cannot be determined.
### Comprehension Quiz 7-03
• The Dot product can be used to find all of the following except ____
• A) sum of two vectors
• B) angle between two vectors
• C) vector component parallel to a line
• D) vector component perpendicular to a line
### Comprehension Quiz 7-04
• Find the dot product (PQ) for
• A) -12 m
• B) 12 m
• C) 12 m2
• D) -12 m2
• E) 10 m2
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Example: marketing
# SAT Math Facts & Formulas Numbers, Sequences, …
SAT math Facts & Formulas Numbers, Sequences, Factors Integers: .. , -3, -2, -1, 0, 1, 2, 3, .. Reals: integers plus fractions, decimals, and irrationals ( 2, 3, , etc.). Order Of Operations: PEMDAS. (Parentheses / Exponents / Multiply / Divide / Add / Subtract). Arithmetic Sequences: each term is equal to the previous term plus d Sequence: t1 , t1 + d, t1 + 2d, .. The nth term is tn = t1 + (n 1)d Number of integers from in to im = im in + 1. Sum of n terms Sn = (n/2) (t1 + tn ) (optional). Geometric Sequences: each term is equal to the previous term times r Sequence: t1 , t1 r, t1 r 2 , .. The nth term is tn = t1 r n 1. Sum of n terms Sn = t1 (r n 1)/(r 1) (optional). Prime Factorization: break up a number into prime factors (2, 3, 5, 7, 11, .. ). 200 = 4 50 = 2 2 2 5 5. 52 = 2 26 = 2 2 13. Greatest Common Factor: multiply common prime factors 200 = 2 2 2 5 5. 60 = 2 2 3 5. GCF(200, 60) = 2 2 5 = 20.
SAT Math Facts & Formulas Averages, Counting, Statistics, Probability average = sum of terms number of terms average speed = total distance total time
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1 SAT math Facts & Formulas Numbers, Sequences, Factors Integers: .. , -3, -2, -1, 0, 1, 2, 3, .. Reals: integers plus fractions, decimals, and irrationals ( 2, 3, , etc.). Order Of Operations: PEMDAS. (Parentheses / Exponents / Multiply / Divide / Add / Subtract). Arithmetic Sequences: each term is equal to the previous term plus d Sequence: t1 , t1 + d, t1 + 2d, .. The nth term is tn = t1 + (n 1)d Number of integers from in to im = im in + 1. Sum of n terms Sn = (n/2) (t1 + tn ) (optional). Geometric Sequences: each term is equal to the previous term times r Sequence: t1 , t1 r, t1 r 2 , .. The nth term is tn = t1 r n 1. Sum of n terms Sn = t1 (r n 1)/(r 1) (optional). Prime Factorization: break up a number into prime factors (2, 3, 5, 7, 11, .. ). 200 = 4 50 = 2 2 2 5 5. 52 = 2 26 = 2 2 13. Greatest Common Factor: multiply common prime factors 200 = 2 2 2 5 5. 60 = 2 2 3 5. GCF(200, 60) = 2 2 5 = 20.
2 Least Common Multiple: check multiples of the largest number LCM(200, 60): 200 (no), 400 (no), 600 (yes!). Percentages: use the following formula to find part, whole, or percent percent part = whole 100. pg. 1. SAT math Facts & Formulas Averages, Counting, Statistics, Probability sum of terms average =. number of terms total distance average speed =. total time sum = average (number of terms). mode = value in the list that appears most often median = middle value in the list (which must be sorted). Example: median of {3, 10, 9, 27, 50} = 10. Example: median of {3, 9, 10, 27} = (9 + 10)/2 = Fundamental Counting Principle: If an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in N M ways. (Extend this for three or more: N1 N2 N3 .. ). Permutations and Combinations (Optional): The number of permutations of n things taken r at a time is n Pr = n!
3 /(n r)! . The number of combinations of n things taken r at a time is n Cr = n!/ (n r)! r! Probability: number of desired outcomes probability =. number of total outcomes The probability of two different events A and B both happening is P (A and B) = P (A) P (B), as long as the events are independent (not mutually exclusive). Powers, Exponents, Roots xa xb = xa+b xa /xb = xa b 1/xb = x b (xa )b = xa b (xy)a = xa y a . n +1, if n is even;. ( 1) =. 0. x =1 xy = x y 1, if n is odd.. If 0 < x < 1, then 0 < x3 < x2 < x < x< 3. x < 1. pg. 2. SAT math Facts & Formulas Factoring, Solving (x + a)(x + b) = x2 + (b + a)x + ab FOIL . a2 b2 = (a + b)(a b) Difference Of Squares . a2 + 2ab + b2 = (a + b)(a + b). a2 2ab + b2 = (a b)(a b). x2 + (b + a)x + ab = (x + a)(x + b) Reverse FOIL . You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and b which add to the number in front of the x, and which multiply to give the constant.
4 For example, to factor x2 + 5x + 6, the numbers add to 5 and multiply to 6, , a = 2 and b = 3, so that x2 + 5x + 6 = (x + 2)(x + 3). To solve a quadratic such as x2 +bx+c = 0, first factor the left side to get (x+a)(x+b) = 0, then set each part in parentheses equal to zero. For example, x2 +4x+3 = (x+3)(x+1) = 0. so that x = 3 or x = 1. To solve two linear equations in x and y: use the first equation to substitute for a variable in the second. , suppose x + y = 3 and 4x y = 2. The first equation gives y = 3 x, so the second equation becomes 4x (3 x) = 2 5x 3 = 2 x = 1, y = 2. Functions A function is a rule to go from one number (x) to another number (y), usually written y = f (x). For any given value of x, there can only be one corresponding value y. If y = kx for some number k (example: f (x) = x), then y is said to be directly proportional to x. If y = k/x (example: f (x) = 5/x), then y is said to be inversely proportional to x.
5 The graph of y = f (x h) + k is the translation of the graph of y = f (x) by (h, k) units in the plane. For example, y = f (x + 3) shifts the graph of f (x) by 3 units to the left. Absolute value: . +x, if x 0;. |x| =. x, if x < 0. |x| < n n < x < n |x| > n x < n or x > n pg. 3. SAT math Facts & Formulas Parabolas: A parabola parallel to the y-axis is given by y = ax2 + bx + c. If a > 0, the parabola opens up. If a < 0, the parabola opens down. The y-intercept is c, and the x-coordinate of the vertex is x = b/2a. Lines (Linear Functions). Consider the line that goes through points A(x1 , y1 ) and B(x2 , y2 ). p Distance from A to B: (x2 x1 )2 + (y2 y1 )2.. x1 + x2 y 1 + y 2. Mid-point of the segment AB: , 2 2. y2 y1 rise Slope of the line: =. x2 x1 run Point-slope form: given the slope m and a point (x1 , y1 ) on the line, the equation of the line is (y y1 ) = m(x x1 ). Slope-intercept form: given the slope m and the y-intercept b, then the equation of the line is y = mx + b.
6 Parallel lines have equal slopes. Perpendicular lines ( , those that make a 90 angle where they intersect) have negative reciprocal slopes: m1 m2 = 1. a b . l a b a . b b . a a . b . m b a . Intersecting Lines Parallel Lines (l k m). Intersecting lines: opposite angles are equal. Also, each pair of angles along the same line add to 180 . In the figure above, a + b = 180 . Parallel lines: eight angles are formed when a line crosses two parallel lines. The four big angles (a) are equal, and the four small angles (b) are equal. pg. 4. SAT math Facts & Formulas Triangles Right triangles: 45 . x 2. c 2x 60 . x b x 30 45 . a x x 3. a2 + b2 = c2 Special Right Triangles A good example of a right triangle is one with a = 3, b = 4, and c = 5, also called a 3 4 5. right triangle. Note that multiples of these numbers are also right triangles. For example, if you multiply these numbers by 2, you get a = 6, b = 8, and c = 10 (6 8 10), which is also a right triangle.
7 All triangles: h b 1. Area = b h 2. Angles on the inside of any triangle add up to 180 . The length of one side of any triangle is always less than the sum and more than the difference of the lengths of the other two sides. An exterior angle of any triangle is equal to the sum of the two remote interior angles. Other important triangles: Equilateral: These triangles have three equal sides, and all three angles are 60 . Isosceles: An isosceles triangle has two equal sides. The base angles (the ones opposite the two sides) are equal (see the 45 triangle above). Similar: Two or more triangles are similar if they have the same shape. The corresponding angles are equal, and the corresponding sides are in proportion. For example, the 3 4 5 triangle and the 6 8 10. triangle from before are similar since their sides are in a ratio of 2 to 1. pg. 5. SAT math Facts & Formulas Circles Arc r r n . (h, k).
8 Sector Area = r 2. Length Of Arc = (n /360 ) 2 r Circumference = 2 r Area Of Sector = (n /360 ) r 2. Full circle = 360 . Equation of the circle (above left figure): (x h)2 + (y k)2 = r 2 . Rectangles And Friends l w h w l Rectangle Parallelogram (Square if l = w) (Rhombus if l = w). Area = lw Area = lh Regular polygons are n-sided figures with all sides equal and all angles equal. The sum of the inside angles of an n-sided regular polygon is (n 2) 180 . Solids r h h w l Rectangular Solid Right Cylinder Volume = lwh Volume = r 2 h Area = 2(lw + wh + lh) Area = 2 r(r + h). pg. 6.
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Wednesday
February 10, 2016
# Posts by Cheryl
Total # Posts: 299
Math
An employee has an annual salary of \$26965. After he arranges to have deducted from his salary 12% for the purchase of bonds 17% for federal witholding tax and 3% for a retirement fund what is the amount of his monthly check
February 7, 2016
algebra
State whether the growth (or decay) is linear or exponential, and answer the associated question. The population of Jamestown is increasing at a rate of 2.5% each year. If the population is 25,000 today, what will it be in 2 years?
February 1, 2016
math
a theater sold 404 tickets for a total of \$1708. Children tickets are \$3.50 and adult tickets are \$5.50. How many adult tickets were sold
November 9, 2015
math
adult tickets for a play cost \$5.50 and a child ticket cost \$3.50. They sold 550 tickets and collected \$2139.00. How many adult tickets sold
November 8, 2015
william dick
If nuts are \$6lb how much 1 3/4
November 5, 2015
math
I don't know how to write these 2 sentences. One number is twice another number. Twice the lesser number increased by three times the greater number is 75
November 2, 2015
Math
On a test of motor coordination, the population of average bowlers has a mean score of 24, with a standard deviation of 6. A random sample of 30 bowlers at Fred's Bowling Alley has a sample mean of 26. A second random sample of 30 bowlers at Ethel's Bowling Alley has a...
October 8, 2015
Consumer Math
Thank you very much!
February 24, 2014
Consumer Math
1. Ken bought a car last year to drive back and forth to work. Last year he spent \$1,098 on gas. This year, it was \$1,562. What is the inflation rate? a. About 51% b. About 42% c. About 20% d. About 39% I believe it is b is this correct? 2.Carnie had an income of \$29,520 last ...
February 24, 2014
Finance
Calculate the project’s initial Time 0 cash flow, taking into account all side effects.? This will be a five-year project. The company bought some land three years ago for \$4.2 million in anticipation of using it as a toxic dump site for waste chemicals, but it built a ...
November 23, 2013
statistics
A group of 49 randomly selected construction workers has a mean age of 22.4 years with a standard deviation of 3.8. According to a recent survey, the mean age should be mu=21.9. Test this hypothesis by constructing a 98% confidence interval for the population mean.
November 19, 2013
chemistry
When doing an experiment with a calorimeter why is important to keep the lid on?
October 11, 2013
chemistry
When 5 grams of salt is dissolved in 95 grams of water the temperature of the solution increases 10°C how much heat is evolved the heat capacity of the solution is 1.1 cal/g°C
October 7, 2013
Math
Finding the critical value(s) of x^2, and use a x^2 distribution table to find the limits containing the P-value. Determine whether there is sufficient evidence to support the given alternative hypothesis. Test Ho:theta=2.63,Versus H1:theta does not equal2.62, given that theta...
September 29, 2013
biology
This is a Independent and Dependent Variable questions. If leaf color change is related to temperature, then exposing plants to low temperatures will result in changes in leaf color.
September 26, 2013
Geometry Honors
You and your friend each draw 3 points. Your points lie in only 1 plane, but your friend's 3 points lie in more than one plane. Explain how this is possible.
August 27, 2013
algebra
how to solve this question? to print 6000 post cards suppose it costs 2 cents to print a large postcard and 1 cent to print a small postcard. If \$97 dolllars is allocated for printing the postcards how many of each type can be printed
August 9, 2013
Math
How many square yards are in a room that measures 14 feet by 11.6 feet
June 28, 2013
Math
Mr.Wood went to the stores to buy school supplies for his children. He spent half of his money plus \$5 at Store A. He then spent half of the remaining money plus \$10 at Store B. At the end he had \$25 left. How much money did Mr. Wood start with?
May 6, 2013
Math
Mr. Wood went to the stores to buy school supplies for his children. He spent half of his money plus \$5 at Store A. He then spent half of his remaining money plus \$10 at Store B. At the end he had \$25 left. How much money did Mr. Wood start with?
May 6, 2013
chemistry
A nitric acid solution containing 71.0% by mass of the acid has a density of 1.42g/ml how many grams of nitric acid are present in 1.0L of this solution
May 3, 2013
Math
A Competitive Coup in the In-Flight Magazine. When the manager for market intelligence of AutoCorp, a major automotive manufacturer, boarded the plane in Chicago, her mind was on shrinking market share and late product announcements. As she settled back to enjoy the remains of...
April 29, 2013
economics
Calculate the percentage return on the security if the payoff to the security in one year is \$1,000, ... security in one year is \$1,000, \$1,500, \$2,000, or \$2,500.
April 14, 2013
English Comp I
What would be a good thesis statement for a proposal essay
April 11, 2013
statistics
The frequency of color blindness in the Caucasian American male population is about 8%. If a random sample of 125 individuals is chosen from this population, what is the probability that 5-10 individuals in the sample are color blind?
March 21, 2013
English
Thank you so much (:
March 11, 2013
English
Aboard at a ship’s helm, A young steersman steering with care. Through fog on a sea-coast dolefully ringing, An ocean-bell—O a warning bell, rock’d by the waves. O you give good notice indeed, you bell by the sea-reefs ringing, Ringing, ringing, to warn the ship...
March 11, 2013
Life Skills
The Civil War was an example of which type of conflict? a. interpersonal b. intrapersonal c. intergroup d. intragroup I believe its C, is that correct?
February 22, 2013
financial acct
Crowns issed 40,000 shares of \$5 par value common stock for \$14 per share. Prepare journal entries using a chart of acct.
January 24, 2013
science
when the beaker of water is heated what will happen to the level of water in the tube. this thermometer would be no use to measure the temperature in the freezer.write 2 sentences to explain why
November 8, 2012
science
what science word can be formed using these letters ECREFENINE?
October 15, 2012
Math
What is the weight or volume of solute in 5.5% NaOH solution?
October 3, 2012
finite math
for the previous ? it as to be set up in five steps I just can't figure out how to set it up to get the answers
September 22, 2012
finite math
A company installs different POS computer systems. POS system A requires two hours to configure and one hour for assemble. POS B requires three hours to configure and one hour to assemble. POS C requires two hours to configure and two hours to assemble. The company has up to ...
September 22, 2012
finite math
A company installs different POS computer systems. POS system A requires two hours to configure and one hour for assemble. POS B requires three hours to configure and one hour to assemble. POS C requires two hours to configure and two hours to assemble. The company has up to ...
September 22, 2012
statistics
If a bag contains a set of 20 red balls, 10 green balls, and 15 black balls, then what is the possiblity of selecting (in this order) a red ball, a green ball, and a red ball. Without Replacement of the items selected. Can you explain how you get the answer? Thank you
September 12, 2012
Algebra
15 less than x divded by 9
August 29, 2012
science
need examples of solutions and suspensions
July 21, 2012
Basic Algebra
A new cruise ship line has just launched 3 new ships: the Pacific, the Caribbean, and the Mediterranean. The Caribbean has 15 more deluxe staterooms than the Pacific. The Mediterranean has 26 fewer deluxe staterooms than four times the number of deluxe staterooms on the ...
July 15, 2012
arithmetic
what is 4.2% of \$24,000?
June 26, 2012
history
•Review “Origins of the Cold War” in Ch. 27 of American History and summarize the revisionist and post-revisionist interpretations of the Cold War. Does our interpretation of history ever change
May 29, 2012
history
•Do you think that the Korean War was merely a civil war in which the United States supported one side, or should it be considered an international war in which the United States used the United Nations to further its anticommunist policies? Why?
May 29, 2012
Math
what is (7.21-3.13)4th power divided by 0.6-0.4?
May 22, 2012
chemistry
If 48.3 g of an unknown gas occupies 10.0 L at 40 degrees celsius and 3.10 atm, what is the molar mass of the gas?
April 13, 2012
pre-algebra 2
April 6, 2012
Science
1. Which of the following is an example of energy transfer from the sun to a producer and then to a consumer? a. sun warms rocks; squirrels climb over rocks; squirrels are eaten by other animals b. sunlight helps plants to grow and deer eat plants c. sun heats pond water; fish...
March 1, 2012
computer science
myprogramming lab- the bane of my existance- code works in c++ but not in myprogramminglab so I wondered if I was missing something in the question
February 24, 2012
computer science
my code works in c++ so the logical I am getting error must be in the reading of the problem, but I can't see it: Given a type Money that is a structured type with two int fields, dollars and cents. Assume that an array named monthlySales with 12 elements, each of type ...
February 24, 2012
Math
lol, plug 5x-2 into the second equation 6x + 3(5x-2) = 15 solve for x
February 24, 2012
algebra
sure you wrote that right?
February 23, 2012
ALGEBRA
well you didn't list the choices so I'm going to say isolating x or y would be your first step.
February 23, 2012
computer science
Declare a structure whose tag name is Emp and that contains these fields (in the following order): a double field named d, a character pointer named str, and an array of eleven integers named arr. * I'm good till here. In addition, declare a array named emp of 30 of these ...
February 19, 2012
computer science
after the due date my teacher told me what was wrong with it apparently the entire problem was the .txt which I erroneously thought you had to have, ty for your help
February 18, 2012
computer science
9.10: Reverse Array Write a function that accepts an int array and the array’s size as arguments. The function should create a copy of the array, except that the element values should be reversed in the copy. The function should return a pointer to the new array. ...
February 18, 2012
math
n = 17 N= (17*2) - 17
February 18, 2012
computer science
those are 2 different codes above, I wrote them that way because what you wrote was similar to what I had before but myprogramminglab does not like it. Still does not like it. thank you though I think i give up, at least on this one.
February 18, 2012
computer science
The variable cp_arr has been declared as an array of 26 pointers to char. Allocate 26 character values, initialized to the letters 'A' through 'Z' and assign their pointers to the elements of cp_arr (in that order). int *arr1[26]={'A','B','...
February 18, 2012
Algebra
What's the question
February 18, 2012
computer science
lol I've had so many codes for this problem. int *arr1[26]={'A','B','C','D','E','F','G','H','I','J','K','L','M','N','O','P','Q','R&#...
February 17, 2012
computer science
The variable cp_arr has been declared as an array of 26 pointers to char. Allocate 26 character values, initialized to the letters 'A' through 'Z' and assign their pointers to the elements of cp_arr (in that order).
February 17, 2012
algebra
f(x) is the same thing as y so if you put y = 2x, and x = -1 that would make y = ?
February 17, 2012
computer science
whole problem, whole program sorry... 9.10: Reverse Array Write a function that accepts an int array and the array’s size as arguments. The function should create a copy of the array, except that the element values should be reversed in the copy. The function should ...
February 17, 2012
computer science
K I have gotten it to work in visual c++ by getting rid of the inner loop but it doesn't work on myprogramminglab so I don't know what else to do. The problem was extensive as well as the entire program, it works up until this function.
February 17, 2012
computer science
int *reverse(const int *, int); int *temp; temp = reverse(arr1,N); int *reverse(const int *arr1, int N) { int *arr2; arr2 = new int[N]; for(int count = 0; count < N; count++) { for(int index = (N-1); index <= 0; index--) arr2[count] = arr2[index]; } return arr2; } I need...
February 17, 2012
English
animals should not - instead of no animal should 1st line. ; after cruelty your first paragraph - you want to state what your arguments are without examples closing paragraph needs to be split between your other paragraphs, your closing argument should be a summation of your ...
February 17, 2012
mortgage rates
what would the monthly payment be on a \$115,000 home with a 4.4 percent annual interest?
January 26, 2012
Presentation
After completing the Unit 4 Live Presentation, discuss each of the following topics in a 3 – 4 paragraph essay: Discuss at least two specific aspects of delivering the Unit 4 Live Presentation that were easier than you thought they would be. Explain the reasons why you ...
January 23, 2012
science
how to draw a delivery device for golf balls
January 14, 2012
Science
1. Symbiotic relationships may evolve a. very quickly to remove all benefits acquired by the relationship. b. rapidly over a short period of time throughout all species. c. over time by co-evolution of adaptations that reduce the harm or improve the benefit of the relationship...
January 6, 2012
Environmental Science
Thank you so much
January 6, 2012
Environmental Science
1. The main provisions of the Endangered Species Act include all of the following EXCEPT: a. individuals may not destroy the habitats of endangered species b.one may not catch or kill, uproot, sell or trade any endangered or threatened species. c. the compilation of a list of ...
January 6, 2012
Math
How do you solve 2t - 4t to the 2nd power? 2t-4t to the 2nd power Identify the greatest common factor Factor this expression
December 24, 2011
physics
Julie carries an 8.0-kg suitcase as she walks 18 m along an inclined walkway to her hotel room at a constant speed of 1.5 m/s. The walkway is inclined 15 degrees above the horizontal. How much work does Julie do in carrying her suitcase?
December 15, 2011
phsyics
A 975-kg car accelerates. The magnitude of the average net force acting on the car is 3478 N, what is the acceleration of the car?
December 15, 2011
math
What is the scientific notation of 200,400
December 7, 2011
math word problem
The area of a square garden is 324 square yards find the amount of material needed to fence the garden
December 5, 2011
Math
Thanks but I still don't understand. can someone do a sample for me, thanks
November 21, 2011
Math
\$10000 x 10% x (90/365) don't know what to do with the number in the brackets
November 21, 2011
math
what is the concentration of a sugar solution that contains 8g sugar dissolved in 500 ml water? answer in the form of a ratio
November 20, 2011
science
what is the concentration of a sugar solution that contains 8 g sugar dissolved in 500 ml of water? explain in ration form
November 20, 2011
MACROECONOMICS
Suppose that there are 10 million workers in Canada and South Korea and each worker in Canada and South Korea can produce 4 cars per year. A Canadian worker can produce 10 tonnes of grain a year, whereas a South Korean worker can produce 5 tonnes of grain a year. The following...
November 13, 2011
STATISTICS
Pure Plastics Inc. manufacture molded plastic pieces. A sample of their raw material inventory is shown in the table below: Item Average Inventory (units) Value (\$/unit) 128 400 3.75 234 300 4.00 233 120 2.50 236 75 1.50 239 60 1.75 240 30 2.00 678 20 1.15 784 12 2.05 821 8 1....
November 13, 2011
math
8-5x<2x-20
November 12, 2011
statistics
Suppose that there are 10 million workers in Canada and South Korea and each worker in Canada and South Korea can produce 4 cars per year. A Canadian worker can produce 10 tonnes of grain a year, whereas a South Korean worker can produce 5 tonnes of grain a year. The following...
November 11, 2011
operations management
Pure Plastics Inc. manufacture molded plastic pieces. A sample of their raw material inventory is shown in the table below: Item Average Inventory (units) Value (\$/unit) 128 400 3.75 234 300 4.00 233 120 2.50 236 75 1.50 239 60 1.75 240 30 2.00 678 20 1.15 784 12 2.05 821 8 1....
November 11, 2011
statistics
2. Punkey Electronics, a small manufacturer of electronic research equipment, has approximately 7,000 items in its inventory. They just hired Joe Detail to manage the inventory. Joe has determined that 10% of the items in inventory are A items, 35% are B items and 55% are C ...
November 11, 2011
Will you please answer the following questions about your company products advertised in the October 2011 Dwell magazine? We are interested in ordering some of your snack products . 1. If we finally make a deal is it possible to start importing your products at the beginning ...
November 11, 2011
Chemistry
if you add 0.056 moles of HC2H3O2 to 0.064 moles of NH3, how many moles of each are actually neutralized?
November 7, 2011
physical science
A certain radioactive substance has a half-life of 1 hour. If you start with 1 gram of the material at noon, how much will be left at 3:00 P.M.? At 6:00 P.M.? At 10:00 P.M.?
November 2, 2011
College
borrowed \$42,400 for 5 years interest rate 2.5%
October 30, 2011
library science
Explain the purpose of AACR2 and Dublin Core Element Set then describe and discuss two significant similarities and two significant differences between them.
October 27, 2011
algebra
October 19, 2011
chem
ok my professor lost me when explaining this material so please bare with me and help me step by step one of the questions is: after absorbing 1850J of energy as heat, the temperature of a .500-kg block of copper is 37C what was its initial temperature (the specific heat ...
October 12, 2011
college chem
sodium sulfide, Na2S, is used in the leather industry to remove hair from hides. The Na2SO4 + 4C -> Na2S +4CO(g) so suppose you mix 15(g) of Na2SO4 and 7.5g of C. which is the limiting reactant AND what is mass of Na2S that is produced?
October 9, 2011
chemistry 2
the balanced equation for the reduction of iron ore to the metal using CO is Fe2O3+3CO(g) -> 2Fe+3CO so what is the maximum of iron in grams that can bbe obtained from 454g (1.00 lb) of iron (III) oxide? AND what mass of CO is required to react with the 454g of Fe2O3? ...
October 9, 2011
college chem
what volume of 0.955 M HCl, in milliliters, is required to titrate 2.152g of Na2CO3 to the equivalent point? Na2CO3(aq) + 2HCl(aq) 2NaCl(aq) + CO2(g) + H2O(1) can this please be explained in simple step by steps? i am sick and i am having trouble understanding anything.
October 6, 2011
chemistry
can someone please show me the steps of setting up the emirical formula for NaN3
October 5, 2011
chemistry
a room is 18ft long, 15ft wide and from floor to ceiling is 8ft6inches. what is the rooms volume in cubic meters? i have 6.498716 is this correct? what is the mass of air in the room in kilograms assuming the density of the air is 1.2g/l and the room is empty of furniture. i ...
October 3, 2011
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Lisp Fundamentals
NOTE: for all my non-computer friends and readers. If technical topics bother you or make you tense, perhaps todays post is not for you. If you are not given cold sweats when facing a new topic with the flavor of computers or programming, by all means please join me.
There are many things that Lisp, the programming language does right. It never ceases to amaze me and I am going to once again take a few minutes to discuss exactly what some of these things are and why they are so important.
Lisp was not originally conceived of as a programming language. It was invented by John Backus as a notation to enable discussion about Alonzo Church’s lambda calculus.
Lisp is characterized by the structure of its expressions, called forms. The simplest of these is the atom. An atom is a singular symbol or literal and represents a value. For instance, 42 is a numeric literal whose value is the number 42. Similarly, “Hello, world!” is a string literal that represents itself as its value. There are also symbols, which are strings of unquoted characters that are used as fundamental elements of Lisp. There are rules for how a valid symbol is form but for now it is sufficient to know that a symbol starts with a letter and then is composed of zero or more additional characters that can be either a letter, a number, or one of a collection of certain punctuation characters. Since the exact list of other characters varies among the dialects of Lisp, we will leave them unspecified at present.
The other type of form is a list. A list is comprised of a left parenthesis followed by zero or more forms and ending in a right parenthesis. Notice I said forms instead of symbols. The implication here is that you can have lists embedded in other lists as deeply nested as you like. This proves to be an interesting trait as we will soon see.
There is one more fundamentally interesting aspect of Lisp. That is that in a typical Lisp form the first element in a list after the left parenthesis is taken to be an operator. The subsequent elements in the list are considered arguments. The operator is either a function, a macro, or a special form. Macros and special forms, while extremely important and interesting are beyond the scope of this discussion.
That leaves us the operator as function. A typical Lisp function form is evaluated as follows. The first element is examined to determine what kind of form the list is. If it is a function, the rest of the arguments in the list are evaluated and collected in a list and the function is applied to them. If another list is encountered as one of the arguments, it is evaluated in exactly the same way.
For example, consider the expression (+ 4 (* 8 7) (/ (-26 8) 9)). The first operator is +. + is a symbol bound to the function that represents addition. The second item in the list is 4. It is a number that represents itself. The next element in the list is the list (* 8 7). When evaluated, the 8 and 7 are arguments to *, the multiplication function and the value returned by that function is 56. The final element in the top level list is (/ (- 26 8) 9). The / is taken as the division function and is applied to the evaluation of (- 26 8) which is the subtraction function that returns 18. When you divide 18 by 9. you get the value 2. Thus the top level argument list consists of 4, 56, and 2. When you add all three of those numbers you get 62 which is the value that the expression ultimately returns.
This simple mathematical expression illustrates another fundamental aspect of Lisp. It is expressed as a list form which, given a set of bindings to arithmetic functions expresses a simple program. This identical representation of both data and programming in Lisp, called homoiconicity by the way, is at the heart of much of Lisp’s power. Since Lisp’s programs are indistinguishable from Lisp’s data, they can be manipulated by Lisp programs to great advantage.
Think of it like this. Lisp can, in some sense, think about how it is thinking and modify it as it desires. This is why artificial intelligence investigators like using Lisp so much, it is so similar to the simplified models of intelligence that they are building that the boundary begins to blur.
Sweet dreams, don’t forget to tell the ones you love that you love them, and most important of all, be kind.
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# Why do the liquid levels on both the containers match?
When we take two containers and fill them both with same or different liquids and put one in the other and let the container reach equilibrium, why do we observe that both the liquid levels match, supposing the containers are ideal and are massless? (Diagram for reference)
• In general, the two levels will not match. It depends on how much water is in there, and what the weight of the container is. If you pour a bit of water in a boat, the boat will not suddenly rise to equalise the water levels inside and outside the boat. Commented Jan 23, 2022 at 11:31
• @fishinear I think you can post this as an answer. Commented Jan 23, 2022 at 14:53
• @fishinear What if the containers were massless, do you still think that the levels wouldn't match? Commented Jan 23, 2022 at 15:11
In general, you just need to use Archimedes's law: $$V_{\mathrm{displaced}} \rho_{\mathrm{outside}} g = M g$$, where $$M$$ is the total mass of the inner vessel and liquid contained in it.
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# Find the centre and radius of the following spheres :$x^{2}+y^{2}+z^{2}+4x-8x+2z=5$
## 1 Answer
Toolbox:
• General equation of a sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ centre $(-u, -v, -w)$ radius $= \sqrt{u^2+v^2+w^2-d}$
$x^2+y^2+z^2=4x-8y+2z=5$
The centre is at $( -u, -v, -w)$ where $2u=4, \: 2v=-8, \: 2w=2$ the centre is at $(2, -4, -1)$
The radius = $\sqrt{u^2+v^2+w^2-d}$ with $d=-5$
$\therefore r = \sqrt{4+16+1+5} = \sqrt{26}$ units
answered Jun 18, 2013
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# Boundary|Definition & Meaning
## Definition
A boundary is a line that outlines an object’s shape or a polygon. The total length of the boundary is called the perimeter. The boundary is different for all shapes.
A boundary allows us to identify the shape of an object or image and is necessary for separating one shape from another.
## Visualizing a Boundary
Figure 1 below shows a rectangle ABCD. The lines between the points ABCD is known as the boundary. With the help of the boundary, we can easily distinguish the shape.
Figure 1 – Boundary of a rectangle.
Figure 2 below shows the points EFG; as in the previous figure, these points are connected with a boundary that gives it its shape. The figure is called a triangle.
Figure 2 – Boundary of a triangle.
## What Is a Perimeter?
The perimeter of any two-dimensional geometric shape is the length of its outline or boundary. Depending on the dimensions, the perimeters of different figures can be equal in size. Consider a triangle made of a wire that is 12 cm long. If all the sides are the same length, the same wire can be used to create a square.
Figure 3 – Boundary of shapes.
Now that we know the perimeter of a geometric shape refers to its outer boundary, how would one determine its value? Let us look at an example. To prevent his cattle from straying off the farm, Jake wants to erect a fence around it. He is curious about the amount of wire required to fence his farm. His farm has a rectangular shape, which means:
• The farm has four sides.
• The length of the opposing sides is equal.
• Each angle is 90 degrees.
Now to find out the perimeter of his farm, Jack only needs to measure all four sides and add them together to get the perimeter of his farm.
### Units of a Perimeter
The length that the shape’s perimeter covers are known as the perimeter. Therefore, the perimeter’s units will be the same as its length units. Perimeter is one-dimensional, as we can say. As a result, it can be measured in meters, kilometers, centimeters, etc. Inches, feet, yards, and miles are some additional perimeter measurement units that are recognized on a global scale.
### The Formula of a Perimeter
The formula for calculating a perimeter changes according to our provided shape. The general formula for calculating a perimeter is as follows:
Perimeter = Sum of all borders
However, different formulas exist for various shapes. Below we can look at some of the commonly used formulas to calculate a perimeter:
• Square: The perimeter of the square = 4 x L, where L is the length of one boundary
• Rectangle: The perimeter of the rectangle = 2 x (L + B), where L and B are the Length and Breath of the rectangle.
• Triangle: The perimeter of a triangle = l + m + n, where l, m, and n are the lengths of a triangle.
• Circle: The perimeter of a Circle = $2 \pi r$, where $\pi$ is a constant and r is the circle’s radius.
## Area of a Shape
The area of a 2-dimensional figure is the amount of space it takes up. In other words, it refers to the number of unit squares covering the surface of a shape enclosed by a boundary. The length and width of a shape are used to calculate its area. The units used to measure length are unidimensional and include meters (m), centimeters (cm), inches (in), etc.
However, a shape’s area can only be measured in two dimensions. It is measured in square units, for example, meters or (m2), square centimeters or (cm2), square inches or (in2), and so on. The majority of the objects or shapes have edges and corners. These edges’ length and width are considered when calculating the area of a specific shape.
### Calculating the Area of a Rectangle
The area is the space inside the boundary of a shape. The following figure shows a rectangle with a length of 5cm and a width of 3 cm. This formula can calculate the area of the rectangle: Area of Rectangle = l x w, where l is the length of the rectangle and w is the width.
Figure 4 – Perimeter of a rectangle.
Using the formula given, we can find the area of the rectangle. The area calculated is 3 cm x 5 cm = 15 cm.
### Calculating the Area of a Circle
The shape of a circle is curved. The amount of space within a circle’s boundary is known as its area. The formula: $\pi$r2, where r is the circle’s radius, and $\pi$ is a mathematical constant with a value roughly 3.14 or 22/7, is used to determine the area of a circle.
Looking at the figure below, we can see that the circle’s radius is 10cm.
Figure 6 – Radius of a circle.
We can use the formula above to find the area of the circle. The area is calculated as shown below:
Area of Circle = $\pi$r2
Area of Circle = $\pi$ x (10 cm)2
Area of Circle = 100 x $\pi$ cm2 $\approx$ 314.16 cm2
## Example of Boundary
The following example on boundaries will help you understand the concept more easily.
In the following figures, distinguish the rectangular shape and find its perimeter.
Figure 6 – Representation of boundaries.
### Solution
We must find the rectangle from the given shapes in the first step. After closer inspection of the shapes, we can conclude that the shape ABCD is a rectangle. Now to calculate the perimeter of the rectangle, we use the following formula:
Perimeter of rectangle = 2(l x w)
Now we know the length of the rectangle is 6cm and the width is 2cm, we put these values in the formula.
Perimeter of rectangle = 2(6 cm + 2 cm)
After solving this formula, we get 18 cm2.
All images/tables are created using GeoGebra.
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# Lemi
Popular questions and responses by Lemi
Alice works part-time after school. On Monday, Alice records that she worked three and two-sixth hours. On Tuesday, Alice records that she worked one and seventy-five hundredths hours. On Wednesday, Alice records that she worked two and two-thirds hours.
2. ## Go Math grade 6
The Webster family is planning a barbecue for one hundred people. The Webster family needs one plastic plate for each person at the barbecue. The Stop and Shop store and Market Place Store both sell plastic plates in packages of fifteen for five dollars
3. ## Go Math grade 6
What is the result of the problem below? Thanks What is the problem system?The Webster family is planning a barbecue for one hundred people. The Webster family needs one plastic plate for each person at the barbecue. The Stop and Shop store and Market
Alice works part-time after school. On Monday, Alice records that she worked three and two-sixth hours. On Tuesday, Alice records that she worked one and seventy-five hundredths hours. On Wednesday, Alice records that she worked two and two-thirds hours.
The New York Subway Bakery is famous for selling large "black and white cookies." The top of each cookie has one-half chocolate icing and one-half vanilla icing. Mario, the baker, bakes at night after the customers leave. Mario wants to bake two hundred
6. ## Chemistry
What is the mass of MgCl2 required to make 500mL if 1.200 M of MgCl2 Solution? I'm having trouble figuring out what the equation looks like. I believe the answer is 57.127g but if so i would like to know why. I did (1.200M*.500mL)/1000* mass of MgCl2. How
Mr. Lincoln is building a rectengular play area in the local park. The length of the play area is ninety-seven and one-half feet. The width of the play area is fifty-six feet. To keep xhildren from falling on a hard surface, the play area will be covered
The New York Subway Bakery is famous for selling large "black and white cookies." The top of each cookie has one-half chocolate icing and one-half vanilla icing. Mario, the baker, bakes at night after the customers leave. Mario wants to bake two hundred
Some students are making muffins to sell to earn money to go on a school trip.The students use two and two- thirds cups of flour and three eggs for every dozen muffins they make. The students make eight dozen muffins. How many cups of flour and how many
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On this page, you will find Quadratic Equations Class 10 Notes Maths Chapter 4 Pdf free download. CBSE NCERT Class 10 Maths Notes Chapter 4 Quadratic Equations will seemingly help them to revise the important concepts in less time.
## CBSE Class 10 Maths Chapter 4 Notes Quadratic Equations
### Quadratic Equations Class 10 Notes Understanding the Lesson
1. Quadratic Equation: A quadratic equation in the variable x is of the form ax2 + bx + c = 0, where a, b, c are real number and a ≠ 0.
2. Roots (or zeroes of a quadratic equation): A real number a is called the root of the quadratic equation
ax2 + bx + c = 0 if aα2 + bα + c = 0.
Alternatively, any equation of the form p(x) = 0, where p(x) is a quadratic polynomial is a quadratic equation and if p(α) = 0 for any real number a; the a is said to be the root (or zero) of p(x).
Solution of a quadratic equation by factorization
Finding the roots of a quadratic equation by the method of factorization means finding out the linear factors of the quadratic equation and equating it to zero, the roots can be found. i.e. ax2 + bx + c = 0
(Ax + B) (Cr + D) = 0
where A, B, C and D are real numbers, A, C≠ 0.
We get Ax + B = 0 or Cx + D = 0
x =$$-\frac{B}{A}$$ or x =$$-\frac{D}{C}$$
x =$$-\frac{\mathrm{B}}{\mathrm{A}},-\frac{\mathrm{D}}{\mathrm{C}}$$ are the two roots of quadratic equation.
Solution of a quadratic equation by completing the square
For given quadratic equation ax2+ bx + c = 0
Divide the equation by a, so that the coefficient of x2 becomes 1.
$$x^{2}+\frac{b}{a} x+\frac{c}{a}=0$$
Adding and subtracting $$\left(\frac{b}{2 a}\right)^{2}$$ i.e., square of the half of the coefficient of x.
This formula is known as quadratic formula.
If α and β are roots of the given equation, then
ax2 + bx + c = 0,
a ≠ 0, a, b, c ∈ R
Discriminant D = b2 – 4ac
Condition exists Nature of roots
(i) b2 – 4ac > 0 Real and unequal
(ii) b2 – 4ac = 0 Real and equal
(iii) b2 – 4ac < 0 No real roots
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$hide=mobile$type=ticker$c=12$cols=3$l=0$sr=random$b=0 ĐẶT MUA TẠP CHÍ / PURCHASE JOURNALS Algebra 1. Let$a, b, c$be positive real numbers such that$abc = 8$. Prove that $$\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6.$$ 2. Let$a,b,c$be positive real numbers. Prove that $$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \\ \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$$ 3. Find all the pairs of integers$ (m, n)$such that $$\sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$$ 4. If the non-negative reals$x,y,z$satisfy$x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur? 5. Let$x,y,z$be positive real numbers such that $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$ Prove that $x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .$ Combinatorics 1. Let$S_n$be the sum of reciprocal values of non-zero digits of all positive integers up to (and including)$n$. For instance, $$S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}.$$ Find the least positive integer$k$making the number$k!\cdot S_{2016}$an integer. 2. The natural numbers from$1$to$50$are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime? 3. A$5 \times 5$table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly one in every$2 \times 2$subtable. The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible regular tables, computes their total sums and counts the distinct outcomes.Determine the maximum possible count. 4. A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer$n \ge 3$, determine the largest integer$m$such that no planar$n$-gon splits into less than$m$triangles. Geometry 1. Let${ABC}$be an acute angled triangle, let${O}$be its circumcentre, and let$D$,$E$,$F$be points on the sides$BC$,$CA$,$AB$, respectively. The circle${(c_1)}$of radius${FA}$, centered at${F}$, crosses the segment${OA}$at${A'}$and the circumcircle${(c)}$of the triangle${ABC}$again at${K}$. Similarly, the circle${(c_2)}$of radius$DB$, centered at$D$, crosses the segment$\left( OB \right)$at${B}'$and the circle${(c)}$again at${L}$. Finally, the circle${(c_3)}$of radius$EC$, centered at$E$, crosses the segment$\left( OC \right)$at${C}'$and the circle${(c)}$again at${M}$. Prove that the quadrilaterals$BKF{A}'$,$CLD{B}'$and$AME{C}'$are all cyclic, and their circumcircles share a common point. 2. Let${ABC}$be a triangle with$\angle BAC={{60}^{{}^\circ }}$. Let$D$and$E$be the feet of the perpendiculars from${A}$to the external angle bisectors of$\angle ABC$and$\angle ACB$, respectively. Let${O}$be the circumcenter of the triangle${ABC}$. Prove that the circumcircles of the triangles${ADE}$and${BOC}$are tangent to each other. 3. A trapezoid$ABCD$($AB || CF$,$AB > CD$) is circumscribed. The incircle of the triangle$ABC$touches the lines$AB$and$AC$at the points$M$and$N$, respectively. Prove that the incenter of the trapezoid$ABCD$lies on the line$MN$. 4. Let${ABC}$be an acute angled triangle whose shortest side is${BC}$. Consider a variable point${P}$on the side${BC}$, and let${D}$and${E}$be points on${AB}$and${AC}$, respectively, such that${BD=BP}$and${CP=CE}$. Prove that, as${P}$traces${BC}$, the circumcircle of the triangle${ADE}$passes through a fixed point. 5. Let$ABC$be an acute angled triangle with orthocenter${H}$and circumcenter${O}$. Assume the circumcenter${X}$of${BHC}$lies on the circumcircle of${ABC}$. Reflect O across${X}$to obtain${O'}$, and let the lines${XH}$and${O'A}$meet at${K}$. Let$L,M$and$N$be the midpoints of$\left[ XB \right]$,$\left[ XC \right]$and$\left[ BC \right]$, respectively. Prove that the points$K$,$L$,$M$and$K$,$L$,$M$,$N$are cocyclic. 6. Given an acute triangle${ABC}$, erect triangles${ABD}$and${ACE}$externally, so that$\angle ADB= \angle AEC=90^\circ$and$\angle BAD= \angle CAE$. Let${{A}_{1}}\in BC$,${{B}_{1}}\in AC$and${{C}_{1}}\in AB$be the feet of the altitudes of the triangle${ABC}$, and let$K$and$K$,$L$be the midpoints of$[ B{{C}_{1}} ]$and${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles$AKL$,${{A}_{1}}{{B}_{1}}{{C}_{1}}$and${DEA_1}$are collinear. 7. Let${AB}$be a chord of a circle${(c)}$centered at${O}$, and let${K}$be a point on the segment${AB}$such that${AK<BK}$. Two circles through${K}$, internally tangent to${(c)}$at${A}$and${B}$, respectively, meet again at${L}$. Let${P}$be one of the points of intersection of the line${KL}$and the circle${(c)}$, and let the lines${AB}$and${LO}$meet at${M}$. Prove that the line${MP}$is tangent to the circle${(c)}$. Number Theory 1. Determine the largest positive integer$n$that divides$p^6 - 1$for all primes$p > 7$. 2. Find the maximum number of natural numbers$x_1,x_2, ... , x_m$satisfying the conditions • No$x_i - x_j , 1 \le i < j \le m$is divisible by$11$, and • The sum$x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$is divisible by$11$. 3. Find all positive integers$n$such that the number$A_n =\frac{ 2^{4n+2}+1}{65}$is a) an integer, b) a prime. 4. Find all triplets of integers$(a,b,c)$such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$ is a power of$2016$. (A power of$2016$is an integer of form$2016^n$where$n$is a non-negative integer.) 5. Determine all four-digit numbers$\overline{abcd} $such that $$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}$$ $hide=mobile$type=ticker$c=36$cols=2$l=0$sr=random$b=0
Name
Abel,5,Albania,2,AMM,2,Amsterdam,4,An Giang,45,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Lan,1,Bà Rịa Vũng Tàu,77,Bắc Bộ,2,Bắc Giang,62,Bắc Kạn,4,Bạc Liêu,18,Bắc Ninh,53,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,41,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,72,Benelux,16,Bình Định,65,Bình Dương,38,Bình Phước,52,Bình Thuận,42,Birch,1,BMO,41,Booklet,12,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,British,16,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,2,BxMO,15,Cà Mau,22,Cần Thơ,27,Canada,40,Cao Bằng,12,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,515,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,30,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,675,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,28,Đà Nẵng,50,Đa Thức,2,Đại Số,20,Đắk Lắk,76,Đắk Nông,15,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2249,Đề Thi JMO,1,DHBB,30,Điện Biên,15,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,5,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đồng Nai,64,Đồng Tháp,63,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,30,E-Book,31,EGMO,30,ELMO,19,EMC,11,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,30,Gauss,1,GDTX,3,Geometry,14,GGTH,30,Gia Lai,40,Gia Viễn,2,Giải Tích Hàm,1,Giới hạn,2,Goldbach,1,Hà Giang,5,Hà Lan,1,Hà Nam,45,Hà Nội,255,Hà Tĩnh,91,Hà Trung Kiên,1,Hải Dương,70,Hải Phòng,57,Hậu Giang,14,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,33,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,126,HSG 10 2010-2011,4,HSG 10 2011-2012,7,HSG 10 2012-2013,8,HSG 10 2013-2014,7,HSG 10 2014-2015,6,HSG 10 2015-2016,2,HSG 10 2016-2017,8,HSG 10 2017-2018,4,HSG 10 2018-2019,4,HSG 10 2019-2020,7,HSG 10 2020-2021,3,HSG 10 2021-2022,4,HSG 10 2022-2023,11,HSG 10 2023-2024,1,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,HSG 10 Bình Định,1,HSG 10 Bình Dương,1,HSG 10 Bình Thuận,4,HSG 10 Chuyên SPHN,5,HSG 10 Đắk Lắk,2,HSG 10 Đồng Nai,4,HSG 10 Gia Lai,2,HSG 10 Hà Nam,4,HSG 10 Hà Tĩnh,15,HSG 10 Hải Dương,10,HSG 10 KHTN,9,HSG 10 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 PTNK,10,HSG 10 Quảng Nam,1,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,9,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,135,HSG 11 2009-2010,1,HSG 11 2010-2011,6,HSG 11 2011-2012,10,HSG 11 2012-2013,9,HSG 11 2013-2014,7,HSG 11 2014-2015,10,HSG 11 2015-2016,6,HSG 11 2016-2017,8,HSG 11 2017-2018,7,HSG 11 2018-2019,8,HSG 11 2019-2020,5,HSG 11 2020-2021,8,HSG 11 2021-2022,4,HSG 11 2022-2023,7,HSG 11 2023-2024,1,HSG 11 An Giang,2,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,3,HSG 11 Bắc Ninh,2,HSG 11 Bình Định,12,HSG 11 Bình Dương,3,HSG 11 Bình Thuận,1,HSG 11 Cà Mau,1,HSG 11 Đà Nẵng,9,HSG 11 Đồng Nai,1,HSG 11 Hà Nam,2,HSG 11 Hà Tĩnh,12,HSG 11 Hải Phòng,1,HSG 11 Kiên Giang,4,HSG 11 Lạng Sơn,11,HSG 11 Nghệ An,6,HSG 11 Ninh Bình,2,HSG 11 Quảng Bình,12,HSG 11 Quảng Nam,1,HSG 11 Quảng Ngãi,9,HSG 11 Quảng Trị,3,HSG 11 Sóc Trăng,1,HSG 11 Thái Nguyên,8,HSG 11 Thanh Hóa,3,HSG 11 Trà Vinh,1,HSG 11 Tuyên Quang,1,HSG 11 Vĩnh Long,3,HSG 11 Vĩnh Phúc,11,HSG 12,668,HSG 12 2009-2010,2,HSG 12 2010-2011,39,HSG 12 2011-2012,44,HSG 12 2012-2013,58,HSG 12 2013-2014,53,HSG 12 2014-2015,44,HSG 12 2015-2016,37,HSG 12 2016-2017,46,HSG 12 2017-2018,55,HSG 12 2018-2019,43,HSG 12 2019-2020,43,HSG 12 2020-2021,52,HSG 12 2021-2022,35,HSG 12 2022-2023,42,HSG 12 2023-2024,23,HSG 12 2023-2041,1,HSG 12 An Giang,8,HSG 12 Bà Rịa Vũng Tàu,13,HSG 12 Bắc Giang,18,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,HSG 12 Bình Định,17,HSG 12 Bình Dương,8,HSG 12 Bình Phước,9,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,7,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,11,HSG 12 Đà Nẵng,3,HSG 12 Đắk Lắk,21,HSG 12 Đắk Nông,1,HSG 12 Điện Biên,3,HSG 12 Đồng Nai,20,HSG 12 Đồng Tháp,18,HSG 12 Gia Lai,14,HSG 12 Hà Nam,5,HSG 12 Hà Nội,17,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,16,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,4,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,10,HSG 12 Khánh Hòa,4,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,3,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,11,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,17,HSG 12 Long An,18,HSG 12 Nam Định,7,HSG 12 Nghệ An,13,HSG 12 Ninh Bình,12,HSG 12 Ninh Thuận,7,HSG 12 Phú Thọ,18,HSG 12 Phú Yên,13,HSG 12 Quảng Bình,14,HSG 12 Quảng Nam,11,HSG 12 Quảng Ngãi,6,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,10,HSG 12 Sóc Trăng,4,HSG 12 Sơn La,5,HSG 12 Tây Ninh,6,HSG 12 Thái Bình,11,HSG 12 Thái Nguyên,13,HSG 12 Thanh Hóa,17,HSG 12 Thừa Thiên Huế,19,HSG 12 Tiền Giang,3,HSG 12 TPHCM,13,HSG 12 Tuyên Quang,3,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,20,HSG 12 Yên Bái,6,HSG 9,573,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,42,HSG 9 2012-2013,41,HSG 9 2013-2014,35,HSG 9 2014-2015,41,HSG 9 2015-2016,38,HSG 9 2016-2017,42,HSG 9 2017-2018,45,HSG 9 2018-2019,41,HSG 9 2019-2020,18,HSG 9 2020-2021,50,HSG 9 2021-2022,53,HSG 9 2022-2023,55,HSG 9 2023-2024,15,HSG 9 An Giang,9,HSG 9 Bà Rịa Vũng Tàu,8,HSG 9 Bắc Giang,14,HSG 9 Bắc Kạn,1,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,12,HSG 9 Bến Tre,9,HSG 9 Bình Định,11,HSG 9 Bình Dương,7,HSG 9 Bình Phước,13,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,2,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,2,HSG 9 Đà Nẵng,11,HSG 9 Đắk Lắk,12,HSG 9 Đắk Nông,3,HSG 9 Điện Biên,5,HSG 9 Đồng Nai,8,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,9,HSG 9 Hà Giang,4,HSG 9 Hà Nam,10,HSG 9 Hà Nội,15,HSG 9 Hà Tĩnh,13,HSG 9 Hải Dương,16,HSG 9 Hải Phòng,8,HSG 9 Hậu Giang,6,HSG 9 Hòa Bình,4,HSG 9 Hưng Yên,11,HSG 9 Khánh Hòa,6,HSG 9 Kiên Giang,16,HSG 9 Kon Tum,9,HSG 9 Lai Châu,2,HSG 9 Lâm Đồng,14,HSG 9 Lạng Sơn,10,HSG 9 Lào Cai,4,HSG 9 Long An,10,HSG 9 Nam Định,9,HSG 9 Nghệ An,21,HSG 9 Ninh Bình,14,HSG 9 Ninh Thuận,4,HSG 9 Phú Thọ,13,HSG 9 Phú Yên,9,HSG 9 Quảng Bình,14,HSG 9 Quảng Nam,12,HSG 9 Quảng Ngãi,13,HSG 9 Quảng Ninh,17,HSG 9 Quảng Trị,10,HSG 9 Sóc Trăng,9,HSG 9 Sơn La,5,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,11,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,12,HSG 9 Thừa Thiên Huế,9,HSG 9 Tiền Giang,7,HSG 9 TPHCM,11,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,6,HSG 9 Vĩnh Long,12,HSG 9 Vĩnh Phúc,12,HSG 9 Yên Bái,5,HSG Cấp Trường,80,HSG Quốc Gia,113,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,43,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,58,IMT,2,IMU,2,India - Ấn Độ,47,Inequality,13,InMC,1,International,349,Iran,13,Jakob,1,JBMO,41,Jewish,1,Journal,30,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,30,KHTN,64,Kiên Giang,74,Kon Tum,24,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,12,Lâm Đồng,47,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,37,Langlands,1,Lào Cai,35,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Hồng Phong,5,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Viết Hải,1,Lê Việt Hưng,2,Leibniz,1,Long An,52,Lớp 10 Chuyên,709,Lớp 10 Không Chuyên,355,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lưu Giang Nam,2,Lưu Lý Tưởng,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,13,Menelaus,1,Metropolises,4,Mexico,1,MIC,1,Michael Atiyah,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,MYM,25,MYTS,4,Nam Định,45,Nam Phi,1,National,276,Nesbitt,1,Newton,4,Nghệ An,73,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Minh Hà,1,Nguyễn Minh Tuấn,9,Nguyễn Nhất Huy,1,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,2,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Song Thiên Long,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,61,Ninh Thuận,26,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,134,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,122,Olympic 12,52,Olympic 23/3,2,Olympic 24/3,10,Olympic 24/3 Quảng Nam,10,Olympic 27/4,24,Olympic 30/4,61,Olympic KHTN,8,Olympic Sinh Viên,78,Olympic Tháng 4,12,Olympic Toán,344,Olympic Toán Sơ Cấp,3,Ôn Thi 10,2,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Quang Đạt,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,32,Phú Yên,42,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,64,Putnam,27,Quảng Bình,64,Quảng Nam,57,Quảng Ngãi,49,Quảng Ninh,60,Quảng Trị,42,Quỹ Tích,1,Riemann,1,RMM,14,RMO,24,Romania,38,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,70,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia - Ả Rập Xê Út,9,Scholze,1,Serbia,17,Sharygin,28,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,28,Sóc Trăng,36,Sơn La,22,Spain,8,Star Education,1,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,17,Tập San,3,Tây Ban Nha,1,Tây Ninh,37,Thái Bình,45,Thái Nguyên,61,Thái Vân,2,Thanh Hóa,69,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,Thông Tin Toán Học,43,THPT Chuyên Lê Quý Đôn,1,THPT Chuyên Nguyễn Du,9,THPTQG,16,THTT,31,Thừa Thiên Huế,56,Tiền Giang,30,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,158,Trà Vinh,10,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,39,Trại Hè Hùng Vương,30,Trại Hè Phương Nam,7,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,12,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trường Đông,23,Trường Hè,10,Trường Thu,1,Trường Xuân,3,TST,544,TST 2008-2009,1,TST 2010-2011,22,TST 2011-2012,23,TST 2012-2013,32,TST 2013-2014,29,TST 2014-2015,27,TST 2015-2016,26,TST 2016-2017,41,TST 2017-2018,42,TST 2018-2019,30,TST 2019-2020,34,TST 2020-2021,30,TST 2021-2022,38,TST 2022-2023,42,TST 2023-2024,23,TST An Giang,8,TST Bà Rịa Vũng Tàu,11,TST Bắc Giang,5,TST Bắc Ninh,11,TST Bến Tre,10,TST Bình Định,5,TST Bình Dương,7,TST Bình Phước,9,TST Bình Thuận,9,TST Cà Mau,7,TST Cần Thơ,6,TST Cao Bằng,2,TST Đà Nẵng,8,TST Đắk Lắk,12,TST Đắk Nông,2,TST Điện Biên,2,TST Đồng Nai,13,TST Đồng Tháp,12,TST Gia Lai,4,TST Hà Nam,8,TST Hà Nội,12,TST Hà Tĩnh,15,TST Hải Dương,11,TST Hải Phòng,13,TST Hậu Giang,1,TST Hòa Bình,4,TST Hưng Yên,10,TST Khánh Hòa,8,TST Kiên Giang,11,TST Kon Tum,6,TST Lâm Đồng,12,TST Lạng Sơn,3,TST Lào Cai,4,TST Long An,6,TST Nam Định,8,TST Nghệ An,7,TST Ninh Bình,11,TST Ninh Thuận,4,TST Phú Thọ,13,TST Phú Yên,5,TST PTNK,15,TST Quảng Bình,12,TST Quảng Nam,7,TST Quảng Ngãi,8,TST Quảng Ninh,9,TST Quảng Trị,10,TST Sóc Trăng,5,TST Sơn La,7,TST Thái Bình,6,TST Thái Nguyên,8,TST Thanh Hóa,9,TST Thừa Thiên Huế,4,TST Tiền Giang,6,TST TPHCM,14,TST Trà Vinh,1,TST Tuyên Quang,1,TST Vĩnh Long,7,TST Vĩnh Phúc,7,TST Yên Bái,8,Tuyên Quang,14,Tuyển Sinh,4,Tuyển Sinh 10,1064,Tuyển Sinh 10 An Giang,18,Tuyển Sinh 10 Bà Rịa Vũng Tàu,22,Tuyển Sinh 10 Bắc Giang,19,Tuyển Sinh 10 Bắc Kạn,3,Tuyển Sinh 10 Bạc Liêu,9,Tuyển Sinh 10 Bắc Ninh,15,Tuyển Sinh 10 Bến Tre,34,Tuyển Sinh 10 Bình Định,19,Tuyển Sinh 10 Bình Dương,12,Tuyển Sinh 10 Bình Phước,21,Tuyển Sinh 10 Bình Thuận,15,Tuyển Sinh 10 Cà Mau,5,Tuyển Sinh 10 Cần Thơ,10,Tuyển Sinh 10 Cao Bằng,2,Tuyển Sinh 10 Chuyên SPHN,19,Tuyển Sinh 10 Đà Nẵng,18,Tuyển Sinh 10 Đại Học Vinh,13,Tuyển Sinh 10 Đắk Lắk,21,Tuyển Sinh 10 Đắk Nông,7,Tuyển Sinh 10 Điện Biên,5,Tuyển Sinh 10 Đồng Nai,18,Tuyển Sinh 10 Đồng Tháp,23,Tuyển Sinh 10 Gia Lai,10,Tuyển Sinh 10 Hà Giang,1,Tuyển Sinh 10 Hà Nam,16,Tuyển Sinh 10 Hà Nội,80,Tuyển Sinh 10 Hà Tĩnh,19,Tuyển Sinh 10 Hải Dương,17,Tuyển Sinh 10 Hải Phòng,15,Tuyển Sinh 10 Hậu Giang,3,Tuyển Sinh 10 Hòa Bình,15,Tuyển Sinh 10 Hưng Yên,12,Tuyển Sinh 10 Khánh Hòa,12,Tuyển Sinh 10 KHTN,21,Tuyển Sinh 10 Kiên Giang,31,Tuyển Sinh 10 Kon Tum,6,Tuyển Sinh 10 Lai Châu,6,Tuyển Sinh 10 Lâm Đồng,10,Tuyển Sinh 10 Lạng Sơn,6,Tuyển Sinh 10 Lào Cai,10,Tuyển Sinh 10 Long An,18,Tuyển Sinh 10 Nam Định,21,Tuyển Sinh 10 Nghệ An,23,Tuyển Sinh 10 Ninh Bình,20,Tuyển Sinh 10 Ninh Thuận,10,Tuyển Sinh 10 Phú Thọ,18,Tuyển Sinh 10 Phú Yên,12,Tuyển Sinh 10 PTNK,37,Tuyển Sinh 10 Quảng Bình,12,Tuyển Sinh 10 Quảng Nam,15,Tuyển Sinh 10 Quảng Ngãi,13,Tuyển Sinh 10 Quảng Ninh,12,Tuyển Sinh 10 Quảng Trị,7,Tuyển Sinh 10 Sóc Trăng,17,Tuyển Sinh 10 Sơn La,5,Tuyển Sinh 10 Tây Ninh,15,Tuyển Sinh 10 Thái Bình,17,Tuyển Sinh 10 Thái Nguyên,18,Tuyển Sinh 10 Thanh Hóa,27,Tuyển Sinh 10 Thừa Thiên Huế,24,Tuyển Sinh 10 Tiền Giang,14,Tuyển Sinh 10 TPHCM,23,Tuyển Sinh 10 Trà Vinh,6,Tuyển Sinh 10 Tuyên Quang,3,Tuyển Sinh 10 Vĩnh Long,12,Tuyển Sinh 10 Vĩnh Phúc,22,Tuyển Sinh 2008-2009,1,Tuyển Sinh 2009-2010,1,Tuyển Sinh 2010-2011,6,Tuyển Sinh 2011-2012,20,Tuyển Sinh 2012-2013,65,Tuyển Sinh 2013-2014,77,Tuyển Sinh 2013-2044,1,Tuyển Sinh 2014-2015,81,Tuyển Sinh 2015-2016,64,Tuyển Sinh 2016-2017,72,Tuyển Sinh 2017-2018,126,Tuyển Sinh 2018-2019,61,Tuyển Sinh 2019-2020,90,Tuyển Sinh 2020-2021,59,Tuyển Sinh 2021-202,1,Tuyển Sinh 2021-2022,69,Tuyển Sinh 2022-2023,113,Tuyển Sinh 2023-2024,49,Tuyển Sinh Chuyên SPHCM,7,Tuyển Sinh Yên Bái,6,Tuyển Tập,45,Tuymaada,6,UK - Anh,16,Undergraduate,69,USA - Mỹ,62,USA TSTST,6,USAJMO,12,USATST,8,USEMO,4,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,6,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,32,Vĩnh Long,41,Vĩnh Phúc,86,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,58,VNTST,25,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Xác Suất,1,Yên Bái,25,Yên Thành,1,Zhautykov,14,Zhou Yuan Zhe,1,
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CC-MAIN-2024-30
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latest
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en
| 0.653006 |
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