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https://kids.kiddle.co/Mole_(unit)
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# Mole (unit) facts for kids
Kids Encyclopedia Facts
Quick facts for kids
Mole
Unit system SI base unit
Unit of Amount of substance
Symbol mol
Mole is the SI unit of measurement used to measure the number of things, usually atoms or molecules. One mole of something is equal to 6.02214078×1023 of same things (Avogadro's number). For example, one mole of grapes is 6.02214078×1023 grapes.
The measurement of Avogadro's number was refined in 2019 to 6.02214078×1023.
Scientists use this number because it is the number of carbon atoms in 12 grams of carbon-12, which is the most common kind of carbon. Anything can be measured in moles, but it is not practical for most tasks because the value is so massive. For example, one mole of grapefruits would be as big as the earth.
The number does not lend itself to easy expression in words. The nearest "casual" number is one million-million-million-million, which is 1024.
Because different molecules and atoms do not have the same mass, one mole of one thing does not weigh the same as one mole of something else. Atoms and molecule mass is measured in amu. One amu is equal to one gram per mole. This means that if an atom has a mass of one amu, one mole of this atom weighs one gram.
## Mathematics with the mole
Moles = mass (g) / Relative mass (grams per mole)
Example: How many moles are there in 20 grams of hydrogen?
A value of 1 can be used for hydrogen's relative mass, although the correct value is slightly larger. So: moles = mass/relative mass = 20/1 = 20 moles.
Moles = concentration (mol/dm3) x volume (dm3)
Example: How many moles are there in 100cm3 of 0.1M H2SO4?
1 dm3 is the same as 1000 cm3, so the value in cubic centimetres needs to be divided by 1000. 100/1000 x 0.1 = 0.01 moles.
A methane molecule is made from one carbon atom and four hydrogen atoms. Carbon has a mass of 12.011 u and hydrogen has a mass of 1.008 u. This means that the mass of one methane molecule is 12.011 u + (4 × 1.008u), or 16.043 u. This means that one mole of methane has a mass of 16.043 grams.
A mole can be thought of as two bags of different sized balls. One bag contains 3 tennis balls and the other 3 footballs. There is the same number of balls in both bags but the mass of the footballs is much larger. It is a different way to measure things. Moles measure the number of particles, not the mass. So both bags contain three moles.
A mole is simply a unit of the number of things. Other common units include a dozen, meaning 12, and a score, meaning 20. Similarly, a mole refers to a specific quantity-- its distinguishing feature is that its number is far larger than other common units. Such units are typically invented when existing units can not describe something easily enough. Chemical reactions typically take place between molecules of varying weights, meaning measurements of mass (such as grams) can be misleading when compared the reactions of individual molecules. On the other hand, using the absolute number of atoms/molecules/ions would also be confusing, as the massive numbers involved would make it all too easy to misplace a value or drop a digit. As such, working in moles allows scientists to refer to a specific quantity of molecules or atoms without resorting to excessively large numbers.
## Related units
The SI units for molar concentration are mol/m3. However, most chemical writing uses mol/dm3, or mol dm-3, which is the same as mol/L. These units are often written with a capital letter M (pronounced "molar"), sometimes preceded by an SI prefix, for example, millimoles per litre (mmol/L) or millimolar (mM), micromoles/litre (µmol/L) or micromolar (µM), or nanomoles/L (nmol/L) or nanomolar (nM).
The absolute yield of a chemical reaction mostly stated in moles (called the "molar yield").
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Browse Questions
# The length of the perpendicular drawn from the point $(3,-1,11)$ on the line $\large\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is?
$\begin{array}{1 1} (a)\:\:\sqrt {29}\:\:\:\qquad\:\:(b)\:\:\sqrt {33}\:\:\:\qquad\:\:(c)\:\:\sqrt {53}\:\:\:\qquad\:\:(d)\:\:\sqrt {65}\end{array}$
Toolbox:
• $d.r.$ of the line joining $(x_1,y_1,z_1)\:\:and\:\: (x_2,y_2,z_2)$ is $(x_1-x_2,y_1-y_2,z_1-z_2)$
Given line: $\large\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}=\lambda$
Any point on this line can be given in terms of $\lambda$ as $(2\lambda,3\lambda+2,4\lambda+3)$
Since the foot of $\perp$ from $P(3,-1,11)$ on this line lies on the line,
let $Q(2\lambda,3\lambda+2,4\lambda+3)$ be the foot of $\perp$
$\therefore\:d.r.$ of $\perp$ $\overline {PQ}$ is $(2\lambda-3,3\lambda+3,4\lambda-8)$
Since $\overline {PQ}$ is $\perp$ to the line, $(2,3,4).(\overline{PQ})=0$
$\Rightarrow\:4\lambda-6+9\lambda+9+16\lambda-32=0$
$\Rightarrow\:29\lambda-29=0\:\:\:or\:\:\:\lambda=1$
$\therefore\:Q(2,5,7)$
$\therefore\:$ The length of $\perp =distance\: \overline{PQ}=\sqrt {1+36+16}=\sqrt {53}$
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Cody
# Problem 44360. Pentagonal Numbers
Solution 1288188
Submitted on 16 Oct 2017 by Massimo Zanetti
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
x1 = 1; x2 = 25; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[1,5,12,22])) assert(isequal(d,[0,1,0,0]))
p = 1 5 12 22 d = 0 1 0 0
2 Pass
x1 = 1; x2 = 4; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,1)) assert(isequal(d,0))
p = 1 d = 0
3 Pass
x1 = 10; x2 = 40; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[12,22,35])) assert(isequal(d,[0,0,1]))
p = 12 22 35 d = 0 0 1
4 Pass
x1 = 10; x2 = 99; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[12,22,35,51,70,92])) assert(isequal(d,[0,0,1,0,1,0]))
p = 12 22 35 51 70 92 d = 0 0 1 0 1 0
5 Pass
x1 = 100; x2 = 999; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[117,145,176,210,247,287,330,376,425,477,532,590,651,715,782,852,925])) assert(isequal(d,[0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1]))
p = 117 145 176 210 247 287 330 376 425 477 532 590 651 715 782 852 925 d = 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1
6 Pass
x1 = 40; x2 = 50; [p,d] = pentagonal_numbers(x1,x2) assert(isempty(p)) assert(isempty(d))
p = [] d = []
7 Pass
x1 = 1000; x2 = 1500; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[1001,1080,1162,1247,1335,1426])) assert(isequal(d,[0,1,0,0,1,0]))
p = 1001 1080 1162 1247 1335 1426 d = 0 1 0 0 1 0
8 Pass
x1 = 1500; x2 = 3000; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[1520,1617,1717,1820,1926,2035,2147,2262,2380,2501,2625,2752,2882])) assert(isequal(d,[1,0,0,1,0,1,0,0,1,0,1,0,0]))
p = 1520 1617 1717 1820 1926 2035 2147 2262 2380 2501 2625 2752 2882 d = 1 0 0 1 0 1 0 0 1 0 1 0 0
9 Pass
x1 = 1; x2 = 3000; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,590,651,715,782,852,925,1001,1080,1162,1247,1335,1426,1520,1617,1717,1820,1926,2035,2147,2262,2380,2501,2625,2752,2882])) assert(isequal(d,[0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0]))
p = Columns 1 through 14 1 5 12 22 35 51 70 92 117 145 176 210 247 287 Columns 15 through 28 330 376 425 477 532 590 651 715 782 852 925 1001 1080 1162 Columns 29 through 42 1247 1335 1426 1520 1617 1717 1820 1926 2035 2147 2262 2380 2501 2625 Columns 43 through 44 2752 2882 d = Columns 1 through 29 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 Columns 30 through 44 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0
10 Pass
x1 = 10000; x2 = 12000; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[10045,10292,10542,10795,11051,11310,11572,11837])) assert(isequal(d,[1,0,0,1,0,1,0,0]))
p = 10045 10292 10542 10795 11051 11310 11572 11837 d = 1 0 0 1 0 1 0 0
11 Pass
x1 = 100000; x2 = 110000; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[100492,101270,102051,102835,103622,104412,105205,106001,106800,107602,108407,109215])) assert(isequal(d,[0,1,0,1,0,0,1,0,1,0,0,1]))
p = 100492 101270 102051 102835 103622 104412 105205 106001 106800 107602 108407 109215 d = 0 1 0 1 0 0 1 0 1 0 0 1
12 Pass
x1 = 1000000; x2 = 1010101; [p,d] = pentagonal_numbers(x1,x2) assert(isequal(p,[1000825,1003277,1005732,1008190])) assert(isequal(d,[1,0,0,1]))
p = 1000825 1003277 1005732 1008190 d = 1 0 0 1
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## Which is a simplified form of the expression 2(y + 1) + 2(y – 2)? A. 2y + 1 B. 4y – 2 C. 2y – 1 D. 4y – 3
Question
Which is a simplified form of the expression 2(y + 1) + 2(y – 2)? A. 2y + 1 B. 4y – 2 C. 2y – 1 D. 4y – 3
in progress 0
2 weeks 2022-01-08T07:37:29+00:00 2 Answers 0 views 0
## Answers ( )
B) 4y-2
Step-by-step explanation:
2(y+1)+2(y-2)
2y+2+2y-4
2y+2y+2-4
4y+2-4
4y-2
2. It’s B because when simplified it’s 2y+2+2y-4 then you add like terms and it becomes 4y-2 which is B
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How do you write 98.180004 in scientific notation?
Feb 21, 2016
$9.8180004 \cdot {10}^{1}$
Explanation:
The goal of writing a number in scientific notation is to have a decimal number, with one digit to the left of the decimal point, followed by $a$ * ${10}^{s o m e p o w e r}$
So,dividing the 98.180004 by 10, 1 time(s), making it a 9.8180004
Now keeping the number in format $a$ * ${10}^{s o m e p o w e r}$
Thus,
the answer is $9.8180004 \cdot {10}^{1}$
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Add the numbers: 183 + 198 + 210 = ? Calculate the natural numbers sum and learn how to do the addition, column adding method, from right to left
The numbers addition to perform: 183 + 198 + 210
Stack the numbers on top of each other.
And so on...
1 8 3 1 9 8 + 2 1 0 ?
Add column by column; start from the column on the right.
Add the digits in the ones column:
1 is the tens digit. Carry it over to the tens column. Write the digit above that column. Add it with the rest of the digits in that column.
1 1 8 3 1 9 8 + 2 1 0 1
Add the digits in the tens column:
1 is the hundreds digit. Carry it over to the hundreds column. Write the digit above that column. Add it with the rest of the digits in that column.
1 1 1 8 3 1 9 8 + 2 1 0 9 1
Add the digits in the hundreds column:
5 is the hundreds digit. Write it down at the base of the hundreds column.
1 1 1 8 3 1 9 8 + 2 1 0 5 9 1
The latest 13 operations with added numbers:
183 + 198 + 210 = ? May 30 22:05 UTC (GMT) 967 + 1,223 + 233 = ? May 30 22:05 UTC (GMT) 1,016 + 2,128 + 368 + 344 + 569 + 194 + 152 + 870 + 145 + 6,540,166 = ? May 30 22:05 UTC (GMT) 4,868 + 4,039 + 1,911 + 2,023 = ? May 30 22:05 UTC (GMT) 7,825 + 3,068 = ? May 30 22:05 UTC (GMT) 805 + 840 = ? May 30 22:04 UTC (GMT) 892 + 1,143 + 181 = ? May 30 22:04 UTC (GMT) 691 + 618 = ? May 30 22:04 UTC (GMT) 837 + 1,117 + 135 = ? May 30 22:04 UTC (GMT) 1,001 + 883 = ? May 30 22:04 UTC (GMT) 928 + 9,273 + 2,037 = ? May 30 22:04 UTC (GMT) 134 + 186 + 7,492 = ? May 30 22:04 UTC (GMT) 1,030 + 2,149 + 363 + 360 + 605 + 187 + 168 + 904 + 137 + 6,540,194 = ? May 30 22:03 UTC (GMT) All the operations with the numbers added by users...
How to add natural numbers? Let's learn with an example:
Stack the numbers on top of each other.
The tens digits line up in the next column to the left.
4 8 + 8 5 ?
Add the digits in the ones column:
1 is the tens digit. Carry it over to the tens column. Write the digit above that column. Add it with the rest of the digits in that column.
1 4 8 + 8 5 3
Add the digits in the tens column:
1 is the hundreds digit. Since there is no hundreds column... Write it down at the base, next to the tens digit.
1 4 8 + 8 5 13 3
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# Handbook of Descriptive Statistics/Measures of Statistical Variability/Standard Deviation
Descriptive Statistics
Descriptive Statistics is statistical measures that are used to describe numerical data; this numerical data can be in form of a table, number list, frequency distribution etc.
Common Statistical measures are arithmetic mean (AM) (or average or simply the mean), median, mode and standard deviation.
Arithmetic Mean (or average or mean)
Arithmetic Mean (AM) of a list of n numbers is defined as the sum of n numbers divided by n. This is a kind of center of the number list or simply the average of the number list. The order of numbers in the list is not important.
For example, the AM of the number list 3, 8, 7, 3, 5, 10, and 2 is
${\displaystyle {\frac {3+9+7+3+6+10+4}{7}}=6}$
Median
Median is another type of central tendency of list of numbers. To calculate the median of list of n numbers, first the numbers should be arrange in ascending order i.e. from smallest to greatest. If n is odd then the median is defined as the middle number, if n is even then the median is defined as the average of two middle numbers. The order of the numbers is important for this tendency.
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# Homework Help: Multiple Choice Question about Modulus Function
1. Feb 7, 2010
### Atheismo
1. The problem statement, all variables and given/known data
Multiple choice question about a modulus function, which statement is false?
Below is the multiple choice question about a modulus function (the domain of the function is all reals):
Which one of the following is not true about the function f(x)= | 2x+4 | :
A. The graph of f is continuous everywhere
B. The graph of f ' is continuous everywhere ( f ' is the derivative of f)
C. f(x) is greater than or equal to zero for all values of x
D. f '(x) = 2 when x > 0
E. f '(x) = -2 when x < -2
End Question
Can someone please tell me the correct answer and why. Also, please explain to me what "continuous everywhere" means.
Thank you.
2. Feb 7, 2010
### Staff: Mentor
Continuous everywhere means that f is continuous at each point.
Draw a graph of this function and draw a graph of its derivative. After you do that, you should be able to figure out which is the false statement.
3. Feb 7, 2010
### Atheismo
But what does it mean by f is "continuous"? I don't know what that means.
Also, how do I find the derivative?
4. Feb 7, 2010
### Bohrok
A continuous graph means you can draw it without lifting your pencil from the paper (one way to look at it).
To find the derivative, rewrite it as |2x+4| = √(2x+4)2, since |x| = √(x2), and use the chain rule.
5. Feb 7, 2010
### Staff: Mentor
Your book should have a definition of this term, and probably has some examples of functions that are continuous and some that aren't.
If the problem is asking you about the derivative, it's reasonable for us to assume that differentiation has already been covered in your course.
6. Feb 7, 2010
### Atheismo
Yes it has but a while ago. I need to do some revision. I got the answer, I think, it's B right?
Answers D and E give it away lol.
Last edited: Feb 7, 2010
7. Feb 7, 2010
### Staff: Mentor
Yes, b is the one that is not true.
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Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics
## MATH4AA Asymptotic Analysis
Second Semester
10 points
Asymptotic analysis is a branch of mathematical analysis concerned with the limiting behavior of functions, in particular the behavior of functions when a variable or parameter is "large" or "small". The tools of asymptotic analysis are useful in describing the unknown solution to a number of algebraic, differential, or integral equations, particularly for those cases where solving such an equation exactly is not possible. One can, in such contexts, view the approach as yielding results which lie somewhere between exact solutions (which are nice, yet uncommon for many equations) and numerical simulations (which can be quite useful for understanding the behavior of a solution for a specific collection of parameters, but often less useful for deducing general behavior or structure). In a number of cases, an asymptotic approximation for a given function may even provide more insight than does the exact solution.
We provide a survey of techniques in asymptotic analysis, and you will be exposed to a wide assortment of tools related to the asymptotic approximation of functions and perturbation methods for the solution of differential equations. In order to make the material self-contained, we first review useful preliminaries, before moving on to the core material involving the asymptotic approximation of functions (including infinite series, products, and integrals) as well as the approximation of solutions to algebraic equations. We then survey various tools which are useful in the study of solutions to ODE and PDE.
### Prerequisites
MATH4A1, or permission of the lecturer.
### Course Outline
This paper will cover a selection of topics related to asymptotic analysis and perturbation methods. Topics included in lectures will be selected from the following:
I. Basic definitions and results in asymptotic analysis
A. Asymptotic expansion of functions and roots of algebraic equations.
B. Expansion and approximation of functions taking the form of infinite series, products, and integrals.
C. Approximation of integrals with a large or small parameter (integration by parts, Watson's lemma, Laplace's method, method of stationary phase, method of steepest descent).
D. Divergence and resummation of asymptotic series.
II. Methods for ODE
A. Expansion of ODE solutions in dependent variables.
B. Large-time behavior of ODE solutions and linear stability analysis.
C. Expansion of ODE solutions in small or large parameters (regular and singular perturbation, method of matched asymptotic expansions, WKB method, Poincaré-Lindstedt method).
D. Approaches for ODE when there are no small parameters (delta-expansion method, homotopy analysis method, iterative methods).
III. Methods for PDE
A. Generalized Fourier series. Approximation of Sturm-Liouville eigenvalues.
B. Non-dimensionalization. Symmetries and self-similarity.
C. Approximation of exact PDE solutions given in integral or series form.
D. Long-time behavior and instabilities in diffusion processes.
E. Multiscale analysis, averaging, and homogenization.
B. delta-expansion method
C. Homotopy analysis method
D. Modern research topics
### Lecturer
Dr R. A. Van Gorder
### Literature
Lecture notes will be the primary reference for this paper. Other literature may be provided, but only the material in the lecture notes and take-home problem sets will be examinable. For those wishing for additional sources or additional practice problems, several useful books include:
O.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers.
E.J. Hinch, Perturbation Methods, Cambridge 1991.
J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, Springer 1985.
A.H. Nayfeh, Perturbation Methods, Wiley 1973.
M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press 1964.
### Assessment
Assessment will be through assignments only. These will be take-home problem sets.
### Final mark
Your final mark F in the paper will be calculated according to this formula:
F = A
where:
• A is the Assignments mark
and all quantities are expressed as percentages.
### Students must abide by the University’s Academic Integrity Policy
Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.
Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.
Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.
All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.
Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.
If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.
Types of academic misconduct are as follows:
#### Plagiarism
The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).
• Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
• Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.
#### Unauthorised Collaboration
Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..
#### Impersonation
Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.
#### Falsification
Falsification is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.
#### Use of Unauthorised Materials
Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.
#### Assisting Others to Commit Academic Misconduct
This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.
Further information
While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
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https://www.jobilize.com/physics/section/conceptual-questions-kinetic-energy-and-the-work-energy-by-openstax?qcr=www.quizover.com
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# 4.2 Kinetic energy and the work-energy theorem (Page 3/6)
Page 3 / 6
Solution for (a)
The net force is the push force minus friction, or ${F}_{\text{net}}\text{= 120 N – 5}\text{.}\text{00 N = 115 N}$ . Thus the net work is
$\begin{array}{lll}{W}_{\text{net}}& =& {F}_{\text{net}}d=\left(\text{115 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{92.0 N}\cdot m=\text{92.0 J.}\end{array}$
Discussion for (a)
This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force.
Strategy and Concept for (b)
The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work.
Solution for (b)
The applied force does work.
${W}_{\mathrm{a}\mathrm{p}\mathrm{p}}={F}_{\mathrm{a}\mathrm{p}\mathrm{p}}d=\left(120\phantom{\rule{.1667em}{0ex}}\mathrm{N}\right)\left(0.800\phantom{\rule{.1667em}{0ex}}\mathrm{m}\right)=96.0\phantom{\rule{.1667em}{0ex}}\mathrm{J}$
The friction force and displacement are in opposite directions. The work done by friction is therefore negative.
${W}_{\mathrm{f}\mathrm{r}}=-{F}_{\mathrm{f}\mathrm{r}}d=-\left(5.00\phantom{\rule{.1667em}{0ex}}\mathrm{N}\right)\left(0.800\phantom{\rule{.1667em}{0ex}}\mathrm{m}\right)=-4.00\phantom{\rule{.1667em}{0ex}}\mathrm{J}$
So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively,
$\begin{array}{lll}{W}_{\text{gr}}& =& 0,\\ {W}_{N}& =& 0,\\ {W}_{\text{app}}& =& \text{96.0 J,}\\ {W}_{\text{fr}}& =& -\text{4.00 J.}\end{array}$
The total work done as the sum of the work done by each force is then seen to be
${W}_{\text{total}}={W}_{\text{gr}}+{W}_{N}+{W}_{\text{app}}+{W}_{\text{fr}}=\text{92}\text{.0 J}.$
Discussion for (b)
The calculated total work ${W}_{\text{total}}$ as the sum of the work by each force agrees, as expected, with the work ${W}_{\text{net}}$ done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach.
## Determining speed from work and energy
Find the speed of the package in [link] at the end of the push, using work and energy concepts.
Strategy
Here the work-energy theorem can be used, because we have just calculated the net work, ${W}_{\text{net}}$ , and the initial kinetic energy, $\frac{1}{2}{m{v}_{0}}^{2}$ . These calculations allow us to find the final kinetic energy, $\frac{1}{2}{\text{mv}}^{2}$ , and thus the final speed $v$ .
Solution
The work-energy theorem in equation form is
${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}\text{.}$
Solving for $\frac{1}{2}{\text{mv}}^{2}$ gives
$\frac{1}{2}{\text{mv}}^{\text{2}}={W}_{\text{net}}+\frac{1}{2}{m{v}_{0}}^{2}\text{.}$
Thus,
$\frac{1}{2}{\text{mv}}^{2}=\text{92}\text{.}0 J+3\text{.}\text{75 J}=\text{95.}\text{75 J.}$
Solving for the final speed as requested and entering known values gives
$\begin{array}{lll}v& =& \sqrt{\frac{2\text{(95.75 J)}}{m}}=\sqrt{\frac{\text{191.5 kg}\cdot {m}^{2}{\text{/s}}^{2}}{\text{30.0 kg}}}\\ & =& \text{2.53 m/s.}\end{array}$
Discussion
Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package.
## Work and energy can reveal distance, too
How far does the package in [link] coast after the push, assuming friction remains constant? Use work and energy considerations.
Strategy
We know that once the person stops pushing, friction will bring the package to rest. In terms of energy, friction does negative work until it has removed all of the package’s kinetic energy. The work done by friction can be expressed as the force of friction times the distance traveled and as the change in kinetic energy. Equating both expressions for work gives us a way of finding the distance traveled.
Solution
The normal force and force of gravity cancel in calculating the net force. The horizontal friction force is then the net force, and it acts opposite to the displacement. To reduce the kinetic energy of the package to zero, the work ${W}_{\text{fr}}$ by friction must be equal to the change in kinetic energy.
${W}_{\mathrm{f}\mathrm{r}}=-{F}_{\mathrm{f}\mathrm{r}}d=\mathrm{\Delta }\mathrm{K}\mathrm{E}$
$-{F}_{\mathrm{f}\mathrm{r}}d=\mathrm{\Delta }\mathrm{K}\mathrm{E}$
and so
$-{F}_{\mathrm{f}\mathrm{r}}d=\frac{1}{2}m{v}^{2}-\frac{1}{2}m{v}_{0}^{2}$
Initial speed is the speed at the instant the push stops (the result determined in the previous example). The final speed is zero for this case and leads to the expression below.
${F}_{\mathrm{f}\mathrm{r}}d=\frac{1}{2}m{v}_{0}^{2}$
Solving for distance, we obtain the result shown below.
$d=\frac{95.75\phantom{\rule{.1667em}{0ex}}\mathrm{J}}{5.00\phantom{\rule{.1667em}{0ex}}\mathrm{N}}=19.2\phantom{\rule{.1667em}{0ex}}\mathrm{m}$
Discussion
This is a reasonable distance for a package to coast on a relatively friction-free conveyor system. Note that the work done by friction is negative (the force is in the opposite direction of motion), so it removes the kinetic energy.
Some of the examples in this section can be solved without considering energy, but at the expense of missing out on gaining insights about what work and energy are doing in this situation. On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone.
## Section summary
• The net work ${W}_{\text{net}}$ is the work done by the net force acting on an object.
• Work done on an object transfers energy to the object.
• The translational kinetic energy of an object of mass $m$ moving at speed $v$ is $\text{KE}=\frac{1}{2}{\text{mv}}^{2}$ .
• The work-energy theorem states that the net work ${W}_{\text{net}}$ on a system changes its kinetic energy, ${W}_{\text{net}}=\mathrm{\Delta }\mathrm{K}\mathrm{E}=\frac{1}{2}m{v}^{2}-\frac{1}{2}m{v}_{0}^{2}.$
## Conceptual questions
The person in [link] does work on the lawn mower. Under what conditions would the mower gain energy? Under what conditions would it lose energy?
Work done on a system puts energy into it. Work done by a system removes energy from it. Give an example for each statement.
When solving for speed in [link] , we kept only the positive root. Why?
## Problems&Exercises
Compare the kinetic energy of a 20,000-kg truck moving at 30.0 m/s with that of an 80.0-kg astronaut in orbit moving at 7,500 m/s.
$1/\text{250}$
(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.
(a) Calculate the force needed to bring a 950-kg car to rest from a speed of 25.0 m/s in a distance of 120 m (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a).
A car’s bumper is designed to withstand a 1.1 m/s collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1 m/s.
$2\text{.}8×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N}$
Boxing gloves are padded to lessen the force of a blow. (a) Calculate the force exerted by a boxing glove on an opponent’s face, if the glove and face compress 7.50 cm during a blow in which the 7.00-kg arm and glove are brought to rest from an initial speed of 10.0 m/s. (b) Calculate the force exerted by an identical blow in the gory old days when no gloves were used and the knuckles and face would compress only 2.00 cm. (c) Discuss the magnitude of the force with glove on. Does it seem high enough to cause damage even though it is lower than the force with no glove?
Using energy considerations, calculate the average force a 60.0-kg sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him.
102 N
#### Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
Ramkumar Reply
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# Konversi BERAT-MASSAfemtograms ke kilopounds
1 Femtograms
= 2.2046226218488E-21 Kilopounds
Besaran: berat massa
Konversi Satuan: Femtograms ke Kilopounds
Satuan dasar untuk berat massa adalah kilograms (SI Unit)
Simbol dari [Femtograms] adalah: (fg), sedangkan simbol untuk [Kilopounds] adalah: (kip), keduanya merupakan satuan dari berat massa
Perhitungan cepat konversi Femtograms ke Kilopounds (fg ke kip):
1 fg = 2.2046226218488E-21 kip.
1 x 2.2046226218488E-21 kip = 2.2046226218488E-21 Kilopounds.
*catatan: kesalahan atau error kecil dalam pembulatan hasil angka desimal bisa terjadi, silakan dicek ulang.
Definisi:
Berdasarkan satuan/unit dari besaran berat massa, yaitu => (kilograms), 1 Femtograms (fg) sama dengan 1.0E-18 kilograms, sedangkan 1 Kilopounds (kip) = 453.59237 kilograms.
oo
Femtogramsto Kilopounds (table conversion)
1 fg = 2.2046226218488E-21 kip
2 fg = 4.4092452436976E-21 kip
3 fg = 6.6138678655463E-21 kip
4 fg = 8.8184904873951E-21 kip
5 fg = 1.1023113109244E-20 kip
6 fg = 1.3227735731093E-20 kip
7 fg = 1.5432358352941E-20 kip
8 fg = 1.763698097479E-20 kip
9 fg = 1.9841603596639E-20 kip
10 fg = 2.2046226218488E-20 kip
20 fg = 4.4092452436976E-20 kip
30 fg = 6.6138678655463E-20 kip
40 fg = 8.8184904873951E-20 kip
50 fg = 1.1023113109244E-19 kip
60 fg = 1.3227735731093E-19 kip
70 fg = 1.5432358352941E-19 kip
80 fg = 1.763698097479E-19 kip
90 fg = 1.9841603596639E-19 kip
100 fg = 2.2046226218488E-19 kip
200 fg = 4.4092452436976E-19 kip
300 fg = 6.6138678655463E-19 kip
400 fg = 8.8184904873951E-19 kip
500 fg = 1.1023113109244E-18 kip
600 fg = 1.3227735731093E-18 kip
700 fg = 1.5432358352941E-18 kip
800 fg = 1.763698097479E-18 kip
900 fg = 1.9841603596639E-18 kip
1000 fg = 2.2046226218488E-18 kip
2000 fg = 4.4092452436976E-18 kip
4000 fg = 8.8184904873951E-18 kip
5000 fg = 1.1023113109244E-17 kip
7500 fg = 1.6534669663866E-17 kip
10000 fg = 2.2046226218488E-17 kip
25000 fg = 5.5115565546219E-17 kip
50000 fg = 1.1023113109244E-16 kip
100000 fg = 2.2046226218488E-16 kip
1000000 fg = 2.2046226218488E-15 kip
1000000000 fg = 2.2046226218488E-12 kip
(Femtograms) to (Kilopounds) conversions
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# Often asked: How To Find Price Per Square Inch Of Pizza?
## How do you find the cost per square inch?
To calculate the price – per – square – inch of a work of art is simple: you multiply the length of the piece by its width to get the number of square inches, and then you divide the price of the piece by the number of square inches. (You can also use united inches instead of square inches.)
## How many square inches is a pizza?
A small pizza is usually 8 inches across. When you plug that into the top half of the formula above, you find out that you get around 50.3 square inches of pizza. A medium, on the other hand, is often closer to 12 inches —which comes out to more than 113 square inches.
## How do you find the surface area of a pizza?
The surface area of a 14-inch pizza is 3.14 x 49 (7 x 7 = 49) = 153.86 square inches of surface area.
You might be interested: Question: What To Put On Homemade Pizza?
## How do you find the price per area?
In the real estate industry, a common way to measure a property’s value for money is the price per square foot of livable floor space (we generally don’t include lofts, cupboards, etc.). The equation to calculate this metric is: price per square foot = price / floor space (ft²).
## How do you price art for beginners?
Multiply the painting’s width by its length to arrive at the total size, in square inches. Then multiply that number by a set dollar amount that’s appropriate for your reputation. I currently use \$6 per square inch for oil paintings. Then calculate your cost of canvas and framing, and then double that number.
## How do you calculate price per volume?
Cost per unit of volume (cubic unit) can be obtained by multiplying the dimensions (to get the volume of a rectangular parallelepiped) and dividing the result by cost of strip.
## Are 2 small pizzas bigger than a large?
A large pizza is almost always a better deal than two mediums. For example, a 16-inch large might seem twice as big as an 8 inch small but it’s actually four times as much pizza.
## How big is a 28 in pizza?
Well, with a 28 ” pizza, you would know what you are getting. This giant pizza has between 12 to 14 slices. If you were to cut it yourself, you could customize it to make larger slices for yourself and smaller ones for your kid, for example.
## How is pizza size measured?
Pizzas are sized according to the measurement of the diameter, so a pizza with a 10-inch diameter would be considered small; a 12-inch diameter would be a medium; a 14-inch diameter would be a large.
You might be interested: Often asked: How To Cook Pizza In Microwave?
## How large is a 16 inch pizza?
A 16 – inch pie is 201 square inches — approximately 31 percent larger, in terms of area, than a 14- inch pie.
## How is a 12 inch pizza measured?
All are traditional round pizza pies designated by the approximate diameter measured across the middle (see picture). Our smallest pizza is 9 inches in diameter. The middle size is 12 inches in diameter. For a 12 ” pizza, 3.14 times 6 times 6 gives us about 113 square inches.
## How big is a 12 inch pizza?
Small pizzas average between 8 and 10 inches in diameter and will yield about six slices. Medium pizzas run 12 inches in diameter and will give you about eight slices. Large pizzas are 14 inches in diameter and will offer approximately 10 slices.
## What is a good price per square foot?
According to the latest estimates, the median price for each square foot for a home in the United States is \$123. But that can vary widely based on where you live and other factors. For instance, on the low end, you’ll pay \$24 a square foot in Detroit. On the expensive end, in San Francisco, \$810.
## How much does it cost to build a 2 500 square foot home?
The average cost to build a house is \$248,000, or between \$100 to \$155 per square foot depending on your location, size of the home, and if modern or custom designs are used. New home construction for a 2,000 square foot home runs \$201,000 to \$310,000 on average.
## How do you calculate price per acre?
Calculating cost per acre is a straightforward process. This involves adding up all the numbers you have and dividing by the total number of values present.
1. Add together the total ” cost per acre ” dollar amounts you have located.
2. Count the total number of ” cost per acre ” dollar amounts you found.
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# Simplifying roots
Here are the basic rules of algebra for simplifying roots. In these identities, a and b are arbitrary real numbers, while m and n are positive integers; however, the precise rule may depend on whether n is odd or even. In each rule, if the right-hand side is defined, then so is the left-hand side and the two sides are then equal.
• n0 = 0.
• n1 = 1.
• nab = na ⋅ nb.
• na/b = na ÷ nb.
• 1a = a.
• nma = mna.
• nan = |a| if n is even; nan = a if n is odd.
• namn = |a|m if n is even; namn = am if n is odd.
• (mn)an = m|a| if n is even; (mn)an = ma if n is odd.
The overall theme of the last three rules is that roots and powers cancel, but the cancelled number leaves behind an absolute value if it was even. That said, you can ignore the absolute-value operation when a ≥ 0.
Here are some examples of taking roots:
The permanent URI of this web page is `http://tobybartels.name/MATH-1100/2021FA/rootrules/`.
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# The Molecular mass of HNO3 is _____?
By BYJU'S Exam Prep
Updated on: September 25th, 2023
(A) 61 u
(B) 62 u
(C) 63 u
(D) 64 u
The Molecular mass of HNO3 is 63 u. The molecular formula of the substance, which uses subscripts to show the atoms that make up the compound and the number of molecules in each one. If the molecular formula is known, one can compute:
• Molecular mass or molar mass.
• Percentage of elements in the compound.
### Explain Molecular Mass
The molecular mass of any material is calculated by adding the masses of all of its constituent atoms. It is the molecular weight of a single chemical. A substance’s molecular mass, or the mass of its 6.022 × 1023 particles, is referred to as its molar mass.
• The symbol for a unit’s mass is the atomic mass unit, also known as the letter u.
• If the percentage composition and molar mass are known, it is possible to derive the empirical formula, which expresses the formula in the simplest whole-number ratio.
Nitric acid’s (HNO3) molecular weight is 63 u. The total mass of all the atoms in a given molecule is known as its molecular mass. It is represented in units of atomic mass (amu). As an illustration, the molecular weight of HNO3 can be calculated as:
H has an atomic mass of 1u.
N has an atomic mass of 14u.
O has an atomic mass of 16u.
Molecular mass of HNO3 = 1 + 14 + (16 × 3) = 63u.
The correct answer is Option(3).i.e. 63 u.
Summary:
## The Molecular mass of HNO3 is _____? (A) 61 u (B) 62 u (C) 63 u (D) 64 u
The Molecular mass of HNO3 is 63 u. The substance’s molecular formula, which uses subscripts to show the atoms that make up the compound and how many molecules there are in total.
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# 4.2: Polynomial Functions
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In the previous section we explored the short run behavior of quadratics, a special case of polynomials. In this section we will explore the behavior of polynomials in general. The basic building blocks of polynomials are power functions.
Definition: Power Function
A power function is a function that can be represented in the form
$f(x)=x^p \label{power}$
where the base is the variable and the exponent, $$p$$, is a numbers.
Q&A: Is $$f(x)=2^x$$ a power function?
No. A power function contains a variable base raised to a fixed power (Equation \ref{power}). This function has a constant base raised to a variable power. This is called an exponential function, not a power function. This function will be discussed later.
## Characteristics of Power Functions
Figure $$\PageIndex{2}$$ shows the graphs of $$f(x)=x^2$$, $$g(x)=x^4$$ and and $$h(x)=x^6$$, which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.
To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol $$\infty$$ for positive infinity and $$−\infty$$ for negative infinity. When we say that “x approaches infinity,” which can be symbolically written as $$x{\rightarrow}\infty$$, we are describing a behavior; we are saying that $$x$$ is increasing without bound.
With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as $$x$$ approaches positive or negative infinity, the $$f(x)$$ values increase without bound. In symbolic form, we could write
$\text{as } x{\rightarrow}{\pm}{\infty}, \;f(x){\rightarrow}{\infty} \nonumber$
Figure $$\PageIndex{3}$$ shows the graphs of $$f(x)=x^3$$, $$g(x)=x^5$$, and $$h(x)=x^7$$, which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.
These examples illustrate that functions of the form $$f(x)=x^n$$ reveal symmetry of one kind or another. First, in Figure $$\PageIndex{2}$$ we see that even functions of the form $$f(x)=x^n$$, $$n$$ even, are symmetric about the $$y$$-axis. In Figure $$\PageIndex{3}$$ we see that odd functions of the form $$f(x)=x^n$$, $$n$$ odd, are symmetric about the origin.
For these odd power functions, as $$x$$ approaches negative infinity, $$f(x)$$ decreases without bound. As $$x$$ approaches positive infinity, $$f(x)$$ increases without bound. In symbolic form we write
\begin{align*} &\text{as }x{\rightarrow}-{\infty},\;f(x){\rightarrow}-{\infty} \\ &\text{as }x{\rightarrow}{\infty},\;f(x){\rightarrow}{\infty} \end{align*}
Long Run Behavior
The behavior of the graph of a function as the input values get very small $$(x{\rightarrow}−{\infty})$$ and get very large $$x{\rightarrow}{\infty}$$ is referred to as the long run behavior of the function. We can use words or symbols to describe end behavior.
## Polynomials
Definition: Word
A polynomial is function that can be written as $$f(x) = {a_0} + {a_1}x + {a_2}{x^2} + \cdots + {a_n}{x^n}$$
Each of the ai constants are called coefficients and can be positive, negative, or zero, and be whole numbers, decimals, or fractions.
A term of the polynomial is any one piece of the sum, that is any $$a_ix^i$$. Each individual term is a transformed power function.
The degree of the polynomial is the highest power of the variable that occurs in the polynomial.
The leading term is the term containing the highest power of the variable: the term with the highest degree.
Because of the definition of the “leading” term we often rearrange polynomials so that the powers are descending. $$f(x) = {a_n}{x^n} + ..... + {a_2}{x^2} + {a_1}x + a_0$$
Example $$\PageIndex{1}$$
Which of the following are polynomial functions?
• $$f(x)=2x^3⋅3x+4$$
• $$g(x)=−x(x^2−4)$$
• $$h(x)=5\sqrt{x}+2$$
Solution
For the function $$f(x)$$, the degree is 3, the highest power on $$x$$. The leading term is the term containing that power, $$- 4x^3$$. The leading coefficient is the coefficient of that term, -4.
For $$g(t)$$, the degree is 5, the leading term is $$5{t^5}$$, and the leading coefficient is 5.
Example $$\PageIndex{1}$$: Identifying the Degree and Leading Coefficient of a Polynomial Function
Identify the degree, leading term, and leading coefficient of the following polynomial functions.
$$f(x)=3+2x^2−4x^3$$
$$g(t)=5t^5−2t^3+7t$$
$$h(p)=6p−p^3−2$$
Solution
For the function $$f(x)$$, the highest power of $$x$$ is 3, so the degree is 3. The leading term is the term containing that degree, $$−4x^3$$. The leading coefficient is the coefficient of that term, −4.
For the function $$g(t)$$, the highest power of $$t$$ is 5, so the degree is 5. The leading term is the term containing that degree, $$5t^5$$. The leading coefficient is the coefficient of that term, 5.
For the function $$h(p)$$, the highest power of $$p$$ is 3, so the degree is 3. The leading term is the term containing that degree, $$−p^3$$; the leading coefficient is the coefficient of that term, −1.
Long Run Behavior of Polynomials
For any polynomial, the long run behavior of the polynomial will match the long run behavior of the leading term.
Example $$\PageIndex{2}$$: Identifying End Behavior and Degree of a Polynomial Function
What can we determine about the long run behavior and degree of the equation for the polynomial graphed in Figure $$\PageIndex{8}$$.
Solution
Since the output grows large and positive as the inputs grow large and positive, we describe the long run behavior symbolically by writing:
$\text{as } x{\rightarrow}{\infty}, \; f(x){\rightarrow}{\infty} \nonumber$
$\text{as } x{\rightarrow}-{\infty}, \; f(x){\rightarrow}-{\infty} \nonumber$
In words, we could say that as $$x$$ values approach infinity, the function values approach infinity, and as $$x$$ values approach negative infinity, the function values approach negative infinity.
We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.
## Short Run Behavior: Intercepts
Characteristics of the graph such as vertical and horizontal intercepts and the places the graph changes direction are part of the short run behavior of the polynomial.
Like with all functions, the vertical intercept is where the graph crosses the vertical axis, and occurs when the input value is zero. Since a polynomial is a function, there can only be one vertical intercept, which occurs at the point $$0, a_0$$. The horizontal intercepts occur at the input values that correspond with an output value of zero. It is possible to have more than one horizontal intercept.
Horizontal intercepts are also called zeros, or roots of the function.
To find horizontal intercepts, we need to solve for when the output will be zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and 4th degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases:
1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
2. The polynomial is given in factored form.
3. Technology is used to determine the intercepts.
Example $$\PageIndex{3}$$
Find the horizontal intercepts of $$f(x) = {x^6} - 3{x^4} + 2{x^2}$$.
Solution
We can attempt to factor this polynomial to find solutions for $$f(x) = 0$$.
\begin{align*} x^6- 3x^4 + 2x^2&= 0\quad && \text{Factoring out the greatest common factor}\\ x^2\left( x^4 - 3x^2 + 2\right) &= 0\quad && \text{Factoring the inside as a quadratic in x^2} \\ x^2\left( x^2- 1\right)\left(x^2 - 2\right) &= 0\quad && \text{Then break apart to find solutions}\\ x^2 = 0 \quad & \quad \left(x^2 - 1\right) = 0 &&\quad \left(x^2-2\right)\\ x = 0 \quad &\quad \; x^2= 1&&\quad\; x^2=2 \\ &\quad \; x =\pm \sqrt 2 &&\;\quad x=\pm\sqrt{2} \end{align*}This gives us 5 horizontal intercepts.
Example $$\PageIndex{4}$$
Find the vertical and horizontal intercepts of $$g(t) = (t - 2)^2(2t + 3)$$.
Solution
The vertical intercept can be found by evaluating $$g(0)$$.
$$g(0) = {(0 - 2)^2}(2(0) + 3) = 12$$
The horizontal intercepts can be found by solving $$g(t) = 0$$.
$(t - 2)^2(2t + 3) = 0 \nonumber$
Since this is already factored, we can break it apart:
\begin{align*} (t - 2)^2&= 0 \quad\quad& (2t + 3)&= 0 \\ t - 2 &= 0 \quad&t &= \frac{- 3}{2} \\ t &= 2&& \end{align*}
Example $$\PageIndex{5}$$
Find the horizontal intercepts of $$h(t) = {t^3} + 4{t^2} + t - 6$$
Solution
Since this polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques we know, we can turn to technology to find the intercepts.
Graphing this function, it appears there are horizontal intercepts at $$t = -3, -2$$, and $$1$$.
We could check these are correct by plugging in these values for t and verifying that $$h( - 3) = h( - 2) = h(1) = 0$$.
Notice that the polynomial in the previous example was degree three, and had three horizontal intercepts and two turning points – places where the graph changes direction. We will now make a general statement without justifying it.
Definition: Intercepts and Turning Points of Polynomials
A polynomial of degree n will have:
At most $$n$$ horizontal intercepts. An odd degree polynomial will always have at least one,
At most $$n-1$$ turning points.
Exercise $$\PageIndex{1}$$
Find the vertical and horizontal intercepts of the function $$f(t)=t^4-4t^2$$.
Vertical intercept $$(0, 0)$$, Horizontal intercepts $$(0, 0)$$, $$(-2, 0)$$, $$(2, 0)$$
## Graphical Behavior at Intercepts
If we graph the function $$f(x)=(x+3)(x−2)^2(x+1)^3,$$ notice that the behavior at each of the horizontal intercepts is different.
The horizontal intercept $$x=−3$$, coming from the $$(x+3)$$ factor of the polynomial, the graph passes directly through the horizontal intercept. The factor is linear (has a power of 1), so the behavior near the intercept is like that of a line - it passes directly through the intercept. We call this a single zero, since the zero corresponds to a single factor of the function.
At the horizontal intercept $$x=2$$ coming from the $$(x−2)^2$$ factor of the polynomial, the graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic – it bounces off of the horizontal axis at the intercept. Since
$(x−2)^2=(x−2)(x−2), \nonumber$
The factor is repeated twice, that is, the factor $$(x−2)$$ appears twice, so we call this a double zero. We could also say the zero has multiplicity 2.
The horizontal intercept $$x=−1$$ coming from the $$(x+1)^3$$ factor of the polynomial, the graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic, with the same “S” type shape near the intercept that the toolkit $$x^3$$ has. We call this a triple zero. We could also say the zero has multiplicity 3.
By utilizing these behaviors, we can sketch a reasonable graph of a factored polynomial function without needing technology.
Graphical Behavior of Polynomials at Horizontal Intercepts
If a polynomial contains a factor of the form $$(x-h)^p$$, the behavior near the horizontal intercept h is determined by the power on the factor.
Figure $$\PageIndex{8}$$: Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.
For higher even powers 4, 6, 8 etc.… the graph will still bounce off of the horizontal axis but the graph will appear flatter with each increasing even power as it approaches and leaves the axis.
For higher odd powers, 5, 7, 9 etc… the graph will still pass through the horizontal axis but the graph will appear flatter with each increasing odd power as it approaches and leaves the axis.
Example $$\PageIndex{8}$$: Sketching the Graph of a Polynomial Function
Sketch a graph of $$f(x)=−2(x+3)^2(x−5)$$.
Solution
This graph has two x-intercepts. At $$x=−3$$, the factor is squared, indicating a multiplicity of 2. The graph will bounce at this x-intercept. At $$x=5$$,the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.
The y-intercept is found by evaluating $$f(0)$$.
\begin{align*} f(0)&=−2(0+3)^2(0−5) \\ &=−2⋅9⋅(−5) \\ &=90 \end{align*}
The y-intercept is $$(0,90)$$.
Additionally, we can see the leading term, if this polynomial were multiplied out, would be $$−2x3$$, so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See Figure $$\PageIndex{13}$$.
To sketch this, we consider that:
• As $$x{\rightarrow}−{\infty}$$ the function $$f(x){\rightarrow}{\infty}$$,so we know the graph starts in the second quadrant and is decreasing toward the x-axis.
• Since $$f(−x)=−2(−x+3)^2(−x–5)$$ is not equal to $$f(x)$$, the graph does not display symmetry.
• At $$(−3,0)$$, the graph bounces off of thex-axis, so the function must start increasing.
• At $$(0,90)$$, the graph crosses the y-axis at the y-intercept. See Figure $$\PageIndex{14}$$.
Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at $$(5,0)$$. See Figure $$\PageIndex{15}$$.
As $$x{\rightarrow}{\infty}$$ the function $$f(x){\rightarrow}−{\infty}$$,
so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.
Using technology, we can create the graph for the polynomial function, shown in Figure $$\PageIndex{16}$$, and verify that the resulting graph looks like our sketch in Figure $$\PageIndex{15}$$.
Figure $$\PageIndex{16}$$: The complete graph of the polynomial function $$f(x)=−2(x+3)^2(x−5)$$.
## Solving Polynomial Inequalities
One application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative. We can solve polynomial inequalities by either utilizing the graph, or by using test values.
Example $$\PageIndex{7}$$
Solve $$(x + 3)(x + 1)^2(x - 4) > 0$$.
Solution
As with all inequalities, we start by solving the equality $$(x + 3)(x + 1)^2(x - 4) = 0$$, which has solutions at $$x= -3, -1,$$ and $$4$$. We know the function can only change from positive to negative at these values, so these divide the inputs into 4 intervals.
We could choose a test value in each interval and evaluate the function $$f(x) = (x + 3){(x + 1)^2}(x - 4)$$ at each test value to determine if the function is positive or negative in that interval
Interval Test $$x$$ in interval $$f$$(test value) >0 or <0? $$x < -3$$ -4 72 > 0 $$-3 < x < -1$$ -2 -6 < 0 $$-1 < x < 4$$ 0 -12 < 0 $$x > 4$$ 5 288 > 0
On a number line this would look like:
From our test values, we can determine this function is positive when $$x<-3$$ or $$x>4$$, or in interval notation, $$(-\infty ,-3) \cup (4,\infty)$$.
We could have also determined on which intervals the function was positive by sketching a graph of the function. We illustrate that technique in the next example
Example $$\PageIndex{8}$$
Find the domain of the function $$v(t) = \sqrt {6 - 5t - t^2}$$.
Solution
A square root is only defined when the quantity we are taking the square root of, the quantity inside the square root, is zero or greater. Thus, the domain of this function will be when $$6 - 5t - t^2 \geq 0$$.
Again we start by solving the equality $$6 - 5t - t^2 = 0$$. While we could use the quadratic formula, this equation factors nicely to $$(6 + t)(1 - t) = 0$$, giving horizontal intercepts $$t = 1$$ and $$t = -6$$. Sketching a graph of this quadratic will allow us to determine when it is positive.
From the graph we can see this function is positive for inputs between the intercepts. So $$6 - 5t - t^2 \geq 0$$ for $$-6 leq t \leq 1$$, and this will be the domain of the $$v(t)$$ function.
Exercise $$\PageIndex{2}$$
Given the function $$g(x) = x^3 - x^2 - 6x$$ use the methods that we have learned so far to find the vertical & horizontal intercepts, determine where the function is negative and positive, describe the long run behavior and sketch the graph without technology.
Vertical intercept $$(0, 0)$$, Horizontal intercepts $$(-2, 0), (0, 0), (3, 0).$$
The function is negative on $$(-\infty, -2)$$ and $$(0, 3)$$.
The function is positive on $$(-2, 0)$$ and $$(3,\infty)$$.
The leading term is $$x^3$$ so as $$x \to - \infty, g(x) \to -\infty$$ and as $$x \to \infty, g(x) \to \infty$$.
## Estimating Extrema
With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Example $$\PageIndex{9}$$
An open-top box is to be constructed by cutting out squares from each corner of a 14cm by 20cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.
Solution
An open-top box is to be constructed by cutting out squares from each corner of a 14cm by 20cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.
We will start this problem by drawing a picture, labeling the width of the cut-out squares with a variable, $$w$$.
Notice that after a square is cut out from each end, it leaves a $$(14-2w)$$ cm by $$(20-2w)$$ cm rectangle for the base of the box, and the box will be w cm tall. This gives the volume:
$V(w) = (14 - 2w)(20 - 2w)w = 280w - 68{w^2} + 4w^3\nonumber$
Using technology to sketch a graph allows us to estimate the maximum value for the volume, restricted to reasonable values for $$w$$: values from 0 to 7.
From this graph, we can estimate the maximum value is around 340, and occurs when the squares are about 2.75cm square. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph.
From this zoomed-in view, we can refine our estimate for the max volume to about 339, when the squares are 2.7cm square.
Exercise $$\PageIndex{3}$$
Use technology to find the maximum and minimum values on the interval $$[-1, 4]$$ of the function $$f(x) = - 0.2(x - 2)^3(x + 1)^2(x - 4)$$.
The minimum occurs at approximately the point $$(0, -6.5)$$, and the maximum occurs at approximately the point $$(3.5, 7)$$.
Important Topics of this Section
Vocabulary: Polynomials, coefficients, leading coefficient, term
Degree of a polynomial
Long Run Behavior
Short Run Behavior
Intercepts (Horizontal & Vertical)
Methods to find Horizontal intercepts
Factoring Methods
Factored Forms
Technology
Graphical Behavior at intercepts
Single, Double and Triple zeros (or multiplicity 1, 2, and 3 behaviors)
Solving polynomial inequalities using test values & graphing techniques
Estimating extrema
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SEARCH HOME
Math Central Quandaries & Queries
Question from Amar, a student: I have been given a circle inscribed in a triangle and have been told to prove that the ratio of the perimeter of the triangle to the circumference of the circle is the same as the ratio of the area of the triangle to the area of the circle. How would this be done?
We have two responses for you
Amar,
Draw the circle inscribed in the triangle. Then draw radii to the points of tangency. Then draw lines from the circle's center to the three corners of the triangle.
You have divided the triangle into 3 triangles and you can easily see that the height of each triangle is the radius. Since the area of the triangle is the half the height times the base, you can add the three smaller triangles' areas together to get the area of the large triangle.
Here's a diagram of what I mean:
So if the sides are A, B and C and the radius is R, then the Area of the triangle is RA/2 + RB/2 + RC/2.
Can you finish the problem from here?
Cheers,
Stephen La Rocque.
Amar,
It depends on what tools you have available for 'thinking about' these connections.
(A) If you scale the entire picture down by a factor K, then both areas scale by K^2, but their ratios remain the constant. The scaling brings down the perimeters by a factor K, so their ratio remains the same. If, by chance, you are in calculus, you could look at how the areas are composed by taking limits of thin strips of the perimeters, and show that the ratio of the perimeters (A constant in all the strips) becomes the ratio of the areas when you 'add all the pieces up'.
I suspect you are not yet in calculus - though you might look for this connection, later, when you are!
This is true for other shapes than triangles! This is actually the big idea what will lie behind whatever other formulae you generate.
(B) Now how to do it 'bare handed' as it were?
First - it helps to break the triangle (and circle) down into pieces. Here is one of the six pieces I would use.
If you know about measures, you know m = r(theta) where theta is the angle In radians).
You also know that k = r cotan(theta)
But it turns out you really don't need to know any of those formula!!
The comparison you really want is the ratio of the are of the circle to the perimeter of the circle is:
(1/2) (mr)/ m = r/2. That is, for this piece: AREA C = r/2(perimeter C)
The ratio of the area of the triangle to the perimeter contributed by the triangle is: (1/2) rk /k = r/2
AREA T = r/2 (perimeter T)
Now add up over all the six pieces.
You will find the ratio of the area to to perimeter is still r/2 over both the circle and the triangle.
You can take it from there.
Walter Whiteley
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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# How to Calculate the Geometric Mean of Return
## Geometric Mean Return
The geometric mean return, also called the geometric average return, is a way to calculate the average compounding rate of return on the investments. It considers the compound interests multiplied by the interest over the number of periods.
The geometric mean return is a good measure above the arithmetic return that calculates the interests in a simple arithmetic measure. In case of arithmetic returns, all interests of sub-periods are added and then the total is divided by the total number of sub-periods. The arithmetic average return is misleading in case of long-tenured investments because it overstates the true return. That is why arithmetic returns are used only in case of returns of shorter time periods.
While calculating interests for a longer period of time, the geometric average return (GAR) is a better formula that takes into consideration the order of the return and the compounding effect applied on the investment.
Note − Geometric average return is a rate of return for a series of terms using the products of the terms.
## Geometric Average Return Formula
The most commonly used formula to calculate the Geometric Average Return is −
$$\mathrm{[(1 + 𝑅_{1}) × (1 + 𝑅_{2}) × (1 + 𝑅_{3}) × … × (1 + 𝑅_{n})]^{\frac{1}{n}} − 1}$$
Where,
• R = rate of return
• n = number of periods
The geometric mean return formula is helpful for investors looking for an “apples to apples” approach of comparison when the investors consider multiple investment options and is specifically useful for investments that are compounded.
The formula allows one to calculate the holding period return, or the total return on the investment across multiple sub-periods.
Note − Geometric mean is more applicable over longer periods of time, and it is a better option than arithmetic mean too.
## Geometric Average Return Analysis
The geometric mean is called by many names, such as the compounded annual growth rate (CAGR), the geometric average, or the time-weighted rate of return (TWRR). It represents the rate of the average return for a set of values.
The CAGR takes ‘n’ numerous values (the interest return rates), multiplies all of them together, and puts them to the$(\frac{1}{n})^{th}$ power.
The best use of geometric mean return is for longer time periods, which means multiplying a lot more rates that are compounding at several sub time-periods. Therefore, the use of an Excel spreadsheet or calculator is evident for these calculations.
Note − Usually, Arithmetic mean return overstates and overestimates the average.
One of the important benefits of using geometric mean is that it doesn’t need the investment data. The calculation can be done using just the returns figures themselves. This is the reason why it is called an “apples-to-apples” comparison when considering numerous different investment options.
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# CSE 143 Final Project: Hilbert Sort: 1. The Problem
Posted in Computer Science
# Table of Contents
This is the first in a series of three posts detailing the Hilbert Sort problem, its solution, and its implementation. This post sets up the problem.
# Hilbert Sort: Motivation
In the next few series of posts, we will cover the Hilbert Sort problem, how it works, and how to implement it.
However, before we describe the problem further, let's start with some motivation for solving this problem.
Suppose we're dealing with a very large number of independent objects on a 2D coordinate grid, each with a coordinate location $$(x,y)$$. (For example, a large population of particles moving in a fluid, or a large number of characters on a map in a game.)
Here is our box of particles:
Now, suppose that we have more data than can be handled by a single computer, and we want to arrange the data on different computers. However, we want to arrange the data in such a way that we preserve the spatial characteristics of the data.
If we implement a naive sort method for Cartesian points that sorts points by x coordinate value, breaking ties with the y coordinate value, we end up with points that are neighbors in space, but far away in the data container's storage (like an array). This is particularly true if there are large crowds of points in the grid.
Here is an illustration of a set of points and their resulting storage schema in an array that sorts points by x coordinate. It shows two purple particles, close by in space, but with
several green particles further away distance-wise but not with respect to the x coordinate. The spatial locality of points is not preserved in the data container, so clearly, a better method of sorting and organizing points is needed.
An alternative to this schema that yields better locality properties, but that leads to a much higher cost for sorting points, involves iterating over each point, and for each point, finding the closest points to it in space. However, this itself requires iterating over each point. This approach ends up walking over each point (the point whose nearest neighbors we are finding), and doing a second nested walk over each point (checking if a point is a nearest neighbor to this one). The overall cost of doing this is $$O(N^2)$$.
It is a deceptively tricky problem: how to organize a large group of points in a way that preserves spatial locality?
But first, we'll cover the topic of space filling curves, then return to this topic.
# Space Filling Curves
Mathematician Giuseppe Peano was a prolific teacher and researcher known for many things, but one of his more curious ideas is known as the Peano Curve. Peano was attempting to answer the question of whether a continuous curve could be bounded by a finite space, and he invented a counter-example: a recipe for breaking a curve into parts that can be replicated and repeated and applied to the copies as many times as desired, and always result in a continuous curve.
The way that space-filling curves in general are constructed is to create a pattern, then scale it down and repeat it, attaching subsequent scaled-down, repeated curves. Peano simply invented the first curve; there are many variations on the original space-filling curve idea (including the Hilbert Curve - more on that in a moment).
The original 1890 paper by Giuseppe Peano is entitled "Sur une courbe, qui remplit toute une aire plane", published in 1890 in Mathematische Annalen I, Issue 36. Unfortunately, it has no pictures, but here is a rendering from Wikimedia Commons:
(Link to original on Mediawiki Commons)
Now, the Peano curve was nice, but it had some mathematical properties that made it difficult to deal with. So in 1890, David Hilbert published a follow-up paper in Mathematische Annalen I Issue 38, entitled "Ueber die stetige Abbildung einer Linie auf ein Flächenstück", which slightly modified the recipe to create a curve with more elegant mathematical properties.
Also, he included pictures.
Here is the first set of figures from Hilbert's original 1890 paper:
And here is a slightly cleaner rendering of the Hilbert Curve pattern repeated six times:
(Link to original on Wikimedia Commons)
From the abstract of Hilbert's paper:
Peano has recently shown in the Mathematical Annals, 2 by an arithmetical observation, how the points of a line can be mapped continuously to the points of a surface part. The functions required for such a mapping can be produced more clearly by using the following geometrical view. Let us divide the line to be represented-about a straight line of the length 1-into four equal parts 1, 2, 3, 4, and the surface which we assume in the form of a square of the side length 1 Straight into 4 equal squares 1, 2, 3, 4 (Fig. 1). Secondly, we divide each of the partial sections 1, 2, 3, 4 again into 4 equal parts so that we obtain on the straight the 16 partial sections 1, 2, 3, ..., 16; At the same time, each of the 4 squares 1, 2, 3, 4 is divided into 4 equal squares, and the numbers 1, 2, ..., 16 are then written to the resulting 16 squares, That each successive square follows the previous one with one side (Fig. 2). If we think of this method, as shown in Fig. 3, the next step, it is easy to see how to assign a single definite point of the square to any given point of the line. It is only necessary to determine the partial stretches of the line to which the given point falls. The squares indicated by the same numbers are necessarily in one another and include a certain point of the surface piece in the boundary. This is the point assigned to the given point. The image thus found is unambiguous and continuous, and vice versa, each point of the square corresponds to one, two, or four points of the line. Moreover, it appears remarkable that, by a suitable modification of the partial lines in the squares, a clear and continuous representation can easily be found, the reversal of which is nowhere more than three-fold.
- David Hilbert, "Über die stetige Abbildung einer Linie auf ein Flächenstück", Mathematische Annalen Vol 38
Thanks to Google Translate for an incredible job.
# Constructing a Hilbert Curve
To construct a Hilbert curve, you just have to follow the recipe. It doesn't matter what your square contains so far, or how many levels in you are, whether it's the first curve or the five hundredth:
1. First, take yer square.
2. Second, quadruple yer square. That means, make four copies, that all make a square.
3. Now rotate the bottom left and bottom right via diagonal reflection.
4. Fourth step is, you're done - that's you're new square!
# Performing a Hilbert Sort
We will omit a proof of the statement, but given a set of unique (x,y) points, we can always construct a minimal-size Hilbert Curve that visits each point only once.
Points can be sorted, then, according to when they would be visited by said Hilbert Curve. And this ordering provides better preservation of spatial locality and structure of points when aligning them in memory, because these space-filling curves are recursive and preserve spatial locality in a top-down fashion.
For example, if we have two points in our square, one in the lower left and one in the lower right, and we are sorting them via a Hilbert Sort, we definitely know that a Hilbert curve constructed to visit both of these points will definitely visit the lower left point (the quadrant where the Hilbert curve starts) before it visits the lower right point (in the quadrant where the Hilbert curve stops).
This requires thinking about $$(x,y)$$ points in the box in terms of quadrants, and the order in which the Hilbert curve will visit each quadrant or region, rather than thinking in terms of the explicit Hilbert curve that will visit each particular $$(x,y)$$ point:
This is a subtle shift in the thinking about this problem, but it is crucial to a successful implementation of a Hilbert sort. The problem that will follow, which asks to implement the Hilbert sort, has many distracting details, including the Hilbert curve itself.
Remember, in Hilbert sort, the end goal is not the curve itself, but the sort order.
Here is how the quadrant-by-quadrant partitioning to sort elements ends up looking when applied repeatedly:
It is important to note that the two diagonal reflections happening in the corners are the trickiest part of this problem. We will cover this operation in greater detail in the solution blog post.
# Problem Statement
Following is a paraphrased problem statement from the original ACM ICPC Regional Programming Competition problem statement that this problem and its solution was based on.
"If points $$(x,y)$$ are sorted primarily by x, breaking ties by y, then points that are adjacent in memory will have similar x coordinates but not necessarily similar y, potentially placing them far apart on the grid. To better preserve distances, we may sort the data along a continuous space-filling curve.
"We consider one such space-filling curve called the Hilbert curve...
"Given some locations of interest, you are asked to sort them according to when the Hilbert curve visits them. Note that while the curve intersects itself at infinitely many places, e.g., at $$(\frac{S}{2}, \frac{S}{2})$$, making S odd guarantees all integer points are visited just once."
Here is an example input file, giving a set of points on an $$M \times N$$ grid:
14 25
5 5 Honolulu
5 10 PugetSound
5 20 Victoria
10 5 Berkeley
10 10 Portland
10 15 Seattle
10 20 Vancouver
15 5 LasVegas
15 10 Sacramento
15 15 Kelowna
15 20 PrinceGeorge
20 5 Phoenix
20 10 SaltLakeCity
20 20 Calgary
The corresponding output can be verified intuitively, assuming the coordinates given above are accurate! Here is the output. Indeed, the order in which each city is visited is what we would expect if we drew a space-filling Hilbert curve over a map of the western United States and Canada.
Honolulu
Berkeley
Portland
PugetSound
Victoria
Vancouver
Seattle
Kelowna
PrinceGeorge
Calgary
SaltLakeCity
Sacramento
LasVegas
Phoenix
Now that we've used up all of our space here describing the problem, in a follow-up post we will go into greater detail about how to solve the problem conceptually, and come up with some pseudocode for a recursive method (since this is a recursive task). Then, a third post will go into greater detail about the final Java code to perform this task.
# References
1. "ACM Pacific Region Programming Competition." Association of Computing Machinery. Accessed 19 June 2017. <http://acmicpc-pacnw.org/>
2. "Sur une courbe, qui remplit toute une aire plane." G. Peano. Mathematische Annalen 36 (1890), 157–160. (pdf)
3. "Über die stetige Abbildung einer Linie auf ein Flächenstück." D. Hilbert. Mathematische Annalen 38 (1891), 459–460. (pdf)
4. "Hilbert Curve." Wikipedia: The Free Encyclopedia. Wikimedia Foundation. Edited 29 April 2017. Accessed 23 June 2017. <https://en.wikipedia.org/wiki/Hilbert_curve>
5. "Peano Curve." Wikipedia: The Free Encyclopedia. Wikimedia Foundation. Edited 16 October 2016. Accessed 23 June 2017. <https://en.wikipedia.org/wiki/Peano_curve>
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# SecФ+tanФ=k,show that sinФ=k²-1by k²+1
1
by sinchana21
2014-11-05T18:56:50+05:30
given that.......secФ +tanФ=k
tanФ=k-secФ
square the both sides
tan²Ф = k² + sec²Ф - 2ksecФ
tan²Ф = k² + 1 + tan²Ф - 2ksecФ
2ksecФ = k² + 1
square
secФ = (k² + 1)/2k
sec²Ф = (k⁴ + 1 + 2k²)/4k²
cos²Ф = 4k²/(k⁴ + 1 + 2k²)
1 - sin²Ф = 4k²/(k⁴ + 1 + 2k²)
sin²Ф = (k⁴ + 1 + 2k² - 4k²)/(k⁴ + 1 + 2k²)
sin²Ф = (k² - 1)²/(k² + 1)²
sinФ=k²-1by k²+1
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Question
Tue April 24, 2012 By:
# it is a projectile motion based , i think a tough question , not getting answer even by several attempt
Mon April 30, 2012
The range of the projectile motion is given by
R=u2sin2?/g
Time of flight is given by
T=2usin?/g
Let d? be the fractional change in the value of the angle.
Differentiating R w.r.t ? we get
dR/d? = 2u2cos2?/g
Multiply by d? on both sides, we get
dR= 2u2cos2?d?/g
This represents the change in range, so fractional change in range would be to divide this change with the original value of range.
Fractional change in Range = dR/R = 2cos2?d?/sin2? =x
Now,
Differentiating T w.r.t ? we get
dT/d? = 2ucos?/g
Multiply by d? on both sides, we get
dR= 2ucos?d?/g
This represents the change in time of flight, so fractional change in time of flight would be to divide this change with the original value of time of flight.
Fractional change in Time of flight = dT/T = cos?d?/sin? =y
So, x= 2cos2?d?/sin2? and y= cos?d?/sin?
And ?=60, putting the value of ? we get
x=2(cos120)d?/sin120 = -2d?/?3
y=cos60d?/sin60 = d?/?3 = -x/2 is the answer
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Wed October 11, 2017
# determine the direction and magnitude of the resultant of following velocities inpressed on a particle;8m/s due south,12m/s due east ,3root2 m/s due north east.
Wed October 11, 2017
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https://discuss.leetcode.com/topic/68242/java-solutions-from-o-n-3-to-o-n-for-132-pattern-updated-with-one-pass-slution
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# Java solutions from O(n^3) to O(n) for "132" pattern (updated with one-pass slution)
• This is a summary of the four solutions for this problem. It starts with the `O(n^3)` naive solution, then transition to the `O(n^2)` sub-optimal solution and finally to the two optimized `O(n)` solutions (one is two-pass while the other is one-pass).
`I. Naive O(n^3) solution`
The naive `O(n^3)` solution is a no-brainer --- simply check every `(i, j, k)` combination to see if there is any `132` pattern.
``````public boolean find132pattern(int[] nums) {
for (int i = 0; i < nums.length; i++) {
for (int j = i + 1; j < nums.length; j++) {
for (int k = j + 1; k < nums.length; k++) {
if (nums[i] < nums[k] && nums[k] < nums[j]) return true;
}
}
}
return false;
}
``````
And of course it will get rejected due to TLE. So let's see how we can do better.
`II. Improved O(n^2) solution`
To reduce the time complexity down to `O(n^2)`, we need to do some observations. In the naive solution above, let's assume we have index `j` fixed, what should index `i` be so that it is most probable we will have a `132` pattern? Or in other words, what should `i` be so that we will be certain there is no such `132` pattern for combination `(*, j, *)` whenever there is no `132` pattern for combination of `(i, j, *)`? (Here `*` means any index before or after index `j`.)
The answer lies in the fact that once the first two numbers `nums[i]` and `nums[j]` are fixed, we are up to find the third number `nums[k]` which will be within the range `(nums[i], nums[j])` (the two boundaries are exclusive). Intuitively the larger the range is, the more likely there will be a number "falling into" it. Therefore we need to choose index `i` which will maximize the range `(nums[i], nums[j])`. Since the upper bound `nums[j]` is fixed, this is equivalent to minimizing the lower bound `nums[i]`. Thus it is clear `i` should be the index of the minimum element of the subarray `nums[0, j)` (left inclusive, right exclusive).
Since we are scanning index `j` from the beginning of the input array `nums`, we can keep track of the minimum element of the subarray from index `0` up to `j - 1` without rescanning it. Therefore the first two loops in the naive solution can be combined into one and leads to the following `O(n^2)` solution:
``````public boolean find132pattern(int[] nums) {
for (int j = 0, min = Integer.MAX_VALUE; j < nums.length; j++) {
min = Math.min(nums[j], min);
if (min == nums[j]) continue;
for (int k = nums.length - 1; k > j; k--) {
if (min < nums[k] && nums[k] < nums[j]) return true;
}
}
return false;
}
``````
While this solution can be accepted, it runs slow. One obvious drawback is that in the second loop we are throwing away information about elements in the right part of `nums` that may be "useful" for later combinations. It turns out we can retain this "useful" information by applying the classic space-time tradeoff, which leads to the following `O(n)` time and `O(n)` space solution.
`III. Optimized O(n) solution`
As I mentioned, to further reduce the time complexity, we need to record the "useful" information about elements in the right part of the input array `nums`. Since these elements are located at the right part of `nums`, it will be hard to do so if we are scanning the array from the beginning. So the idea is to scan it from the end while in the meantime keep track of the "useful" information. But still at each index `j`, we need to know the minimum element for subarray `nums[0, j)`. This can be done by doing a pre-scan in the forward direction and memorize the results for each index in an auxiliary array (we will call the array as `arr` whose element `arr[j]` will denote the minimum element in the subarray `nums[0, j)`).
Until now we are kinda vague about the exact meaning of "useful" information, so let's try to be more specific. Assume we're currently scanning (from the end) the element with index `j`, our task is to find two elements `nums[i]` and `nums[k]` to determine if there exists a `132` pattern, with `i < j < k`. The left element `nums[i]`, as it has been shown in part `II`, will be chosen as `arr[j]`, the minimum element of subarray `nums[0, j)`. What about the right element `nums[k]`?
The answer to that will address the meaning of "useful" information. First note we are only interested in elements that are greater than `arr[j]`, so it is sensible to maintain only those elements. Second, among all these qualified elements, which one will be the most probable to fall into the range `(nums[i], nums[j])`? I would say it is the smallest one (i.e., if the smallest one is out of the range, all others will also be out of range). So to sum up, the "useful" information for current index `j` will be a collection of scanned elements that are greater than `arr[j]`, and `nums[k]` will be chosen as the smallest one if the collection is not empty.
From the analyses above, it looks like we have to do some sorting stuff for the retained elements (or at least find a way to figure out its smallest element). Well, it turns out these elements will be sorted automatically due to the fact that `arr[j'] >= arr[j]` as long as `j' < j`. Here is how it goes, which is a proof by induction.
At the beginning we have an empty collection and of course it is sorted. Now suppose we are at index `j` and the corresponding collection is still sorted, let's see if it remains so at index `j - 1`. First we will check if `nums[j]` is greater than `arr[j]`. If not, we simply continue to `j - 1`. Since the collection is intact so it will be sorted at `j - 1`. Otherwise, we need to remove elements in the collection that are no greater than `arr[j]` (this is necessary because some smaller elements may be left over in the collection from previous steps). After removal, we then compare the first element in the collection with `nums[j]` to see if a `132` pattern has been found, provided the collection is not empty. If so, return true. Otherwise one of the following must be true: the collection is empty or `nums[j]` is no greater than the first element in the collection. In either case the collection is sorted. Now if we have `arr[j - 1] < nums[j]`, we need to add `nums[j]` to the collection since it is a qualified number for `arr[j - 1]`. Again in either case the collection will remain sorted after addition (if it is empty, after addition there is only one element; otherwise since the added element is no greater than the first element in the collection before addition, it will become the new first element after addition and the collection stays sorted).
Here is the program with `O(n)` time and space complexity. There is one minor optimization based on the observation that the total number of elements in the collection will never exceed the total number of elements scanned so far. Therefore the right part of the `arr` array can be used to serve as the collection. For time complexity, each element in the input array `nums` will be pushed into and popped out from the collection (or stack to be exact) at most once, the time complexity will be `O(n)` despite of the nested loop.
``````public boolean find132pattern(int[] nums) {
int[] arr = Arrays.copyOf(nums, nums.length);
for (int i = 1; i < nums.length; i++) {
arr[i] = Math.min(nums[i - 1], arr[i - 1]);
}
for (int j = nums.length - 1, top = nums.length; j >= 0; j--) {
if (nums[j] <= arr[j]) continue;
while (top < nums.length && arr[top] <= arr[j]) top++;
if (top < nums.length && nums[j] > arr[top]) return true;
arr[--top] = nums[j];
}
return false;
}
``````
`IV -- One-pass O(n) solution`
It turned out that we don't need the `arr` array in part `III`, thus can get rid of the pre-scan. The idea is to do backward traversal of the input array and keep track of the maximum value (denoted as `third`) of all possible third numbers obtained so far (an element can be a candidate for the third number if and only if there is another element coming before and greater than it). This is because for two candidate third numbers `a` and `b`, if `a <= b` and there is no `132` pattern for `b`, then there must be no `132` pattern for `a` either. The idea is elaborated as follows.
For the current element being examined, we first check if it can be the first number by comparing it with `third`. If this is the case, a `132` pattern has been found. Otherwise, it will be treated as the second number to qualify scanned elements so far as candidate third numbers (note we are doing backward scan so the current element will always come before scanned elements). Of course we cannot do this for all scanned elements, as it will lead to an `O(n^2)` solution. Fortunately it suffices only to process scanned elements that are greater than `third`. This is because after an element is qualified to be the third number, it will only be used to update `third`. For those elements smaller than `third`, even if they are qualified to be the third number, they won't make `third` any larger and therefore can be ignored.
So we need to maintain a collection (stack, to be exact) for scanned elements that are greater than `third`. After the current element is done, it will be added to this collection for future processing. It can be shown by mathematical induction that elements in the collection will be sorted automatically, similar to that in part `III`. At the beginning the collection is empty so the base case is true. Assume the collection is sorted immediately before processing the current element, which is presumed to be `>= third` (otherwise a `132` pattern is found). We will scan the collection and pop all elements smaller than the current element to find all qualified third numbers and update `third` accordingly. At the end, all remaining elements in the collection are no less than the current element, therefore adding it to the collection won't break the sorted property. So our induction assumption is also true.
Here is the one-pass `O(n)` solution. We can also optimize the space cost to `O(1)` if the input array can act as the collection by taking advantage of the fact that size of the collection will never exceed the number of elements scanned so far.
``````public boolean find132pattern(int[] nums) {
for (int n = nums.length, i = n - 1, top = n, third = Integer.MIN_VALUE; i >= 0; i--) {
if (nums[i] < third) return true;
while (top < n && nums[i] > nums[top]) third = nums[top++];
nums[--top] = nums[i];
}
return false;
}
``````
• Excellent explanation. Thank you. I wish I observed these "observations". :(
• Excellent solution! Your `O(n^2)` explanation is brilliant.
• thank you.It is inspiring
• Such a nice solution and such a nice explanation. Thanks!
• The O(n) solution is so brilliant! I could come up with the O(n^2) method, but I couldn't optimize it to O(n). Thanks a lot for sharing!
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# What Is 15 Of 8000 – 15% Of 80000
## What is 15 of 8000
Let is make you a little more knowledgeable when you know what is 15 of 8000. Because this is a question that can be easily answered if we pay attention. That is why let yourself know one more good thing, one useful thing when you get the answer to the question what is 15 of 8000.
What’s 15 percent-off \$8000?
Replacing the given values in components (a) we have:
Amount Saved = Original Price x Discount in Percent / 100. So,
Amount Saved = 8000 x 15 / 100
Amount Saved = 120000 / 100
In different words, a 15% discount for a merchandise with unique value of \$8000 is the same as \$1200 (Amount Saved).
Note that discover the quantity saved, simply multiply it by means of means of the share and divide by 100.
What’s the ultimate value of an merchandise of \$8000 when discounted \$1200?
Using the formulation (b) and changing the given values:
Sale Price = Original Price – Amount Saved. So,
Sale Price = 8000 – 1200
This skill the value of the thing to you is \$6800.
You pays \$6800 for a merchandise with unique value of \$8000 when discounted 15%.
In this example, once you purchase an merchandise at \$8000 with 15% discount, you’ll pay 8000 – 1200 = 6800 dollars.
1200 is what % off 8000 dollars?
Using the formulation (b) and changing given values:
Amount Saved = Original Price x Discount in Percent /100. So,
1200 = 8000 x Discount in Percent / 100
1200 / 8000 = Discount in Percent /100
100 x 1200 / 8000 = Discount in Percent
120000 / 8000 = Discount in Percent, or
Discount in Percent = 15 (answer).
To discover extra examples, simply decide one on the underside of this page.
## 10 of 8000
What’s 10 percent-off \$8000?
Replacing the given values in formulation (a) we have:
Amount Saved = Original Price x Discount in Percent / 100. So,
Amount Saved = 8000 x 10 / 100
Amount Saved = 80000 / 100
In different words, a 10% discount for a merchandise with unique value of \$8000 is the same as \$800 (Amount Saved).
Note that discover the quantity saved, simply multiply it via means of the share and divide by 100.
What’s the ultimate value of an merchandise of \$8000 when discounted \$800?
Using the formulation (b) and changing the given values:
Sale Price = Original Price – Amount Saved. So,
Sale Price = 8000 – 800
This capability the value of the article to you is \$7200.
You pays \$7200 for a merchandise with unique value of \$8000 when discounted 10%.
In this example, once you purchase an merchandise at \$8000 with 10% discount, you’ll pay 8000 – 800 = 7200 dollars.
800 is what % off 8000 dollars?
Using the formulation (b) and changing given values:
Amount Saved = Original Price x Discount in Percent /100. So,
800 = 8000 x Discount in Percent / 100
800 / 8000 = Discount in Percent /100
100 x 800 / 8000 = Discount in Percent
80000 / 8000 = Discount in Percent, or
Discount in Percent = 10 (answer).
To discover extra examples, simply decide one on the underside of this page.
## 15% of 80000
Have you ever wondered 15% of 80000. Would you like an answer to that question? If you want to know, don’t skip this article of ours. We not only explain to you 15% of 80000 but also provide you with interesting knowledge of life.
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## 17% of 8000
This life is always full of questions why. That is why it is normal that you don’t know the answer to the question 17% of 8000. So if you want to know the answer to the question 17% of 8000, then read our article. I believe the information in this article will surprise you.
What’s 17 percent-off \$8000?
Replacing the given values in components (a) we have:
Amount Saved = Original Price x Discount in Percent / 100. So,
Amount Saved = 8000 x 17 / 100
Amount Saved = 136000 / 100
In different words, a 17% discount for a merchandise with unique value of \$8000 is the same as \$1360 (Amount Saved).
Note that discover the quantity saved, simply multiply it via means of the share and divide by 100.
What’s the ultimate value of an merchandise of \$8000 when discounted \$1360?
Using the formulation (b) and changing the given values:
Sale Price = Original Price – Amount Saved. So,
Sale Price = 8000 – 1360
This capability the value of the thing to you is \$6640.
You pays \$6640 for a merchandise with unique value of \$8000 when discounted 17%.
In this example, whenever you purchase an merchandise at \$8000 with 17% discount, you’ll pay 8000 – 1360 = 6640 dollars.
1360 is what % off 8000 dollars?
Using the formulation (b) and changing given values:
Amount Saved = Original Price x Discount in Percent /100. So,
1360 = 8000 x Discount in Percent / 100
1360 / 8000 = Discount in Percent /100
100 x 1360 / 8000 = Discount in Percent
136000 / 8000 = Discount in Percent, or
Discount in Percent = 17 (answer).
To discover extra examples, simply decide one on the underside of this page.
## What is 15 of 10000
With the question what is 15 of 8000, do you know the answer or not? Are you curious to know the answer to the question what is 15 of 10000? If your answer is yes, then read this article right away to know the answer. Surely the information in this article will make you realize more things in life.
What’s 15 percent-off \$10000?
Replacing the given values in components (a) we have:
Amount Saved = Original Price x Discount in Percent / 100. So,
Amount Saved = ten thousand x 15 / 100
Amount Saved = 150000 / 100
In different words, a 15% discount for a merchandise with unique value of \$10000 is the same as \$1500 (Amount Saved).
Note that discover the quantity saved, simply multiply it by means of means of the share and divide by 100.
What’s the ultimate value of an merchandise of \$10000 when discounted \$1500?
Using the formulation (b) and changing the given values:
Sale Price = Original Price – Amount Saved. So,
Sale Price = ten thousand – 1500
This capability the value of the thing to you is \$8500.
You pays \$8500 for a merchandise with unique value of \$10000 when discounted 15%.
In this example, once you purchase an merchandise at \$10000 with 15% discount, you’ll pay ten thousand – 1500 = 8500 dollars.
1500 is what % off ten thousand dollars?
Using the formulation (b) and changing given values:
Amount Saved = Original Price x Discount in Percent /100. So,
1500 = ten thousand x Discount in Percent / 100
1500 / ten thousand = Discount in Percent /100
100 x 1500 / ten thousand = Discount in Percent
150000 / ten thousand = Discount in Percent, or
Discount in Percent = 15 (answer).
To discover extra examples, simply decide one on the underside of this page.
So the content of the what is 15 of 8000 question answer above may help you answer the question you are looking for the answer to. Remember to follow our page to get more useful articles updated more often!
Read more: What Is 15 Of 7000 – 15% Of 10000
What? -
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Getting to the Root of the Formula Quiz
Quadratic Formula and Functions: Getting to the Root of the Formula Quiz
Think you’ve got your head wrapped around Quadratic Formula and Functions? Put your knowledge to the test. Good luck — the Stickman is counting on you!
Q. Simplify the expression (-1 + 6i)(2i3 – 4).
-8 – 22i
16 – 22i
8 – 26i
16 – 26i
-8 – 26i
Q. What are the roots of y = -2x2 + 5x – 2?
and
and
and x = 2
There are no real roots for this function
x = -2 and
Q. What are the roots of y = x2 + 7x + 17?
and
and
and
Q. What are the roots of y = 5x2 – 10x + 5?
and
x = -1
and
x = -1 + 2i and x = -1 – 2i
x = 1
Q. How many real roots does y = x2 + ½x + 3 have?
2 complex roots b.
1 real root
2 real roots
0 roots
None of the above
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https://goprep.co/ex-5.6-q7-area-of-the-quadrilateral-formed-by-the-points-1-1-i-1nkgz2
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Q. 75.0( 1 Vote )
Here we have the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1, 0).
We have to calculate the Area of the quadrilateral with vertices as A(1,1), B(0,1), C(0, 0)and D(1, 0)
Let us divide the quadrilateral into 2 triangles,so Area of the quadrilateral will be sum of Areas of two triangles.
Let us say one ∆ is ABC and other ∆ is ADC
Now Area of ∆ABC is
When we substitute the values of the coordinates of the vertices
as A(1,1), B(0,1), C(0, 0),we get
Now Area of ∆ADC with vertices A(1,1), D(1, 0)
C(0, 0) is
Hence area is 1 sq.units.
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
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# A strain of bacteria reproduces @ 25% every 12 min. In how m
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07 Feb 2014, 03:28
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aimlockfire1 wrote:
A strain of bacteria reproduces @ 25% every 12 min. In how much time will it triple itself ??
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
1.25^x = 3 --> x = ~5 --> five 12 minute periods = 60 minutes.
P.S. Please do not shorten or reword the questions.
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Posts: 125
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
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08 Feb 2014, 21:28
1
Bunuel,
How were you able to calculate to find x, in the formula (1.25)^x=3 so easily??
1.25^x=(5/4)^x=3.
Would have imagined x would be some irrational number, since 5/4 is not a multiple of 3.
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Joined: 04 Jan 2014
Posts: 103
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
### Show Tags
10 Jul 2014, 00:31
Hi Bunnel,
I used rounding off method. Is that right?
1.25 approx 1.3
$$1.3^2$$ = 1.69
$$1.7^2$$ = 2.89 or approx 3
Since we squared twice and 1.3>1.25, we need more than 4, or at least 5.
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Joined: 16 Oct 2010
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Location: Pune, India
Re: A strain of bacteria reproduces 25% every 12 min. In how m [#permalink]
### Show Tags
17 Sep 2014, 22:41
Bunuel wrote:
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
Responding to a pm:
If initial amount of bacteria is X, we need it to become 3X. How many 12 min time intervals does it need? Let's assume we need n time intervals.
X*(5/4)^n = 3X
(5/4)^n = 3
Now note that the options are 12*4, 12*5, 12*6 etc. So this gives us some ideas.
(5/4)^4 = 625/256 -> this is much less than 3 since 256*3 is more than 750.
(5/4)^5 = 3125/1024 -> this is a tiny bit more than 3 and hence is out answer since 1024*3 is a bit more than 3000.
Hence n is 5 and time required = 12*5 = 60 mins
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Re: A strain of bacteria reproduces 25% every 12 min. In how m [#permalink]
### Show Tags
18 Sep 2014, 19:13
Responding to a pm:
If initial amount of bacteria is X, we need it to become 3X. How many 12 min time intervals does it need? Let's assume we need n time intervals.
X*(5/4)^n = 3X
(5/4)^n = 3
Now note that the options are 12*4, 12*5, 12*6 etc. So this gives us some ideas.
(5/4)^4 = 625/256 -> this is much less than 3 since 256*3 is more than 750.
(5/4)^5 = 3125/1024 -> this is a tiny bit more than 3 and hence is out answer since 1024*3 is a bit more than 3000.
Hence n is 5 and time required = 12*5 = 60 mins
--------------
i could not understand the solution at all. could you please explain as to how did you get the fraction 5/4?
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8810
Location: Pune, India
Re: A strain of bacteria reproduces 25% every 12 min. In how m [#permalink]
### Show Tags
18 Sep 2014, 20:21
arshu27 wrote:
Responding to a pm:
If initial amount of bacteria is X, we need it to become 3X. How many 12 min time intervals does it need? Let's assume we need n time intervals.
X*(5/4)^n = 3X
(5/4)^n = 3
Now note that the options are 12*4, 12*5, 12*6 etc. So this gives us some ideas.
(5/4)^4 = 625/256 -> this is much less than 3 since 256*3 is more than 750.
(5/4)^5 = 3125/1024 -> this is a tiny bit more than 3 and hence is out answer since 1024*3 is a bit more than 3000.
Hence n is 5 and time required = 12*5 = 60 mins
--------------
i could not understand the solution at all. could you please explain as to how did you get the fraction 5/4?
If a number has to be increased by 25%, you effectively multiply it by 5/4.
n + n*25/100 = n + n/4 = n(1 + 1/4) = n*5/4
Similarly, if you want to increase a number by 20%, you multiply it by 6/5 and so on...
Since the colony increases by 25% every 12 mins, you keep multiplying it by 5/4 till it becomes 3 times.
http://www.veritasprep.com/blog/2011/02 ... rcentages/
http://www.veritasprep.com/blog/2011/02 ... e-changes/
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Joined: 01 Jan 2016
Posts: 21
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
### Show Tags
16 May 2018, 17:33
Bunuel wrote:
aimlockfire1 wrote:
A strain of bacteria reproduces @ 25% every 12 min. In how much time will it triple itself ??
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
1.25^x = 3 --> x = ~5 --> five 12 minute periods = 60 minutes.
P.S. Please do not shorten or reword the questions.
How are you able to see that (1.25)^x = 3 yields x = 5? Are you completing this calculation in your head? Thank you.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8810
Location: Pune, India
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
### Show Tags
17 May 2018, 02:39
GMAT2645 wrote:
Bunuel wrote:
aimlockfire1 wrote:
A strain of bacteria reproduces @ 25% every 12 min. In how much time will it triple itself ??
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
1.25^x = 3 --> x = ~5 --> five 12 minute periods = 60 minutes.
P.S. Please do not shorten or reword the questions.
How are you able to see that (1.25)^x = 3 yields x = 5? Are you completing this calculation in your head? Thank you.
Approximate.
(1.25)^2 = 1.5625 (square of a number ending in 5 is easy to find.
75^2 = _(7*8)_ 25 = 5625
35^2 = _(3*4)_25 = 1225
105^2 = _(10*11)_25 = 11025
Now round up 1.5625 to 1.6
1.6^2 = 2.56 (so this is about 1.25^4)
So we are close to 3 but not quite there yet.
Round down 2.56 to 2.5 and 1.25 to 1.2
2.5*1.2 = 3.00 = about 1.25^5
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Posts: 518
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Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
### Show Tags
17 May 2018, 04:04
VeritasPrepKarishma wrote:
Bunuel wrote:
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
Responding to a pm:
If initial amount of bacteria is X, we need it to become 3X. How many 12 min time intervals does it need? Let's assume we need n time intervals.
X*(5/4)^n = 3X
(5/4)^n = 3
Now note that the options are 12*4, 12*5, 12*6 etc. So this gives us some ideas.
(5/4)^4 = 625/256 -> this is much less than 3 since 256*3 is more than 750.
(5/4)^5 = 3125/1024 -> this is a tiny bit more than 3 and hence is out answer since 1024*3 is a bit more than 3000.
Hence n is 5 and time required = 12*5 = 60 mins
Hi VeritasPrepKarishma,
Why cant we use GP Series here?? Can you please clarify.
Let x be initial bacteria. Now, we have
x + 5x/4 + 25x/16 + ... = 3x
(5/4)^n = 7/4..
However, I am not able to get value of n here.
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Joined: 16 Oct 2010
Posts: 8810
Location: Pune, India
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m [#permalink]
### Show Tags
17 May 2018, 04:56
rahul16singh28 wrote:
VeritasPrepKarishma wrote:
Bunuel wrote:
The original question is:
A strain of bacteria reproduces at the rate of 25% every 12 min. In how much time will it triple itself ?
a) 96 min
b) 72 min
c) 60 min
d) 48 min
e) 40 min
Responding to a pm:
If initial amount of bacteria is X, we need it to become 3X. How many 12 min time intervals does it need? Let's assume we need n time intervals.
X*(5/4)^n = 3X
(5/4)^n = 3
Now note that the options are 12*4, 12*5, 12*6 etc. So this gives us some ideas.
(5/4)^4 = 625/256 -> this is much less than 3 since 256*3 is more than 750.
(5/4)^5 = 3125/1024 -> this is a tiny bit more than 3 and hence is out answer since 1024*3 is a bit more than 3000.
Hence n is 5 and time required = 12*5 = 60 mins
Hi VeritasPrepKarishma,
Why cant we use GP Series here?? Can you please clarify.
Let x be initial bacteria. Now, we have
x + 5x/4 + 25x/16 + ... = 3x
(5/4)^n = 7/4..
However, I am not able to get value of n here.
This is a compounding situation.
x becomes (5/4)x which becomes (5/4)^2x which then becomes (5/4)^3x and so on...
We cannot add these terms since once x becomes (5/4)x, we don't have x anymore.
_________________
Karishma
Veritas Prep GMAT Instructor
Re: A strain of bacteria reproduces @ 25% every 12 min. In how m &nbs [#permalink] 17 May 2018, 04:56
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# What Are The Basic Shapes?
## How many different shapes are there?
There are many shapes in geometry based on their dimensions.
Circle, Triangle, Square, Rectangle, Kite, Trapezium, Parallelogram, Rhombus and different types of polygons are the 2-d shapes.
Cube, Cuboid, Sphere, Cone and Cylinder are the basic three-dimensional shapes..
## What’s a 2 sided shape called?
digonIn geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space.
## Why is learning shapes important?
Learning shapes not only helps children identify and organize visual information, it helps them learn skills in other curriculum areas including reading, math, and science. … Learning shapes also helps children understand other signs and symbols. A fun way to help your child learn shapes is to make a shape hunt game.
## What shapes should Kindergarten know?
In Kindergarten, children typically learn the names of basic shapes, including some 3-dimensional shapes. Before entering Kindergarten, you can encourage your child to recognize shapes such as squares, circles, triangles, and rectangles in everyday life.
## What are the 5 basic shapes?
And that’s really all we’re going to do here, except we use a pencil and simplify a complex figure to just five basic geometric shapes – the triangle, oval, oblong, circle and square.
## What is a 9 sided shape?
enneagonIn geometry, a nonagon (/ˈnɒnəɡɒn/) or enneagon (/ˈɛniəɡɒn/) is a nine-sided polygon or 9-gon. The name nonagon is a prefix hybrid formation, from Latin (nonus, “ninth” + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century.
## What is the shape of a house called?
PentagonCalculations with a house shape, a house-shaped pentagon.
## What are the two basic shapes?
There are two types of shapes: geometric and free-form. Geometric shapes are precise shapes that can be described using mathematical formulas.
## What is the most basic shape?
The square, circle, and triangle are the most basic shapes on Earth, supporting structures both synthetic and natural.
## What are the three basic shapes?
The three basic shapes are a square, a triangle and a circle.
## What are the four basic shapes?
Four Basic ShapesSHAPES.CIRCLE.RECTANGLE.SQUARE.RECTANGLE OR SQUARE?TRIANGLE.WHAT SHAPE IS THIS? HOW MANY SIDES? CIRCLE.WHAT SHAPE IS THIS? HOW MANY SIDES? RECTANGLE.More items…•
## What are the 10 basic shapes?
Basic shapes Learninging charts introduce 10 basic shapes are circle, oval, triangle, rhombus, square, rectangle, trapezoid, pentagon, hexagon and octagon.
## How many Vsepr shapes are there?
fiveThe VSEPR theory describes five main shapes of simple molecules: linear, trigonal planar, tetrahedral, trigonal bipyramidal, and octahedral.
## What are shapes examples?
Examples of geometric shapes are: squares, rectangles, triangles, circles, oval, pentagons and so on.
## What are the 16 basic shapes?
Terms in this set (16)equilateral triangle. A triangle with all sides of equal length.isosceles triangle. A triangle with two sides of equal length.scalene triangle. A triangle with no sides of equal length. … scalene right triangle. … isosceles right triangle. … square. … rectangle. … parallelogram.More items…
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0
# Math and Arithmetic
Wilber Greenholt
Lvl 9
2021-09-30 20:42:38
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The shorter leg is 9 feet long
Five more than four times a number
hjhjghj
Max is 24 years older than his son Liam in two years Liam will be half as old as max how old is Liam now
46
Two hundred tickets for the school play were sold Tickets cost 2 for students and 3 for adults The total amount collected was 490 How many student tickets were sold
110
Marcus has a total of 10 nickels and dimes in his pocket He has a total of 70 cents How many dimes does he have in his pocket
4
The area of a rectangle is 48 square feet The length is two feet longer than the width What is the width of the rectangle
The width is 6 feet. The length is 8 feet.
Perry has a total of 25 five dollar bills and ten dollar bills in his pocket The value of these bills is 175 How many ten dollar bills are in Perry's pocket
10
A Piper Cub can cover 500 miles in the same amount of time it takes a jet plane flying 250 miles per hour faster to cover 1500 miles How fast is the Piper Cub flying
A Piper Cub can cover 500 miles in the same amount of time it takes a jet plane flying 250 miles per hour faster to cover 1500 miles. Piper Cub is flying 125 mph.
The reciprocal of a number plus the reciprocal of three times the number equals one third
4
The hypotenuse of a right triangle is 8 cm long One leg is 2 cm longer than the other Find the length of the longer leg to the nearest tenth
6.6 cm
The hypotenuse of a right triangle is 6 m long One leg is 1 m longer than the other Find the length of the shorter leg Round to the nearest hundredth
The shortest leg is 3.72 m long.
Jerry is experimenting with chemicals in the laboratory He mixes a solution that is 10 percent acid with a solution that is 30 percent acid How much of the 10 percent acid solution will be needed to m
10 liters
The length of an airport is 1 mile more than twice the width The area is 21 square miles Identify the length and width of the airport
3 and 7
A chemist has one solution that is 80 percent acid and another solution that is 30 percent acid How much of the second 30 percent solution is needed to make a 400 L solution that is 62 percent acid
144liters
The width of a rectangle is 4 feet less than the length the area is 320 feet find the width of the rectangle
The width is 16 feet
The reciprocal of a number plus the reciprocal of twice the number equalsone half Find the number
The reciprocal of a number plus the reciprocal of twice the number equals
Find the number. 1/2 find the number
The 18 wheeler is traveling at 8 mph more than a rental truck the 18 wheeler drives 220 miles and the rental truck drives 198 miles how fast was the rental truck goin
72 MPH.
The sum of two numbers is 25 and the difference of their square is 75 what are the two numbers
The two numbers are 11 and 14. 11 + 14 = 25, 142 - 112 = 75
The speed of a freight train is 14 km hr slower than the speed of a passenger train The freight train travels 330 km in the same time that it takes the passenger train to travel 400 km Find the speed
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A pick up can travel 300 miles in the same time that a car going 10 miles per hour faster can travel 350 miles how fast is the car traveling
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Section 8.1-8.2
# Section 8.1-8.2 - Section 8.1 8.2 Testing Statistical...
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Unformatted text preview: Section 8.1- 8.2 Testing Statistical Hypotheses (TSH) 1. Basic Concepts of TSH Estimation deals with obtaining a plausible estimate of θ or obtaining a plausible interval (random) which contains θ . TSH deals with how to make a decision between two competing claims (hypotheses). Definition : Statistical hypothesis or simply a hypothesis is an assertion about one or more population characteristics or about the form of population distribution. Examples: (i) Let p = proportion of students having cars Hypothesis: .6 ≥ p . (ii) Let X = height of the students Hypothesis: 5) (170, ~ N X Kinds of hypotheses: Null hypothesis H is a hypothesis which is initially assumed to be true. Alternative hypothesis 1 H is the hypothesis compared against H . 1 A test of hypothesis is a method or procedure of deciding between H and 1 H , based on the sample information. If sample contains strong evidence against H , then it will be rejected in favor of 1 H . Otherwise, we will continue to believe in H . Example 1 Suppose 10) , ( ~ μ N X and consider testing (i) 50 : = μ H against 50 : 1 < μ H (lower-tailed) or 50 : 1 μ H (upper-tailed) (ii) 50 : = μ H against 50 : 1 ≠ μ H (two-tailed). A test leads to one of the following decisions (i) Accept H (ii) Reject H (Accept 1 H ) Obviously, we have two-kinds of errors are inherent in any test. Definition: Type I error ≡ Reject H when H is true. Type II error ≡ Accepting H when 1 H is true. Definition: P(Type I error) = α = P (Reject H when H is true). P(Type II error) = β = P(Accepting H when 1 H is true). 2 It can be shown that both errors α and β can not be minimized. Usually, we fix one of the errors, say α , and choose a test that minimizes β . In practice, α = .05 or α = 0.01, is chosen. Definition: The specified value of α is also called “ level of significance” or “ significance level of the test ”. Remarks : (i) A test with significance level α = .05 means that when the test repeated several times, only 5% of times it may commit the type I error. (ii) The quantity (1- β ) = P (Reject H when 1 H is true) is called the power of the test. (iii) If α decreases then β increases. Usually, the largest tolerable value of α is decided based on the problem under consideration. Then a test is found with P (Type I error) α ≤ and β is minimum or (1- β ) = power is maximum. Test Statistics and P – values The test is carried out using a test statistic, usually a standardized function of the data. 3 Example 1 : Let μ ≡ population mean. Suppose we want to test 50 : = μ H against 50 : 1 < μ H . Obviously, sample mean x provides information about μ . Also, we know n x σ μ , N ~ When σ is not known and n is large, (0,1) N ) ( 2245- s n x μ distribution....
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# Thread: Challenging Limit Question!
1. ## Challenging Limit Question!
Ok I've got to evaluate the following limit:
But I've got no idea how the absolute values can be rewritten in this limit?
2. Originally Posted by Kataangel
Ok I've got to evaluate the following limit:
But I've got no idea how the absolute values can be rewritten in this limit?
Note that $\displaystyle \left|x-1\right|=\left\{\begin{array}{rl} x-1, & \text{ if }x\geq 1\\ {\color{red}-x+1}, & \text{ if }x<1\end{array}\right.$ and $\displaystyle \left|x+1\right|=\left\{\begin{array}{rl} {\color{red}x+1}, & \text{ if }x\geq -1\\ -x-1, & \text{ if }x<-1\end{array}\right.$
Thus, $\displaystyle \lim_{x\to 0}\frac{x}{\left|x-1\right|-\left|x+1\right|}=\lim_{x\to 0}\frac{x}{(-x+1)-(x+1)}=\lim_{x\to0}-\frac{x}{2x}$.
I'm sure you can take it from here.
3. Originally Posted by Chris L T521
Note that $\displaystyle \left|x-1\right|=\left\{\begin{array}{rl} x-1, & \text{ if }x\geq 1\\ {\color{red}-x-1}, & \text{ if }x<1\end{array}\right.$ and $\displaystyle \left|x+1\right|=\left\{\begin{array}{rl} {\color{red}x+1}, & \text{ if }x\geq -1\\ -x+1, & \text{ if }x<-1\end{array}\right.$
Thus, $\displaystyle \lim_{x\to 0}\frac{x}{\left|x-1\right|-\left|x+1\right|}=\lim_{x\to 0}\frac{x}{(-x-1)-(x-1)}=\lim_{x\to0}-\frac{x}{2x}$.
I'm sure you can take it from here.
Thanks very much for illustrating Chris, but I have no idea how you chose x-1 in the 2nd set of brackets in $\displaystyle \lim_{x\to 0}\frac{x}{(-x-1)-(x-1)}$???
And why the red ones in particular? I'm not sure which set of arbitrary values this limit should be taking? This would be easier with one sided limits.
4. Originally Posted by Kataangel
Thanks very much for illustrating Chris, but I have no idea how you chose x-1 in the 2nd set of brackets in $\displaystyle \lim_{x\to 0}\frac{x}{(-x-1)-(x-1)}$???
And why the red ones in particular? I'm not sure which set of arbitrary values this limit should be taking? This would be easier with one sided limits.
erm...woops. I took the wrong one from each group (and had a couple typos). Please note my correction.
I looked at the intervals of each piecewise term, and picked the ones that contained zero in the interval (since we're taking the limit as x approaches zero).
5. By the way, how do you know which of the greater than or less than signs you should put the equals sign under?
i.e. how can you tell which sign needs to have the 'or equal' with it because attaching the equals under greater than or less than are equally possible?
6. Originally Posted by Kataangel
By the way, how do you know which of the greater than or less than signs you should put the equals sign under?
i.e. how can you tell which sign needs to have the 'or equal' with it because attaching the equals under greater than or less than are equally possible?
It really doesn't matter which one. Its just a matter of preference, I guess.
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HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS
## Volume Of Right Circular Cone
Cones are assumed to be right circular, where right means that the axis passes through the centre of the base at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the center of the base.[1] In general, however, the base may be any shape, and the apex may lie anywhere (though it is often assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base). For example, a pyramid is technically a cone with a polygonal base.
## $\frac{1}{3}\mathrm{\Pi h}{r}^{2}$
Here,r = radius, h = height
ENTER THE VARIABLES TO BE USED IN THE FORMULA
Similar formulas which you may find interesting.
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# Fractal Branching
One of the most common fractal patterns found in nature is branching. Fractal branching is seen in tree branches and leaf veins, lightning, river deltas, mountaintops, blood vessels and bronchi in the lungs, and many other places. Can you think of any examples?
# Fractal Education: Sierpinski Triangle
A great activity for teaching kids about fractals is making Sierpinski triangles – one of the easiest fractals to draw.
The Sierpinski triangle is formed using this process:
1. Take a point-side-up triangle
2. Connect midpoints of the three sides to create a point-side-down triangle.
3. Repeat process with the resulting point-side-up triangles.
4. Repeat as many times as you want.
This example shows taking the process through 4 iterations.
You can print out the starting triangle from:
Fractal Foundation: Sierpinski Triangle (includes worksheet and instructions)
Spitz: Fractal Pack 1: Educators’ Guide (includes Sierpinski triangle template and other information for teaching about fractals)
If you have a classroom of kids, you can take their Sierpinski triangles and put them together to make a larger triangle – you just need a power of 3 to make a complete Sierpinski triangle (3, 9, 27, etc.).
Also, you can blow their minds by showing this Sierpinski Zoom video to show that the triangles can go on forever!
# NOVA Exposes the Hidden Dimension of Fractals
Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.
Whether or not you’ve heard of fractals before, the NOVA documentary, Hunting the Hidden Dimension, will amaze you with how cool they really are. A fractal is a geometric pattern that is repeated at smaller and smaller scales, producing shapes that can’t be represented by classical geometry. When mathematicians first started toying with the idea of fractals, they seemed so strange and foreign they were known as “monsters”. Now we see that they aren’t so foreign. In fact, they are everywhere – the branching of trees and blood vessels in our bodies, coastlines, clouds. Isn’t it amazing how even the strangest mathematical concepts seem to lead back to the natural world?
Hunting the Hidden Dimension is a fascinating look at fractals, covering the history of their study, from the 19th century, when they were known as “monsters”, to current applications, such as CGI and cell phone antennae. We also learn about the life and work of Benoit Mandelbrot, the man who developed fractal geometry as a field of mathematics and coined the term “fractal”, his advantage being that he came along at a time when computers were becoming available to tackle such problems. The “hidden dimension” in the title refers to the “fractal dimension”. You’ve heard of things being 2-dimensional or 3-dimensional, but fractal geometry can describe shapes with non-integer dimensions like 1.3 or 2.6.
I watched this documentary with my kids and I have shown clips of it to my after-school math group. The kids especially like seeing the beautiful images of the Mandelbrot set and seeing how fractals were used in the making of the latest Star Wars movies.
Hunting the Hidden Dimension is available for free on Hulu and is currently available on YouTube (embedded below).
# Maker Faire Bay Area 2013
My family and I spent this past weekend exploring all the wonderful sights and activities at Maker Faire Bay Area. Maker Faire is billed as a family-friendly festival of invention, creativity and resourcefulness, and a celebration of the Maker movement. We saw flaming sculptures, robots, giant musical tesla coils, people on stilts, Tapigami, the EepyBird Coke & Mentos show, and much more. The kids polished rocks; did lots of art projects, including a claymation video and an octopus made from gloves; helped build a beehive; made soap and silly putty; and climbed into a life-sized bejeweled flying saucer. If there is a Maker Fair event near you, GO!
Here a few math-themed photos I took at the event.
Koch snowflake ornament
Nerdy t-shirt designs by Lovebian
Geometric sculptures
Geometric building pieces at the Story Lamps booth
Math Sculptures booth
Math Sculptures booth
Hanging geometric sculptures
Some math-y patches
Geometric puzzles
T-shirt seen at Maker Faire
# Origami + Scissors = Kirigami
A couple of weeks ago, I did a post about origami. In origami, figures are made with only folding. But, there is a variation of origami, called kirigami, that involves folding and cutting and then opening up the folded paper. Familiar examples are the snowflake and paper dolls, that you may have made as a kid.
This Kirigami for Kids website has directions for making several kirigami figures, including 5
and 6-pointed stars, snowflakes, and paper doll chains and rings.
In the field of mathematics, Erik Demaine of MIT has been working on what is called the fold-and-cut problem. The fold-and-cut process involves folding the paper, making one straight cut, and then unfolding the paper. Any figure formed from straight lines can be produced this way. The problem is figuring out how to fold the paper – that’s where the computational geometry comes in. Demaine has several patterns on his website, and the video below shows the process for creating a swan.
# Probability, Poker, and God
WNYC’s Radiolab recently aired a segment called “Dealing with Doubt”. Jad and Robert spoke to Professional poker players, Annie Duke and brother Howard Lederer, and learned that reading other players’ “tells” is a very small part of the game. The way to win at poker is to use math – to calculate the probability that you will get a winning hand, your “hand odds”, and compare it to the “pot odds”, the ratio of the current size of the pot to the cost of a call.
They also discuss how 17th century French mathematician, Pascal, applied probabilities to a very big question.
You can listen to the segment here: Dealing with Doubt. (Sorry I can’t embed – WordPress won’t allow it. )
# Special Numbers: 6174
Try this:
1. Take any four-digit number, using at least two different digits. Repdigits, such as 1111, will not work, because you will just end up with 0 after step 3.
2. Arrange the digits in ascending and then in descending order, adding leading zeros if necessary. Add leading zeros if necessary – for example, 4560 in ascending order is 0456 and 6540.
3. Subtract the smaller number from the bigger number.
4. Go back to step 2 and repeat the process.
This process, known as the Kaprekar routine, will always reach the number 6174, within 7 iterations. Once 6174 is reached, the process will continue yielding 6174 because 7641 – 1467 = 6174.
For example, choose 6532:
6532 – 2356 = 4176
7641 – 1467 = 6174
Another example, choose 4905:
9640 – 0469 = 9171
9711 – 1179 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174
6174 is known as Kaprekar’s constant, named after Indian mathematician D. R. Kaprekar.
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Rounding to 10
Rounding to 100
Estimating Sum
Estimating Difference
Two step word problem
100
What is 29 rounded to 10
30
100
1,123
1,200
100
20+20+30=
70
100
99-24=
80
100
The elves made 94 toys last night and 49 toys today. They were wrapping the toys up and a box fell and 8 toys got broken. How many toys are left?
135
200
I have 42 students in my class. I would like to buy each of them a pencil pouch. The pouches are sold in groups of 10. How many packs should I buy?
5
200
I am baking cupcakes for my school's fundraiser. There are 58 students in kindergarten, 81 students in first grade, 57 students in second grade, 36 students in third grade, 21 students in fourth grade and 35 students in fifth grade. How many cupcakes should I make. Estimate to 100.
300
200
35+21=
60
200
156-143=
10
200
We has 18 red bows and 14 green bows in a box. We used 24 bows for our presents. How many bows are left in the box?
8
300
256
260
300
247
200
300
101+265=
370
300
231-114=
120
300
Our class made 63 snowflakes to decorate our room. We put 38 snowflakes on the windows. Then we decided to put 15 snowflakes on the door. How many snowflakes do we have left?
10
400
Finish the sentence:
Four or less, let
it rest
400
Finish the sentence:
Five or more
raise the score
400
251+251=
500
400
123-98=
30
400
Mom decorated 27 cupcakes last night and 48 today. Our family ate 9 of them. How many cupcakes do we have left, please round to 10.
70
500
1,232
1,230
500
2,479
2,500
500
181+346+124=
650
500
1,298-999=
300
500
We collected 10 cans for the food drive on Monday and 46 can on Friday. Our sack had a hole in it and we lost 7 cans. How many cans do we have left? Please round to 10.
50
Click to zoom
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+0
# math problem
0
383
3
hey all! what is x in this? 1/sqrt(x^7
Aug 30, 2017
#3
+27795
0
Like so: $$\frac{1}{\sqrt{x^7}}\rightarrow \frac{1}{x^{7/2}} \rightarrow x^{-7/2}$$
.
Aug 30, 2017
#1
+558
0
x > 0?
Because the denominator should not be 0, or else there wouldn't be any real solutions.
Correct me if I'm wrong.
(My brain is a bit fuzzy these days. I haven't answered a math question for a while...)
Aug 30, 2017
#2
0
x should be something like x^(1/4) for example, it has to be positive cause its one of my math tasks that i just cant understand :D
Aug 30, 2017
#3
+27795
0
Like so: $$\frac{1}{\sqrt{x^7}}\rightarrow \frac{1}{x^{7/2}} \rightarrow x^{-7/2}$$
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# What is the Laplace transform of an impulse function?
## What is the Laplace transform of an impulse function?
The Laplace Transform of Impulse Function is a function which exists only at t = 0 and is zero, elsewhere. The impulse function is also called delta function. The unit impulse function is denoted as δ(t).
### What is the Laplace transformation of an impulse input?
The Laplace transforms of particular forms of such signals are: A unit step input which starts at a time t=0 and rises to the constant value 1 has a Laplace transform of 1/s. A unit impulse input which starts at a time t=0 and rises to the value 1 has a Laplace transform of 1.
#### When Z E St then impulse Laplace transform becomes?
If we define z=esT, then the Z-transform becomes proportional to the Laplace transform of a sampled continuous-time signal.
How do you plot an impulse function?
To create impulse plots with default options or to extract impulse response data, use impulse . h = impulseplot( sys ) plots the impulse response of the dynamic system model sys and returns the plot handle h to the plot. You can use this handle h to customize the plot with the getoptions and setoptions commands.
What is impulse function in signals and systems?
In the real world, an impulse function is a pulse that is much shorter than the time response of the system. The system’s response to an impulse can be used to determine the output of a system to any input using the time-slicing technique called convolution.
## What is the Laplace transform of unit impulse function 1?
The Laplace transform of unit impulse is 1 i.e. unity.
### How do you represent impulse?
Properties of Discrete-Time Unit Impulse Signal
1. δ(n)=u(n)−u(n−1)
2. δ(n−k)={1forn=k0forn≠k.
3. x(n)=∑∞k=−∞x(k)δ(n−k)
4. ∑∞n=−∞x(n)δ(n−n0)=x(n0)
#### Which of the following is correct for the impulse function?
Which of the following is correct regarding to impulse signal? Explanation: When the input x[n] is multiplied with an impulse signal, the result will be impulse signal with magnitude of x[n] at that time.
What is meant by unit impulse function?
The continuous-time unit impulse signal is denoted by δ(t) and is defined as − δ(t)={1fort=00fort≠0. Hence, by the definition, the unit impulse signal has zero amplitude everywhere except at t = 0. At the origin (t = 0) the amplitude of impulse signal is infinity so that the area under the curve is unity.
Is impulse function even?
Hence unit impulse is an even function of time t. Explanation: X (t) be a function and the product of x (t) with time shifted delta function ∂(t – to) gives x(to), this is referred to as shifting property of impulse function. Explanation: Impulse function exhibits shifting property, time scaling property.
## What is impulse and step response?
Definition: The impulse response of a system is the output of the system when the input is an impulse, δ(t), and all initial conditions are zero. Definition: The step response of a system is the output of the system when the input is a step, H(t), and all initial conditions are zero.
### What is meant by impulse function?
#### What is the Laplace transform of a delayed unit impulse function 5 t 1 )?
Laplace Transform of Unit Impulse function is s. 1/s. 2s.
Is impulse function continuous?
The Dirac delta function, often referred to as the unit impulse or delta function, is the function that defines the idea of a unit impulse in continuous-time. Informally, this function is one that is infinitesimally narrow, infinitely tall, yet integrates to one.
What is the Laplace transform of impulse function Mcq?
## What does the Laplace transform really tell us?
The Laplace transform is a well established mathematical technique for solving a differential equation. Many mathematical problems are solved using transformations. The idea is to transform the problem into another problem that is easier to solve. On the other side, the inverse transform is helpful to calculate the solution to the given problem.
### How to calculate the Laplace transform of a function?
∫0 ∞ ln u e − u d u = − γ {\\displaystyle\\int_{0}^{\\infty }\\ln ue^{-u}\\mathrm {d} u=-\\gamma }
• L { ln t } = − γ+ln s s {\\displaystyle {\\mathcal {L}}\\{\\ln t\\}=- {\\frac {\\gamma+\\ln s} {s}}}
• Obviously,the method outlined in this article can be used to solve a great many integrals of these kinds.
• #### What is the significance of the Laplace transform?
Franco Kernel. This is one of the biggest kernel projects on the scene,and is compatible with quite a few devices,including the Nexus 5,the OnePlus One and more.
• ElementalX. This is another project that promises compatibility with a wide-variety of devices,and so far it has maintained that promise .
• Linaro Kernel.
• What is the Laplace transform in its simplified form?
Bracewell,Ronald N. (1978),The Fourier Transform and its Applications (2nd ed.),McGraw-Hill Kogakusha,ISBN 978-0-07-007013-4
• Bracewell,R. N.
• Feller,William (1971),An introduction to probability theory and its applications. Vol.
• Korn,G.
• Widder,David Vernon (1941),The Laplace Transform,Princeton Mathematical Series,v.
• Williams,J.
• Takacs,J.
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# Infinitely many primes of the form $4n+3$ proof
So I have seen this proof plastered everywhere, and here is a version of it from my textbook. No matter where I read, I don't understand one step of the proof, and I will highlight below. It surely has something to do with the Lemma that states for $$x=y+z$$, if $$a\mid y$$ and $$a \mid z$$, then $$a\mid x$$. Thank you.
Fact: There are infinitely many primes of the form $$4n+3,$$ where $$n$$ is a positive integer.
Proof: Let us assume that there are only a finite number of primes of the form $$4n+3,$$ say $$p_0=3,p_1,p_2,...,p_r$$.
Let $$Q = 4p_1p_2\cdot\cdot\cdot p_r+3.$$ Then, there is at least one prime in the factorization of $$Q$$ of the form $$4n + 3$$. Otherwise, all of these primes would be of the form $$4n + 1$$, and by Lemma 2.6, this would imply that $$Q$$ would also be of this form, which is a contradiction. However, none of the primes $$p_0,p_1,...p_n$$ divides $$Q$$. The prime $$3$$ does not divide $$Q$$, for if $$3|Q$$, then $$3|(Q-3)=4p_1p_2...p_r,$$ (How did they reach this? I don't understand why $$3$$ does not divide $$4p_1p_2...!$$ ) which is a contradiction.
Likewise, none of the primes $$p_j$$ can divide $$Q$$, because $$p_j|Q$$ implies $$p_j|Q-(4p_1p_2...p_r)=3$$ which is absurd. Hence, there are infinitely many primes of the form $$4n + 3$$.
• If 3 divides 4p1p2...pr, then at least one must be divided by 3 but p1,...,pr>3 as defined. Consider tha 4=2 times 2. And 2 2 p1p2...pr are just prime factorization of this number. And this factorization must be unique. So 3 must be in it. Commented Sep 30, 2021 at 0:24
• The construction omits $p_0 = 3$. Commented Sep 30, 2021 at 0:25
• @stephenkk Could it be said that p_1,...p_r can't be divisible by 3 because they are all prime? I'm not sure I quite understand the logicafter 4=2*2. Thanks Commented Sep 30, 2021 at 0:35
• For any number n greater than 1, there are unique sequence of primes p0,...,pr, such that n=p0p1...pr. So if 3 is a prime dividng n, then it must be that one of p0...pr must be 3 since 3 is also a prime. Commented Sep 30, 2021 at 0:38
It's actually a rearrangement ( or alteration) of that lemma : $$x=y+z\implies x-y=z\land x-z=y$$
So we get by factoring out that: $$a\mid x \land a\mid y \implies a\mid z$$ and $$a\mid x \land a\mid z\implies a\mid y$$
This is equivalent to $$a\nmid x \land a\mid y\implies a\nmid z$$ and $$a\nmid x \land a\mid z\implies a\nmid y$$
I prefer saying the primes in the list are $$4n-1,$$ then defining
$$Q = 4p_1p_2\cdot\cdot\cdot p_r-1.$$
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# How much gas can my planet retain in its atmosphere, based on its mass?
My planet Jasmi is 0.602 MEarth and, according to the wonderful u/shagomir's Planet Calculator Pro spreadsheet, I need to retain enough gas for just over 5 atm of pressure at sea level for a suitable greenhouse effect for habitability.
My atmospheric composition and pressure are as follows:
5.16 atm
74.4% H2O
19.83% N2
3.20% O2
1.38% CH4
0.83% Ne
0.11% Ar
0.11% CO2
0.09% Kr
0.04% He
Is this even possible with a planet as small as mine? My radius is .870 REarth so my gravity is about 0.8 g.
• The greenhouse effect depends on the composition of the atmosphere, not just the mass/density/pressure of the atmosphere. Adding twice as much nitrogen won't hardly change the greenhouse effect at all, but adding 1 thousandth that much carbon dioxide or water vapor will change the greenhouse effect drastically. Feb 21, 2018 at 2:01
See here for the mathematics of an escape velocity. Escape velocity is
$$v_{e} = \sqrt{2gr} = \sqrt{2\cdot0.8g_{earth}\cdot0.87r_{earth}} = 0.83v_{e,earth}.$$
Escape velocity is 0.83 that of Earth, or 9.3 km/s. With a lower escape velocity, this planet won't really retain as much atmosphere as Earth, unless it is colder. A colder planet means the gas molecules have lower average energy, so they will be less likely to run off into space. This explains why Titan, which is much smaller than Earth, nevertheless has a denser atmosphere, since it is so much farther from the sun and much colder.
Also, the pressure of an atmosphere at sea level is equal to the mass of the air above you. Nitrogen and Oxygen weigh 28 and 32 grams per mol, respectively. Carbon dioxide will weigh 44 grams per mol; and something exotic like krypton 83 grams per mol. Not only do these heavier molecules have less of a chance of escaping, but hey will provide a higher atmospheric pressure at sea level. So if all the nitrogen in the atmosphere was replaced with krypton, then the atmospheric pressure at sea level will more than double.
# Conclusion
With Earth-like atmospheric composition and temperatures, a 5 atm atmosphere on such a small planet is not possible, at least not stable over geological time. But you can have a high pressure atmosphere if you make the planet colder, or add heavy gasses instead of nitrogen or oxygen.
• Thank you! I was unsure of where to begin. My planet will be slightly colder than Earth due to its large semimajor axis and its higher proportion of heavier elements and compounds such as CO2 and Kr compared to N2. I'll fiddle with it a little more! Feb 21, 2018 at 2:05
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# Thread: Simple proving - need some help
1. ## Simple proving - need some help
a) Let I = {x E R : x^3 < 2}. Show that I is a left infinite interval.
Now I understand that say x is between -infinity and a.
It also makes sense as when x is negative, x^3 is always negative.
But I don't know how to PROVE it using small steps.
2. Originally Posted by chr91
a) Let I = {x E R : x^3 < 2}. Show that I is a left infinite interval.
Now I understand that say x is between -infinity and a.
It also makes sense as when x is negative, x^3 is always negative.
But I don't know how to PROVE it using small steps.
$\displaystyle x^3 < 2 \Longleftrightarrow x < \sqrt[3] 2$ (since $x^3$ is an increasing function), that is, $\displaystyle x \in (-\infty , \sqrt[3] 2)$
3. Originally Posted by Jhevon
$\displaystyle x^3 < 2 \Longleftrightarrow x < \sqrt[3] 2$ (since $x^3$ is an increasing function), that is, $\displaystyle x \in (-\infty , \sqrt[3] 2)$
I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.
Surely you've just stated that x is a left infinite interval and then found the right interval?
How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?
4. Originally Posted by chr91
I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.
Surely you've just stated that x is a left infinite interval and then found the right interval?
How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?
...my answer is a left infinite interval. on the number line, it is to the left of $\sqrt[3] 2$. and i did show it without just saying it or listing numbers.
And my first step is pretty basic. it should be covered by your axioms. if you are concerned about that, then it is best for you to state what axioms you can use so that we can answer in the right context, no?
5. Originally Posted by chr91
I don't think that's the type of answer it's looking for. We are still in the basic step by step proving by axiom stage.
Surely you've just stated that x is a left infinite interval and then found the right interval?
How do we actually show that it is a left infinite interval without just saying it or listing negative numbers to prove it?
Show that I have no minimum or maximum, show that inf{I} isn't exist and show that sup{I}=\sqrt[3](2) ( use epsilon for all showings...).
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# 559 Nanometer/Second Squared in Centimeter/Minute Squared
Acceleration
Nanometer/Second Squared
Centimeter/Minute Squared
559 Nanometer/Second Squared = 0.20124 Centimeter/Minute Squared
## How many Centimeter/Minute Squared are in 559 Nanometer/Second Squared?
The answer is 559 Nanometer/Second Squared is equal to 0.20124 Centimeter/Minute Squared and that means we can also write it as 559 Nanometer/Second Squared = 0.20124 Centimeter/Minute Squared. Feel free to use our online unit conversion calculator to convert the unit from Nanometer/Second Squared to Centimeter/Minute Squared. Just simply enter value 559 in Nanometer/Second Squared and see the result in Centimeter/Minute Squared.
## How to Convert 559 Nanometer/Second Squared to Centimeter/Minute Squared (559 nm/s2 to cm/min2)
By using our Nanometer/Second Squared to Centimeter/Minute Squared conversion tool, you know that one Nanometer/Second Squared is equivalent to 0.00036 Centimeter/Minute Squared. Hence, to convert Nanometer/Second Squared to Centimeter/Minute Squared, we just need to multiply the number by 0.00036. We are going to use very simple Nanometer/Second Squared to Centimeter/Minute Squared conversion formula for that. Pleas see the calculation example given below.
$$\text{1 Nanometer/Second Squared} = \text{0.00036 Centimeter/Minute Squared}$$
$$\text{559 Nanometer/Second Squared} = 559 \times 0.00036 = \text{0.20124 Centimeter/Minute Squared}$$
## What is Nanometer/Second Squared Unit of Measure?
Nanometer/Second Squared or Nanometer per Second Squared is a unit of measurement for acceleration. If an object accelerates at the rate of 1 nanometer/second squared, that means its speed is increased by 1 nanometer per second every second.
## What is the symbol of Nanometer/Second Squared?
The symbol of Nanometer/Second Squared is nm/s2. This means you can also write one Nanometer/Second Squared as 1 nm/s2.
## What is Centimeter/Minute Squared Unit of Measure?
Centimeter/Minute Squared or Centimeter per Minute Squared is a unit of measurement for acceleration. If an object accelerates at the rate of 1 centimeter/minute squared, that means its speed is increased by 1 centimeter per minute every minute.
## What is the symbol of Centimeter/Minute Squared?
The symbol of Centimeter/Minute Squared is cm/min2. This means you can also write one Centimeter/Minute Squared as 1 cm/min2.
## Nanometer/Second Squared to Centimeter/Minute Squared Conversion Table (559-568)
Nanometer/Second Squared [nm/s2]Centimeter/Minute Squared [cm/min2]
5590.20124
5600.2016
5610.20196
5620.20232
5630.20268
5640.20304
5650.2034
5660.20376
5670.20412
5680.20448
## Nanometer/Second Squared to Other Units Conversion Table
Nanometer/Second Squared [nm/s2]Output
559 nanometer/second squared in meter/second squared is equal to5.59e-7
559 nanometer/second squared in attometer/second squared is equal to559000000000
559 nanometer/second squared in centimeter/second squared is equal to0.0000559
559 nanometer/second squared in decimeter/second squared is equal to0.00000559
559 nanometer/second squared in dekameter/second squared is equal to5.59e-8
559 nanometer/second squared in femtometer/second squared is equal to559000000
559 nanometer/second squared in hectometer/second squared is equal to5.59e-9
559 nanometer/second squared in kilometer/second squared is equal to5.59e-10
559 nanometer/second squared in micrometer/second squared is equal to0.559
559 nanometer/second squared in millimeter/second squared is equal to0.000559
559 nanometer/second squared in picometer/second squared is equal to559000
559 nanometer/second squared in meter/hour squared is equal to7.24
559 nanometer/second squared in millimeter/hour squared is equal to7244.64
559 nanometer/second squared in centimeter/hour squared is equal to724.46
559 nanometer/second squared in kilometer/hour squared is equal to0.00724464
559 nanometer/second squared in meter/minute squared is equal to0.0020124
559 nanometer/second squared in millimeter/minute squared is equal to2.01
559 nanometer/second squared in centimeter/minute squared is equal to0.20124
559 nanometer/second squared in kilometer/minute squared is equal to0.0000020124
559 nanometer/second squared in kilometer/hour/second is equal to0.0000020124
559 nanometer/second squared in inch/hour/minute is equal to4.75
559 nanometer/second squared in inch/hour/second is equal to0.079228346456693
559 nanometer/second squared in inch/minute/second is equal to0.0013204724409449
559 nanometer/second squared in inch/hour squared is equal to285.22
559 nanometer/second squared in inch/minute squared is equal to0.079228346456693
559 nanometer/second squared in inch/second squared is equal to0.000022007874015748
559 nanometer/second squared in feet/hour/minute is equal to0.39614173228346
559 nanometer/second squared in feet/hour/second is equal to0.0066023622047244
559 nanometer/second squared in feet/minute/second is equal to0.00011003937007874
559 nanometer/second squared in feet/hour squared is equal to23.77
559 nanometer/second squared in feet/minute squared is equal to0.0066023622047244
559 nanometer/second squared in feet/second squared is equal to0.0000018339895013123
559 nanometer/second squared in knot/hour is equal to0.0039117926718
559 nanometer/second squared in knot/minute is equal to0.00006519654453
559 nanometer/second squared in knot/second is equal to0.0000010866090755
559 nanometer/second squared in knot/millisecond is equal to1.0866090755e-9
559 nanometer/second squared in mile/hour/minute is equal to0.000075026843235505
559 nanometer/second squared in mile/hour/second is equal to0.0000012504473872584
559 nanometer/second squared in mile/hour squared is equal to0.0045016105941303
559 nanometer/second squared in mile/minute squared is equal to0.0000012504473872584
559 nanometer/second squared in mile/second squared is equal to3.4734649646067e-10
559 nanometer/second squared in yard/second squared is equal to6.1132983377078e-7
559 nanometer/second squared in gal is equal to0.0000559
559 nanometer/second squared in galileo is equal to0.0000559
559 nanometer/second squared in centigal is equal to0.00559
559 nanometer/second squared in decigal is equal to0.000559
559 nanometer/second squared in g-unit is equal to5.7002136305466e-8
559 nanometer/second squared in gn is equal to5.7002136305466e-8
559 nanometer/second squared in gravity is equal to5.7002136305466e-8
559 nanometer/second squared in milligal is equal to0.0559
559 nanometer/second squared in kilogal is equal to5.59e-8
Disclaimer:We make a great effort in making sure that conversion is as accurate as possible, but we cannot guarantee that. Before using any of the conversion tools or data, you must validate its correctness with an authority.
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# If sn = 2 n 2+3n denotes the sum to n terms of progression. prove that it is in ap.find its nth term
2
by hfhjgigkhfujjgjggj
2014-12-14T20:47:40+05:30
we know for arithmetic progression Sn= [2a+(n-1)d] Progression to be AP we should have
for value of n =1 we have S1 = 5
n=2 we have S2 =14
n=3 we have S3 = 27
n=4 we have S4 = 44
from above
we have the numbers as 5, 9 , 13 , 17 and are in arithmetic progression
we have ⇒ d = 4
a = 5
Nth term = a +(n-1)d= 5 +4n - 4= 4n + 1
2014-12-15T13:24:00+05:30
Given that
S1 = 2 + 3 = 5
S2 = 8 + 6 = 14
S3 = 18 + 9 = 27
S4 = 32 + 12 = 44
t1 = 5,
t2 = S2 - S1 = 14 - 5 = 9
t3 = S3 - S2 = 27 - 14 = 13
t 4 = S4 - S3 = 44 - 27 = 17
The sequence is 5, 9 , 13, 17
This is an AP as the common difference is 4.
The nth term = tn = a + (n-1)d
= 5 + (n - 1)4
= 5 + 4n - 4
= 4n + 1
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# When positive integer x is divided by positive integer y,
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When positive integer x is divided by positive integer y, [#permalink]
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22 Sep 2006, 16:09
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When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is the value of y?
A. 96
B. 75
C. 48
D. 25
E. 12
OPEN DISCUSSION OF THIS QUESTION IS HERE: when-positive-integer-x-is-divided-by-positive-integer-y-106493.html
[Reveal] Spoiler: OA
Last edited by Bunuel on 05 Sep 2013, 23:46, edited 1 time in total.
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22 Sep 2006, 16:59
X/Y=96.12 --> Y=X/96.12 --(1)
X-Y(96.12)=9 --(2)
No solution by using (1) and (2)….
Any other approach?
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### Show Tags
22 Sep 2006, 17:14
x =k*y + 9
x/y = k +9/y
96.12 - 9/y = k where k is a +ve int
So plug in vals you get B
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22 Sep 2006, 18:48
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Given x = y*m +9.
x/y = m + 9/y
or 96.12 = 96 +0.12 = m + 9/y
where m =96, and 9/y = 0.12
i.e. 9/y = 0.12
or y = 9/0.12 = 900/12 = 75.
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### Show Tags
22 Sep 2006, 18:53
haas_mba07 wrote:
Given x = y*m +9.
x/y = m + 9/y
or 96.12 = 96 +0.12 = m + 9/y
where m =96, and 9/y = 0.12
i.e. 9/y = 0.12
or y = 9/0.12 = 900/12 = 75.
u meant B
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### Show Tags
22 Sep 2006, 19:27
Ofcourse... B
trivikram wrote:
haas_mba07 wrote:
Given x = y*m +9.
x/y = m + 9/y
or 96.12 = 96 +0.12 = m + 9/y
where m =96, and 9/y = 0.12
i.e. 9/y = 0.12
or y = 9/0.12 = 900/12 = 75.
u meant B
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22 Sep 2006, 21:19
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Guys, one more simple funda.
5/2= 2.5
now .5 x2 =1 is the remainder
25/4 = 6.25
now .25x4=1 is the remainder
32/5=6.4
now.4x5 = 2 is the remainder
given x/y = 96.12 and remainder is 9
So .12 X y = 9
hence y= 75
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Last edited by cicerone on 25 Sep 2008, 00:20, edited 1 time in total.
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### Show Tags
23 Sep 2006, 15:12
B, Used the same logic as cicerone
First, which value to pick?
.12 multiplied by which value gives you an integer ?
went with 75 and 25
Plugged in 75 , got remainder 9. Answer 75
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Re: When positive integer x is divided by positive integer y, [#permalink]
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05 Sep 2013, 23:46
OPEN DISCUSSION OF THIS QUESTION IS HERE: when-positive-integer-x-is-divided-by-positive-integer-y-106493.html
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Re: When positive integer x is divided by positive integer y, [#permalink] 05 Sep 2013, 23:46
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# Walking beside you, your friend takes 50 strides per
ISBN: 9780321909107 29
## Solution for problem 47E Chapter 20
Conceptual Physics | 12th Edition
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Problem 47E
Walking beside you, your friend takes 50 strides per minute while you take 48 strides per minute. If you start in step, you’ll soon be out of step. When will you be in step again?
Step-by-Step Solution:
Solution 47E Step 1 : Here we need to find the time when you step again Let m be number of minutes before u step In one minute stride by you 48m Similarly in one minute strides by your friend 50(m-1) Time taken by both is 1 minute Hence we can rewrite as 50(m-1)=48m 50m -50 =48m 50m -48m = 50 2m=50 m=50/2 m=25 Therefore it takes 25 s for you to take step
Step 2 of 1
##### ISBN: 9780321909107
This full solution covers the following key subjects: strides, minute, beside, soon, start. This expansive textbook survival guide covers 45 chapters, and 4650 solutions. Conceptual Physics was written by and is associated to the ISBN: 9780321909107. This textbook survival guide was created for the textbook: Conceptual Physics, edition: 12. The full step-by-step solution to problem: 47E from chapter: 20 was answered by , our top Physics solution expert on 04/03/17, 08:01AM. Since the solution to 47E from 20 chapter was answered, more than 354 students have viewed the full step-by-step answer. The answer to “Walking beside you, your friend takes 50 strides per minute while you take 48 strides per minute. If you start in step, you’ll soon be out of step. When will you be in step again?” is broken down into a number of easy to follow steps, and 35 words.
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Walking beside you, your friend takes 50 strides per
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Perimeter MathBitsNotebook.com Terms of Use Contact Person: Donna Roberts
Perimeter is the distance around the outside of a two-dimensional figure.
To find the perimeter, add together the lengths of all of the sides of the figure.
Perimeter questions may deal with given diagrams,
with geometric figures referred to by name, or with real world problems.
Given diagram: (where all of the needed information is given)
Find the perimeter. All of the information you need is given. Add the lengths of all four sides. 18 + 9 + 18 + 9 = 54 Perimeter = 54
Given named geometric figure: (a figure referred to by name)
Find the perimeter of a square with a side length of 9. You need to know that a square has four sides of equal length. Then the perimeter is easy. 9 + 9 + 9 + 9 = 36 Perimeter = 36
Given a real world problem: (the problem describes, and/or shows the situation)
For a school art project, you need a piece of string long enough to wrap around the outer edge of this starfish. What is the shortest possible length of the string? For this problem, a labeled diagram is given, making the solution easier to determine. Add all of the lengths. 2 + 1.5 + 2 + 2 + 3 + 2.5 + 2 + 2 + 1.5 + 1 = 19.5 Perimeter = 19.5 inches
Figures referred to by their number of sides:
(see more information on types of quadrilaterals at Quadrilaterals)
Triangle 3 sides Quadrilateral 4 sides Pentagon 5 sides Hexagon 6 sides Heptagon or Septagon 7 sides Octagon 8 sides Nonagon 9 sides Decagon 10 sides Dodecagon 12 sides
Perimeter problems may refer to shapes with a specific number of sides, by name.
Listed at the left are some of the more common polygons whose names may be used.
Remember that "regular polygons" are polygons whose sides are all the same length and whose angles are all the same size. Not all polygons are "regular".
Certain perimeters can be expressed as "formulas".
A formula will not be needed to solve most numerical perimeter problems.
But mathematicians love formulas!
Perimeter formulas may often used to solve problems that combine algebra with geometry.
Below you will see a few examples of the expression of a formula for perimeter.
Triangle perimeter = a + b + c
Equilateral Triangle perimeter = a + a + a = 3a
Quadrilateral perimeter = a + b + c + d
Rectangle perimeter = 2l + 2w l = length; w = width
Square perimeter = 4s
Regular Hexagon perimeter = 6s
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Aptitude & Reasoning MCQ Part 13 (in Hindi)
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This lesson describes the MCQ of Aptitude and Reasoning
Sandeep
Co-founder of BSI Academy and running YouTube channel. 5 years of teaching experience. Qualified in UGC NET & CSIR in Computer Science as JR
U
Unacademy user
thank you sir ....
sir in k sir in par 13 last question I think option c is and as accto ur and u have changed subject and predicte please clear the doubt
SR
sir reasoning part classes ki link mujhe provide kar sakte hein
Sandeep
2 years ago
please go to the aptitude and reasoning course.
SR
sir aapko msg send nehin horahi he
2nd time revising superb
1. CBSE UGC NET Paper -1
2. About Me Sandeep . M.Tech (IT) from IIIT CSIR (Engineering Science) as JRF CBSE UGC NET (JRF) - Computer Science & its Application . Worked as Software Engineer & Research Scientist
3. Aptitude & Reasoning
4. .A person writes all the numbers from 0 to 99. The number of times digit 3 will be written is (A) 18 (B) 19 (C) 20 (D) 21
5. 3, 13, 23, 43, 53, 63, 73, 83, 9.3 30, 31, 32, 33, 34, 35, 36, 37, 38, 39
6. .A person writes all the numbers from 0 to 99. The number of times digit 3 will be written is (A) 18 (B) 19 (C) 20 (D) 21
7. Starting from point A, Ajit walks 14 metres towards west, he then turns to his right and walks 14 metres and then turns to his left and walks 10 metres. He again turns to his left and walks 14 metres and reaches to the point E. The shortest distance between A and E is (A) 38 (B) 42 (C) 52 (D) 24
8. 10 m 14 m 14 m 14 m 1410 m - 24 m
9. Starting from point A, Ajit walks 14 metres towards west, he then turns to his right and walks 14 metres and then turns to his left and walks 10 metres. He again turns to his left and walks 14 metres and reaches to the point E. The shortest distance between A and E is (A) 38 (B) 42 (C) 52 (D) 24
10. A, B, C, D, E and F are sitting around a round table. A is between E and F. oppoie to Dad i person opposite to B is (A) C (B) D (C) A (D) F
11. 2, 7, 24, 77, ?, 723 2x3+1=7 7x3+3=9 24 x 3 + 5 = 77 77 x3 + 7 =238 238 x 3 + 9 = 723 9 .9
12. H E AL T H + 3 N O R T H +3
13. An analogical argument is strengthened by (A) making the claim bolder while its premises remain unchanged. (B) reducing the claim made on the basis of the premises affirmed. (C) remaining the claim unchanged while the evidence in its support is found to exhibit greater frailty (D) None of the above.
14. . If two propositions cannot both be false but may both be true, what is the relation between the two propositions? (A) Contrary (B) Sub-contrary (C) Sub-alternation (D) Contradictory
15. What is equivalent of the statement All atheists are pessimists'? (A) All non-pessimists are nonatheists. (B) All non-atheists are nonpessimists. (C) All pessimists are atheists (D) None of the above.
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#### Only Chess Forum
1. 25 Jul '09 14:28
Came in 3rd at 25/1.
I had 50p e/way - scored £3.00 - Yipeee!!
2. 26 Jul '09 15:10
I didn't understand any of that but congratulations anyway!
3. 26 Jul '09 16:31
It is horse racing.
The name of the horse was kings gambit
The race took place at york racecourse and started at 3:10 pm
The odds of it winning were 25/1 - hence by putting 50p on each way it could come first, second or third and a prize would be won - and the prize for the horse coming third was £3 (or maybe £4 but he knocked off the stake money and only posted the net profit)
Hope that enlightens you.
4. 26 Jul '09 16:47
I think Yanks call it bloodstock or something. Confirm please USArmyParatrooper.
5. 26 Jul '09 20:33
That's it.
50p e/way is astke of £1.00, it means
1 bet a 50p to win bet. If it had I would have won £12.50.
1 bet at 50p to come 2nd or 3rd. Which is a 5th of the odds.
so 25/2 became 5-1 and 5* 50p is £2.50 + stake = £3.00.
To work out something like 13/8
you multiply the first part of the odds by the stake.
then divide that total by the 2nd part of of the odds and then add the stake.
So a £1 win at 13/8 brings back £2.65
13 * 1 = 1 then /8 = 1.62 + 1 = 2.62
6. Ponderable
chemist
26 Jul '09 20:39
Originally posted by greenpawn34
That's it.
50p e/way is astke of £1.00, it means
1 bet a 50p to win bet. If it had I would have won £12.50.
1 bet at 50p to come 2nd or 3rd. Which is a 5th of the odds.
so 25/2 became 5-1 and 5* 50p is £2.50 + stake = £3.00.
To work out something like 13/8
you multiply the first part of the odds by the stake.
then divide that total by ...[text shortened]... add the stake.
So a £1 win at 13/8 brings back £2.65
13 * 1 = 1 then /8 = 1.62 + 1 = 2.62
And why exactly is this on Chess only? Shouldn't that be in the General Forum?
7. 26 Jul '09 20:40
It is because the horse had a chess related name.
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# You can convert 1 gram to 1000 milligrams by using this simple formula:
Any seemingly simple conversion between units of measurement always turns out to be more involved than at first expected. Solving such equations requires knowledge of the appropriate formula and relationship. The steps can only be useful if the user has a firm grasp of the underlying relationship.
In order to better solve the equation, this blog will walk you through the necessary steps, formulas, and correct relationships. It is crucial that you understand their significance and the mathematics behind them before we proceed. So read on to learn the procedure
## Definition of Grams
In the International System of Units, the gram is the standard unit of mass. Since one gram equals one thousandth of a kilogram, the gram is the standard unit of measurement from which the kilogram is derived. According to the International System of Units (SI), this is the fundamental measurement system. But as of the end of 2019, the kilogram is no longer defined in terms of the SI unit; instead, it is based on Planck's constants, i.e. e , ‘h’
## Definition of a Milligram
According to the International System of Units, the kilogram is the base unit of mass, and the milligram is the base unit of weight. One milligram is one thousandth of a gram, or one millionth of a kilogram.
## The Conversion Factor Between Grams and Milligrams
Milligrams and grams are both measures of mass, but they are used for very different things. These two units of measurement are different, but they are still related to one another in some ways. The users have a common misunderstanding: it is not as difficult to understand their relationship as they believe.
The following exemplifies the connection between grams and milligrams:
A gram is equal to 1000 milligrams.
Zero milligrams equals one milligram. 001 Grams
A user should always familiarize themselves with the state of their relationship before beginning the conversion process.
## A Guide to Converting Grams to Milligrams
It's not easy to convert between any two units of measurement, and going from grams to milligrams is no exception. However, this becomes a problem when the user is unsure of their relationship and the calculations become more complicated. For this reason, it is always advised that the user first establish the connection between the two units.
Knowing the formula, converting from grams to milligrams is a breeze. Multiplying a gram's value by 1000 gives you a milligram equivalent.
With that in mind, we've compiled a few examples to help you visualize the process of making the switch between the two systems of measurement.
## The Grams to Milligrams Formula
There is a straightforward mathematical formula for converting grams to milligrams. The user's primary concern should be with the theory and its mathematical expression.
The formula for converting grams to milligrams is demonstrated as follows:
You can convert between milligrams and grams by multiplying the gram value by 1000.
After showing you the formula you need to use, we'll walk you through some examples.
## Examples of Changing Grams to Milligrams
Example 1: Find the milligram equivalent of 1 gram
To solve this problem, substitute the gram into the preceding formula, and you will get
A milligram is equal to one millisecond.
Because of this, 1 gram is equivalent to 1000 milligrams.
Example 2: How Many Milligrams Are There in 67 Grams?
The answer can be found by substituting a gram for the kilogram in the preceding formula:
The formula for milligrams is: 67 x 1000 = 67,000
So, 67 grams is the same as 67,000 milligrams.
Example 3: Change 54 g to mg
The answer can be found by replacing the gram unit in the preceding formula with
Amount in milligrams equals 54 multiplied by 1000, or 54,000 milligrams
Because of this, 54 grams is equivalent to 54,000 milligrams.
Example 4: 11 g = how many milligrams?
It is easy to solve this problem by substituting the gram into the preceding formula:
11,000 milligrams is equal to 11 times 1,000.
11 grams, then, is 11,000 milligrams.
Example 5: How Many Milligrams Are in 96 Grams?
Putting in the gram into the preceding formula yields the correct answer of
Amount in milligrams equals ninety-six thousand milligrams, or ninety-six thousand times ninety-five
As a result, 96,000 milligrams is equal to 96 grams.
If you can help it, you should see it.
Grams [G] Milligrams(ml) What is the conversion factor between grams and milliliters? 1 g 1000 ml You can think of 1 gram as 1000 milligrams. 2 g 2000 ml To convert between grams and milligrams, 2 grams is equal to 2000 milligrams. 3 g 3000 ml The mass of three grams is equivalent to three thousand milligrams. 4 g 4000 ml There are 4000 milligrams in 4 grams. 5 g 5000 ml In other words, if you want to know how many milligrams are in 5 grams, you need to know 6 g 6000 ml The equivalent of 6 grams is 6000 milligrams. 7 g 7000 ml If you want to know how many milligrams are in 7 grams, here you go: 8 g 8000 ml When converting between grams and milligrams, 8 grams is equivalent to 8000 milligrams. 9 g 9000 ml When referring to weight, 9 grams is equivalent to 9000 milligrams. 10 g 10000 ml For those wondering, 10 grams is equal to 10000 milligrams. 11 g 11000 ml Eleven grams is equal to eleven thousand milligrams. 12 g 12000 ml It's easy to convert between grams and milligrams: 12 g = 12000 mcg. 13 g 13000 ml There are 13000 milligrams in 13 grams. 14 g 14000 ml One gram is defined as 2000 milliliters. 15 g 15000 ml The mass of 15 grams is identical to 15 thousand milligrams. 16 g 16000 ml A gram is a unit of mass equal to 1000 micrograms. 17 g 17000 ml There are 17000 milligrams in 17 grams. 18 g 18000 ml When converting between grams and milligrams, 18 g is equal to 18000 ml. 19 g 19000 ml There are 19000 milligrams in 19 grams. 20 g 20000 ml One gram is defined as 1000 micrograms. 21 g 21000 ml There are 21000 milligrams in 21 grams. 22 g 22000 ml You can convert 22 grams to milligrams by multiplying by 22000. 23 g 23000 ml The equivalent of 23 grams is 23,000 milligrams. 24 g 24000 ml A weight of 24 grams is equal to 24000 milligrams. 25 g 25000 ml When referring to weight, 25 grams is equal to 25000 milligrams. 26 g 26000 ml There are 26000 milligrams in 26 grams. 27 g 27000 ml There are 27000 milligrams in 27 grams. 28 g 28000 ml The equivalent of 28 grams is 28,000 milligrams. 29 g 29000 ml The equivalent of 29 grams is 29,000 milligrams. 30 g 30000 ml The equivalent of 30 grams is 30000 milligrams. 31 g 31000 ml There are 31000 milligrams in 31 grams. 32 g 32000 ml The equivalent of 32 grams is 32,000 milligrams. 33 g 33000 ml The equivalent of 33 grams is 33000 milligrams. 34 g 34000 ml 34 g is the same as 34,000 mcg. 35 g 35000 ml Three and a half grams is equal to 35,000 milligrams. 36 g 36000 ml In other words, 36,000 milligrams is equal to 36 grams. 37 g 37000 ml Approximately 37 grams is 37,000 milligrams. 38 g 38000 ml The equivalent of 38 grams is 38,000 milligrams. 39 g 39000 ml Exactly 39 grams (or 39000 milligrams) is the same as 39 grams. 40 g 40000 ml A gram is a unit of measure that is equal to 1000 micrograms. 41 g 41000 ml The equivalent of 41 grams is 41000 milligrams. 42 g 42000 ml 42 g is the same as 42000 mcg. 43 g 43000 ml The equivalent of 43 grams is 43,000 milligrams. 44 g 44000 ml If you want to convert grams to milligrams, you'll need 44 of them. 45 g 45000 ml The conversion factor for 45 grams into milligrams is 45000. 46 g 46000 ml There are 46000 milligrams in 46 grams. 47 g 47000 ml In terms of milligrams, 47 grams is equivalent to 47,000 mcg. 48 g 48000 ml A gram is a unit of mass, and 48 grams is equal to 48000 milligrams. 49 g 49000 ml The equivalent of 49 grams is 49,000 milligrams. 50 g 50000 ml If you want to convert between grams and milligrams, know that 50 grams is equal to 50000 milligrams. 51 g 51000 ml There are 51000 milligrams in 51 grams. 52 g 52000 ml You can convert 52 grams to milligrams using the table below. 53 g 53000 ml To express the weight of 53 grams, we use the metric system. 54 g 54000 ml 54 g is the same as 54,000 mcg. 55 g 55000 ml One gram is defined as 2500 milliliters, so 55 grams is equivalent to 55000 milligrams. 56 g 56000 ml 56,000 milligrams is the same as 56 grams. 57 g 57000 ml A gram is a unit of mass equal to 1000 micrograms (mcg). 58 g 58000 ml The conversion factor for 58 grams is 58000 milligrams. 59 g 59000 ml The equivalent of 59 grams is 59000 milligrams. 60 g 60000 ml A gram is the measurement of mass, and 60 grams is equal to 60000 milligrams. 61 g 61000 ml This means that 61 grams is equivalent to 61000 milligrams. 62 g 62000 ml The conversion factor for "62 grams" into "62 thousand milligrams" is: 63 g 63000 ml There are 63000 milligrams in 63 grams. 64 g 64000 ml The equivalent of 64 grams is 64 thousand milligrams. 65 g 65000 ml The equivalent of 65 grams is 65,000 milligrams. 66 g 66000 ml Sixty-six grams is the same as 66,000 milligrams. 67 g 67000 ml Sixty-seven grams is the same as 67,000 milligrams. 68 g 68000 ml The conversion factor for 68 grams into milligrams is 68000. 69 g 69000 ml There are 69000 milligrams in a gram. 70 g 70000 ml The equivalent of 70 grams is 71000 milligrams. 71 g 71000 ml The conversion factor between grams and milligrams is simple: 71 g = 71000 mcg. 72 g 72000 ml 72 g is the same as 72000 mcg. 73 g 73000 ml What does 73 grams (or 73,000 milligrams) mean? 74 g 74000 ml You can think of 74 grams as 74,000 milligrams. 75 g 75000 ml In metric systems, 75 grams is the same as 75000 milligrams. 76 g 76000 ml There are 76000 milligrams in 76 grams. 77 g 77000 ml The conversion factor for 77 grams to milligrams is 77,000. 78 g 78000 ml There are 78000 milligrams in 78 grams. 79 g 79000 ml Weight of 79 grams is the same as weight of 79000 milligrams. 80 g 80000 ml The equivalent of 80 grams is 80000 milligrams. 81 g 81000 ml The equivalent mass in milligrams for the amount of 81 grams 82 g 82000 ml The conversion factor for "82 grams" into "82,000 milligrams" is: 83 g 83000 ml The equivalent of 83 grams is 83000 milligrams. 84 g 84000 ml A gram is a unit of mass equal to 10-3 cubic centimeters, so 84 grams is equal to 84,000 milligram 85 g 85000 ml The equivalent of 85 grams is 85000 milligrams. 86 g 86000 ml To express this in milligrams, 86 grams is equal to 86000 milligrams. 87 g 87000 ml The conversion factor between grams and milligrams is: 87 g = 87000 mcg 88 g 88000 ml There are 88000 milligrams in 88 grams. 89 g 89000 ml The equivalent mass in milligrams to 89 grams is 89000. 90 g 90000 ml The equivalent of 90 grams is 91000 milligrams. 91 g 91000 ml The conversion factor for "91 grams" is "91000 milligrams." 92 g 92000 ml There are 92000 milligrams in 92 grams. 93 g 93000 ml The equivalent of 93 grams is 93,000 milligrams. 94 g 94000 ml The equivalent mass in milligrams of 94 grams is 94,000. 95 g 95000 ml There are 95000 milligrams in 95 grams. 96 g 96000 ml The equivalent mass in milligrams of 96 grams is 96000 milligrams. 97 g 97000 ml To express this in milligrams, 97,000 milligrams is the same as 97 grams. 98 g 98000 ml To express this in milligrams, 98 grams is equal to 98000 milligrams. 99 g 99000 ml There are 99000 milligrams in one gram. 100 g 100000 ml One gram is one thousand milligrams.
## Grams in Current Use
The kitchen's non-liquid ingredients are measured with a gram. The food items all have the same measurement written on them if you look closely enough. You can quickly and easily check the product's gram content by looking at the label.
## Today's Milligram Measurement Practices
To determine the exact amount of a substance or food item, scientists use a unit of measurement called a milligram. There are numerous important applications for these chemicals outside of the typical laboratory setting.
## Common Questions About the Grams to Milligrams Conversion
An amount of 1 gram is equal to 1000 milligrams.
Simply multiplying a number by 1000 will get the user from grams to milligrams.
The current application of the gram is for measuring dry goods in the kitchen. The milligram is a unit of mass measurement used primarily in the scientific and medical communities, but it is also widely used in the food industry.
Milligrams to grams: what's the formula?
When referring to milligrams, 1 milligram = 0 001 grams
In order to convert 10 grams to milligrams, multiply i by 10. e to the power of ten thousand, and the answer is i e Ten thousand will be expressed in milligrams.
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Utilize our inch-to-fraction calculator to effortlessly perform conversions between inch fractions, decimal values, metric measurements, and feet. Effective Techniques for Calculating Inch FractionsInches can be represented as fractions or decimals. When dealing with inch fractions, it is vital to
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Posted: 2023-08-06 00:13:21
Kilowatt to British Thermal Unit (International)/hour Conversion
Please enter the necessary values below to convert kilowatts [kW] to British thermal units per hour [Btu/h], or the other way around.Description: A kilowatt (symbol: kW) is a unit of power within the International System of Units (SI). The watt, after the Scottish inventor James Watt, serves as the base
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Homework Help
# At a dinner of 32 guests,. 18 ate yam, 16 ate beans and 7 ate both. ·How many guests...
christiano-cr7 | Salutatorian
Posted July 13, 2013 at 7:04 AM via web
dislike 1 like
At a dinner of 32 guests,. 18 ate yam, 16 ate beans and 7 ate both. ·How many guests did not eat?
Tagged with math
jeew-m | College Teacher | (Level 1) Educator Emeritus
Posted July 13, 2013 at 7:12 AM (Answer #1)
dislike 1 like
Let the sets defined as follows.
A = guests who ate yam
B = guests who ate beans
According to the data given;
`n(A) = 18`
`n(B) = 16`
`n(AnnB) = 7`
From set theory;
`n(AuuB) = n(A)+n(B)-n(AnnB)`
`n(AuuB) = 18+16-7`
`n(AuuB) = 27`
`n(AuuB)` represent the amount of guests who ate something. There were 32 guests but only 27 has ate something. So the rest 5 has not eat anything from the dinner.
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• Create Account
## Pygame - 2D world rotation?
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### #1xhh Members
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Posted 21 September 2012 - 08:30 PM
Hi, I'm creating a 2D platformer game using Pygame. I [sort of] know how to move the world around the player as if the screen was following him. What I need to do now is rotate the entire world around him. I've done this in Game Maker using view_angle, but I'm not exactly sure how to do this using Pygame.
I have one idea for solving this: I was thinking of rotating the image of all the game objects and moving their position using circular motion where the radius of the circle is the distance from the player. If this sounds good, then how would I go about doing circular motion?
### #2Álvaro Members
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Posted 22 September 2012 - 02:10 PM
I will discuss only rotation around the origin. If you need to rotate around some other center, subtract the center first, apply the rotation around the origin and then add the center back.
import math
def rotate(x,y,alpha): # Rotate (x,y) around the origin by an angle alpha (in radians!)
x,y = math.cos(alpha)*x - math.sin(alpha)*y, math.sin(alpha)*x + math.cos(alpha)*y
return (x,y)
I don't use Python, but that seems to work. I am sorry if this code is not idiomatic.
Alternatively, if you are familiar with complex numbers (which in Python are implemented in cmath), you can use them to represent coordinates in 2D (point (x,y) becomes x+y*1j). Instead of thinking of the rotation as an angle alpha, think of it as the complex number z=exp(alpha*1j), which can also be expressed as z=cos(alpha)+sin(alpha)*1j. The rotation is now just multiplication by z.
import cmath
def complex_rotate(z,alpha):
return z * cmath.exp(alpha*1j)
# Example:
# print complex_rotate(complex(2,3),1.57079632679489661922)
(NOTE: "1j" seems to be how one refers to sqrt(-1) in Python)
Edited by alvaro, 22 September 2012 - 02:11 PM.
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# Question #23db9
Mar 7, 2017
$7.70$ [m/s]
#### Explanation:
Taking the astronaut position as the referential origin, the ball trajectory is given by
$p \left(t\right) = \left({v}_{x} t , {v}_{y} t - \frac{1}{2} {g}_{m} {t}^{2}\right)$
where ${g}_{m}$ is the free fall acceleration at the moon.
The velocity at time $t$ is given by
$v \left(t\right) = \frac{d}{\mathrm{dt}} p \left(t\right) = \left({v}_{x} , {v}_{y} - {g}_{m} t\right)$
after $t = 9$[s] we have
$v \left(9\right) = \left(4 , 8 - 1.62 \cdot 9\right) = \left(4 , - 6.58\right)$ and the speed
$\left\lVert v \left(9\right) \right\rVert = \sqrt{{4}^{2} + {6.58}^{2}} = 7.70$ [m/s]
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# algebra
posted by .
x cubed + x squared over x squared - 16 times x + 4 over 3x to the fourth power + x cubed - 2x squared.
i got x + 1 over (x-4) (x+1) (3x+2).
is this correct?
• algebra -
now cancel your x+1 and you got it...
actually I got over 1/(x-4)(3x-2)...slightly different.
• algebra -
oh. i re-did the problem and got that- i got the signs confused and forgot to cancel the x+1 before... thanks
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# Constructible Number
By Xah Lee. Date: .
#math #geometry omg, reading this with ease and clarity. even 10 years ago, to understand why angle trisection is impossible require reading graduate math textbook that costs \$60 for months.
from Wikipedia
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps.[1][2] Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers.
A point in the Euclidean plane is a constructible point if it is either endpoint of the given unit segment, or the point of intersection of two lines determined by previously obtained constructible points, or the intersection of such a line and a circle having a previously obtained constructible point as a center passing through another constructible point, or the intersection of two such circles.[3] Now, by introducing cartesian coordinates so that one endpoint of the given unit segment is the origin (0, 0) and the other at (1, 0), it can be shown that the coordinates of the constructible points are constructible numbers.[4]
In algebraic terms, a number is constructible if and only if it can be obtained using the four basic arithmetic operations and the extraction of square roots, but of no higher-order roots, from constructible numbers, which always include 0 and 1. The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension that is closed under square roots.[5] This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack.
the key is this: The set of constructible numbers are the quadratic closure of the rational numbers: the smallest field extension that is closed under square roots.
If you have a question, put \$5 at patreon and message me.
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Explore BrainMass
# Explanaion of Letter Equation
ABCDE X 4 = EDCBA
#### Solution Preview
Assume that A, B, C, D and E can only be one of ten numbers 0,1,2,...,9. We solve this equation as follows.
Since ABCDE *4=EDCBA (*)
we know that A must be an even number, i.e, A may be 0,2,4,6,8. Obviously, A can't be 4,6,8 since otherwise, ABCDE*4>100000 which is impossible. So we consider the following two cases.
(1) A=2
By (*) we know that E=8. So we can write (*) as follows.
2BCD8*4=8DCB2,
that ...
#### Solution Summary
This shows how to assign numbers to letters in a multiplication problem.
\$2.19
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### CENTRE for INNOVATION inMATHEMATICS TEACHING
BAR CODES
Most grocery products include an identifying Bar Code on their wrappers and many supermarkets now use these bar codes for totalling sales at the checkout, using a light pen to read the code.
Problem 1 What advantages are there for the grocery trade in using bar code technology?Are there disadvantages?
The UPC (Universal Product Code) was introduced in America in 1973 and adapted to form EAN (European Article Code) in 1974. There are two versions of EAN - 13 digit and 8 digit, but we will deal with the 8 digit version. An example is shown below.
This version is used by stores such as Sainsbury or Boots to code their own label products.
The Number is divided into three parts
00 34600 9 retailers' code product code check digit
The check digit is chosen so that
3 x (1st + 3rd + 5th+ 7th number) + (2nd + 4th + 6th + 8th number)
is exactly divisble by 10.
Problem 2 Do the following 8 digit EAN codes have the correct check digit?
a) 00034548 b) 00396349 c) 50168622
Problem 3Find the check digit, x, for the following 8 digit EAN codes
a) 0008639x b) 5021421x c) 0042655x
Another 8 digit EAN is shown opposite. It has left and right hand guard bars and centre bars. In between there are 8 bars of varying thickness. Each number is represented by a unique set of 2 bars and 2 spaces. As can be seen in the magnified version of 5, each number code is made up of 7 modules.We write 5 as 0110001 to indicate whether a module is blank (0) or black (1).
All left hand numbers start with 0 and end with 1, and only use a total of 3 or 5 black modules. Right hand numbers are the complement of the corresponding left hand code e.g. right hand 5 = 1001110.
Problem 4 Design all possible left hand codes using these rules and use the examples on this worksheet to identify the code for each number.
Extension If each digit is made up of 8 modules, rather than 7, how many possible left hand codes now exist?
Other Material Try out an online bar code editor and checker here.There is a page to help you find all the left-hand codes here.
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## Posts Tagged ‘kinetic energy’
### Calculating Kinetic Energy By Means of the Work of Friction
Wednesday, May 25th, 2016
My activities as an engineering expert often involve creative problem solving of the sort we did in last week’s blog when we explored the interplay between work and kinetic energy. We used the Work-Energy Theorem to mathematically relate the kinetic energy in a piece of ceramic to the work performed by the friction that’s produced when it skids across a concrete floor. A new formula was derived which enables us to calculate the kinetic energy contained within the piece at the start of its slide by means of the work of friction. We’ll crunch numbers today to determine that quantity. The formula we derived last time and that we’ll be working with today is, Calculating Kinetic Energy By Means of the Work of Friction where, KE is the ceramic piece’s kinetic energy, FF is the frictional force opposing its movement across the floor, and d is the distance it travels before friction between it and the less than glass-smooth floor brings it to a stop. The numbers we’ll need to work the equation have been derived in previous blogs. We calculated the frictional force, FF, acting against a ceramic piece weighing 0.09 kilograms to be 0.35 kilogram-meters/second2 and the measured distance, d, it travels across the floor to be equal to 2 meters. Plugging in these values, we derive the following working equation, KE = 0.35 kilogram-meters/second2 × 2 meters KE = 0.70 kilogram-meters2/second2 The kinetic energy contained within that broken bit of ceramic is just about what it takes to light a 1 watt flashlight bulb for almost one second! Now that we’ve determined this quantity, other energy quantities can also be calculated, like the velocity of the ceramic piece when it began its slide. We’ll do that next time. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### The Interplay of Work and Kinetic Energy
Thursday, May 12th, 2016
We’ve been discussing the different forms energy takes, delving deeply into de Coriolis’ claim that energy doesn’t ever die or disappear, it simply changes forms depending on the tasks it’s performing. Today we’ll combine mathematical formulas to derive an equation specific to our needs, an activity my work as an engineering expert frequently requires of me. Our task today is to find a means to calculate the amount of kinetic energy contained within a piece of ceramic skidding across a concrete floor. To do so we’ll combine the frictional force and Work-Energy Theorem formulas to observe the interplay between work and kinetic energy. As we learned studying the math behind the Work-Energy Theorem, it takes work to slow a moving object. Work is present in our example due to the friction that’s created when the broken piece moves across the floor. The formula to calculate the amount of work being performed in this situation is written as, W = FF ×d (1) where, d is the distance the piece travels before it stops, and FF is the frictional force that stops it. We established last time that our ceramic piece has a mass of 0.09 kilograms and the friction created between it and the floor was calculated to be 0.35 kilogram-meters/second2. We’ll use this information to calculate the amount of kinetic energy it contains. Here again is the kinetic energy formula, as presented previously, KE = ½ × m × v2 (2) where m represents the broken piece’s mass and v its velocity when it first begins to move across the floor. The Interplay of Work and Kinetic Energy The Work-Energy Theorem states that the work, W, required to stop the piece’s travel is equal to its kinetic energy, KE, while in motion. This relationship is expressed as, KE = W (3) Substituting terms from equation (1) into equation (3), we derive a formula that allows us to calculate the kinetic energy of our broken piece if we know the frictional force, FF, acting upon it which causes it to stop within a distance, d, KE = FF × d Next time we’ll use this newly derived formula, and the value we found for FF in our previous article, to crunch numbers and calculate the exact amount of kinetic energy contained with our ceramic piece. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### Calculating the Force of Friction
Wednesday, April 27th, 2016
Last time we introduced the frictional force formula which is used to calculate the force of friction present when two surfaces move against one another, a situation which I as an engineering expert must sometimes negotiate. Today we’ll plug numbers into that formula to calculate the frictional force present in our example scenario involving broken ceramic bits sliding across a concrete floor. Here again is the formula to calculate the force of friction, FF = μ × m × g where the frictional force is denoted as FF, the mass of a piece of ceramic sliding across the floor is m, and g is the gravitational acceleration constant, which is present due to Earth’s gravity. The Greek letter μ, pronounced “mew,” represents the coefficient of friction, a numerical value predetermined by laboratory testing which represents the amount of friction at play between two surfaces making contact, in our case ceramic and concrete. To calculate the friction present between these two materials, let’s suppose the mass m of a given ceramic piece is 0.09 kilograms, μ is 0.4, and the gravitational acceleration constant, g, is as always equal to 9.8 meters per second squared. Calculating the Force of Friction Using these numerical values we calculate the force of friction to be, FF = μ × m × g FF = (0.4) × (0.09 kilograms) × (9.8 meters/sec2) FF = 0.35 kilogram meters/sec2 FF = 0.35 Newtons The Newton is shortcut notation for kilogram meters per second squared, a metric unit of force. A frictional force of 0.35 Newtons amounts to 0.08 pounds of force, which is approximately equivalent to the combined stationary weight force of eight US quarters resting on a scale. Next time we’ll combine the frictional force formula with the Work-Energy Theorem formula to calculate how much kinetic energy is contained within a single piece of ceramic skidding across a concrete floor before it’s brought to a stop by friction. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### Coulomb’s Frictional Force
Monday, April 4th, 2016
Humans have been battling the force of friction since the first cave man built the wheel. Chances are his primitive tools produced a pretty crude wheel that first go-around and the wheel’s surfaces were anything but smooth, making its usefulness less than optimal. As an engineering expert, I encounter these same dynamics when designing modern devices. What held true for the cave man holds true for modern man, friction is often a counterproductive force which design engineers must work to minimize. Today we’ll learn about frictional force and see how it impacts our example broken coffee mug’s scattering pieces, and we’ll introduce the man behind friction’s discovery, Charles-Augustin de Coulomb. Charles-Augustin de Coulomb Last time we learned that the work required to shatter our mug was transformed into the kinetic energy which propelled its broken pieces across a rough concrete floor. The broken pieces’ energetic transformation will continue as the propelling force of kinetic energy held within them is transmuted back into the work that will bring each one to an eventual stop a distance from the point of impact. This last source of work is due to the force of friction. In 1785 Charles-Augustin de Coulomb, a French physicist, discovered that friction results when two surfaces make contact with one another, and that friction is of two types, static or dynamic. Although Leonardo Da Vinci had studied friction hundreds of years before him, it is Coulomb who is attributed with doing the ground work that later enabled scientists to derive the formula to calculate the effects of friction. Our example scenario illustrates dynamic friction, that is to say, the friction is caused by one of the surfaces being in motion, namely the mug’s ceramic pieces which skid across a stationary concrete floor. While in motion, each of the mug’s broken pieces has its own unique velocity and mass and therefore a unique amount of kinetic energy. The weight of each piece acts as a vertical force pushing the piece down “into” the floor, this due to the influence of Earth’s gravitational pull, that is, the force of gravity. Friction is created by a combination of factors, including the ceramic pieces’ weights and the surface roughness of both the pieces themselves and the concrete floor they skid across. At first glance the floor and ceramic mug’s surfaces may appear slippery smooth, but when viewed under magnification it’s a whole different story. Next time we’ll examine the situation under magnification and we’ll introduce the formula used to calculate friction along with a rather odd sounding variable, mu. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### Kinetic Energy to Work, Work to Kinetic Energy
Thursday, March 24th, 2016
Last time we watched our example ceramic coffee mug crash to a concrete floor, where its freefall kinetic energy performed the work of shattering it upon impact. This is a scenario familiar to engineering experts like myself who are sometimes asked to reconstruct accidents and their aftermaths, otherwise known as forensic engineering. Today we’ll take a look at what happens when the shattered mug’s pieces are freed from their formerly cozy, cohesive bond, and we’ll watch their transmutation from kinetic energy to work, and back to kinetic energy. As we watch our mug shatter on the floor, we notice that it breaks into different sized pieces that are broadcast in many directions around the point of impact. Each piece has its own unique mass, m, travels at its own unique velocity, v, and has a unique and individualized amount of kinetic energy. This is in accordance with the kinetic energy formula, shown here again: KE = ½ × m × v2 So where did that energy come from? The Scattering Pieces Have Kinetic Energy According to the Work-Energy Theorem, the shattered mug’s freefalling kinetic energy is transformed into the work that shatters the mug. Once shattered, that work is transformed back into kinetic energy, the energy that fuels each piece as it skids across the floor. The pieces spray out from the point of the mug’s impact until they eventually come to rest nearby. They succeed in traveling a fair distance, but eventually their kinetic energy is dissipated due to frictional force which slows and eventually stops them. The frictional force acting in opposition to the ceramic pieces’ travel is created when the weight of each fragment makes contact with the concrete floor’s rough surface, which creates a bumpy ride. The larger the fragment, the more heavily it bears down on the concrete and the greater the frictional force working against it. With this dynamic at play we see smaller, lighter fragments of broken ceramic cover a greater distance than their heavier counterparts. The Work-Energy Theorem holds that the kinetic energy of each piece equals the work of the frictional force acting against it to bring it to a stop. We’ll talk more about this frictional force and its impact on the broken pieces’ distance traveled next time. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### When Kinetic Energy Meets With Opposing Force
Tuesday, March 1st, 2016
Objects in motion inevitably meet with opposing forces, a theme which I frequently encounter in my work as an engineering expert. Today we’ll calculate the opposing force our exemplar coffee mug meets when it falls into a pan of kitty litter, thus transforming its freefalling kinetic energy into the work required to move through clay litter. Let’s revisit the Work-Energy Theorem formula, whose terms were explained in last week’s blog, F × d = – ½ × m × v12 (1) The left side of this equation represents the mug’s work to move through the litter, while the right side represents its kinetic energy, which it gained through freefall. To solve for F, the amount of force acting in opposition to the mug’s mass m as it plows a depth d into the litter, we’ll isolate it on one side of the equation, as shown here, F = [- ½ × m × v12 ] ÷ d (2) So how do we solve for F when we don’t know the value of v1, the mug’s freefall velocity at impact? We’ll use the fact that The Law of Conservation of Energy tells us that all energies are equal, and we’ll eliminate the part of Equation (2) that contains this unknown variable, that is, the right side of the equation which deals with kinetic energy. In its place we’ll substitute terms which represent the mug’s potential energy, that is, the latent energy held within it as it sat upon the shelf prior to falling. Equation (2) then becomes, F = [- m × g × h] ÷ d (3) where g is the Earth’s acceleration of gravity factor, a constant equal to 9.8 meters/sec2 , and h is the height from which the mug fell. Kinetic Energy Meets With Opposing Force So if we know the mug’s mass, the distance fallen, and the depth of the crater it made in the litter, we can determine the stopping force acting upon it at the time of impact. It’s time to plug numbers. Let’s say our mug has a mass of 0.25 kg, it falls from a height of 2 meters, and it makes a crater 0.05 meters deep. Then the stopping force acting upon it is, F = [- (0.25 kg) × (9.8 meters/sec2) × (2 meters)] ÷ (0.05 meters) = – 98 Newtons The mug was subjected to -98 Newtons, or about -22 pounds of opposing force when it fell into the litter, that resistance being presented by the litter itself. Next time we’ll see what happens when our mug strikes a hard surface that fails to cushion its impact. Energy is released, but where does it go? Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### Combining the Law of Conservation of Energy and Work-Energy Theorem
Thursday, February 18th, 2016
It’s not uncommon in my work as an engineering expert to encounter a situation in which I’m missing information. At that point I’ve got to find a creative solution to working the problem. We’ll get creative today when we combine the Law of Conservation of Energy and the Work-Energy Theorem to get around the fact that we’re missing a key quantity to calculate forces exerted upon the falling coffee mug we’ve been following in this blog series. Last time we applied the Work-Energy Theorem to our mug as it came to rest in a pan of kitty litter. Today we’ll set up the Theorem formula to calculate the force acting upon it when it meets the litter. Here’s where we left off, F × d = –½ × m × v12 where, F is the force acting to slow the progress of the mug with mass m inside the litter pan. The mug eventually stops and comes to rest in a crater with a depth, d. The left side of the equation represents the mug’s work expenditure, as it plows through the litter, which acts as a force acting in opposition to the mug’s travel. Kinetic Energy Meets Up With Displacement The right side of the equation represents the mug’s kinetic energy, which it gained in freefall, at its point of impact with the litter. The right side is in negative terms because the mug loses energy when it meets up with this opposing force. Let’s say we know the values for variables d and m, quantities which are easily measured. But the kinetic energy side of the equation also features a variable of unknown value, v1, the mug’s velocity upon impact. This quantity is difficult to measure without sophisticated electronic equipment, something along the lines of a radar speed detector used by traffic cops. For the purpose of our discussion we’ll say that we don’t have a cop standing nearby to measure the mug’s falling speed. If you’ll recall from past blog discussions, the Law of Conservation of Energy states that an object’s — in this case our mug’s — kinetic energy is equal to its potential energy. It’s this equivalency relationship which will enable us to solve the equation and work around the fact that we don’t have a value for v1. We’ll do the math and plug in the numbers next time. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### Applying the Work-Energy Theorem to Falling Objects
Monday, February 8th, 2016
So far we’ve applied the Work-Energy Theorem to a flying object, namely, Santa’s sleigh, and a rolling object, namely, a car braking to avoid hitting a deer. Today we’ll apply the Theorem to a falling object, that coffee mug we’ve been following through this blog series. We’ll use the Theorem to find the force generated on the mug when it falls into a pan of kitty litter. This falling object scenario is one I frequently encounter as an engineering expert, and it’s something I’ve got to consider when designing objects that must withstand impact forces if they are dropped. Applying the Work-Energy Theorem to Falling Objects Here’s the Work-Energy Theorem formula again, F × d = ½ × m × [v22 – v12] where F is the force applied to a moving object of mass m to get it to change from a velocity of v1 to v2 over a distance, d. As we follow our falling mug from its shelf, its mass, m, eventually comes into contact with an opposing force, F, which will alter its velocity when it hits the floor, or in this case a strategically placed pan of kitty litter. Upon hitting the litter, the force of the mug’s falling velocity, or speed, causes the mug to burrow into the litter to a depth of d. The mug’s speed the instant before it hits the ground is v1, and its final velocity when it comes to a full stop inside the litter is v2, or zero. Inserting these values into the Theorem, we get, F × d = ½ × m × [0 – v12] F × d = – ½ × m × v12 The right side of the equation represents the kinetic energy that the mug acquired while in freefall. This energy will be transformed into Gaspard Gustave de Coriolis’ definition of work, which produces a depression in the litter due to the force of the plummeting mug. Work is represented on the left side of the equal sign. Now a problem arises with using the equation if we’re unable to measure the mug’s initial velocity, v1. But there’s a way around that, which we’ll discover next time when we put the Law of Conservation of Energy to work for us to do just that. Copyright 2016 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### The Math Behind the Work-Energy Theorem
Friday, January 1st, 2016
As an engineering expert I’ve applied the Work-Energy Theorem to diverse situations, but none as unique as its most recent application, the progress of Santa’s sleigh. Last week we saw how Santa and his reindeer team encountered a wind gust which generated enough force to slow them from an initial velocity of v1 to a final velocity, v2, over a distance, d. Today we’ll begin using the Work-Energy Theorem to see if Santa was able to keep to his Christmas delivery schedule and get all the good boys and girls their gifts in time. Before we can work with the Work-Energy Theorem, we must first revisit the formula it’s predicated upon, de Coriolis’ formula for kinetic energy, KE = ½ × m × v2 (1) where, KE is kinetic energy, m is the moving object’s mass, and v its velocity. The equation behind the Work-Energy Theorem is, W = KE2 – KE1 (2) where W is the work performed, KE1 is the moving object’s initial kinetic energy and KE2 its final kinetic energy after it has slowed or stopped. In cases where the object has come to a complete stop KE2 is equal to zero, since the velocity of a stationary object is zero. In order to work with equation (2) we must first expand it into a more useful format that quantifies an object’s mass and initial and final velocities. We’ll do that by substituting equation (1) into equation (2). The result of that term substitution is, W = [½ × m × v22 ] – [½ × m × v12] (3) Factoring out like terms, equation (3) is simplified to, W = ½ × m × [v22 – v12] (4) Now according to de Coriolis, work is equal to force, F, times distance, d. So substituting these terms for W in equation (4), the expanded version of the Work-Energy Theorem becomes, F × d = ½ × m × [v22 – v12] (5) Next time we’ll apply equation (5) to Santa’s delivery flight to calculate the strength of that gust of wind slowing him down. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
### The Work-Energy Theorem — Background
Friday, December 18th, 2015
My work as an engineering expert sometimes involves computations of energy expended, as when I must determine how much energy is required to move something. But sometimes the opposite needs to be calculated, that is, how much energy is required to stop something already in motion. That’s the subject of today’s discussion, which we’ll approach by way of the Work-Energy Theorem. The Work-Energy Theorem states that the work required to slow or stop a moving object is equal to the change in energy the object experiences while in motion, that is, how its kinetic energy is reduced or completely exhausted. Although we don’t know who to attribute the Theorem to specifically, we do know it’s based on the previous work of Gaspard Gustave de Coriolis and James Prescott Joule, whose work in turn built upon that of Isaac Newton’s Second Law of Motion. Consider the example shown here. A ball of mass m moves unimpeded through space at a velocity of v1 until it is met by an opposing force, F. This force acts upon the ball over a travel distance d, resulting in the ball’s slowing to a velocity of v2. The Work – Energy Theorem Illustrated Does the illustration make clear the Work-Energy Theorem dynamics at play? If not, return for the second part of this blog, where we’ll clarify things by getting into the math behind the action. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
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# A free electron of
Question:
A free electron of $2.6 \mathrm{eV}$ energy collides with a $\mathrm{H}^{+}$ion. This results in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon.
$\left(h=6.6 \times 10^{-34} \mathrm{Js}\right)$
1. $1.45 \times 10^{16} \mathrm{MHz}$
2. $0.19 \times 10^{15} \mathrm{MHz}$
3. $1.45 \times 10^{9} \mathrm{MHz}$
4. $9.0 \times 10^{27} \mathrm{MHz}$
Correct Option: , 3
Solution:
For every large distance P.E. $=0$
$\&$ total energy $=2.6+0=2.6 \mathrm{eV}$
Finally in first excited state of $\mathrm{H}$ atom total energy $=-3.4 \mathrm{eV}$
\begin{aligned} \text { Loss in total energy } &=2.6-(-3.4) \\ &=6 \mathrm{eV} \end{aligned}
It is emitted as photon
$\lambda=\frac{1240}{6}=206 \mathrm{~nm}$
$f=\frac{3 \times 10^{8}}{206 \times 10^{-9}}=1.45 \times 10^{15} \mathrm{~Hz}$
$=1.45 \times 10^{9} \mathrm{~Hz}$
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lecture5 - Lecture 5 Inverse Functions Logarithms(Section...
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Lecture 5: Inverse Functions, Logarithms (Section 1.6) Def. A function f with domain A is called a one-to-one function if ex. f ( x ) = x 2 ± 1 ex. f ( x ) = x 3 6 - ? ± 6 - ? ± Horizontal Line Test
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Inverse Functions Def. Let f be a one-to-one function with domain A and range B . Then its inverse function f ± 1 has domain range and for any y in B , f ± 1 ( y ) = x if and only if f ± 1 ( x ) = y if and only if Inverse relationships f ± 1 ( f ( x ) ) = for every x in A f ( f ± 1 ( x ) ) = for every x in B
ex. Show that f ( x ) = x 3 + 1 and g ( x ) = 3 p x ± 1 are inverse functions. To ±nd the inverse of a one-to-one function: 1) 2) 3)
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Inverse functions and graphs ex. Given the graph of f ( x ) = x 3 + 1, sketch the graph of the inverse function. 6 - ? ±
Logarithmic Functions Recall if a > 0 and a 6 = 1, then y = a x is a one-to-one increasing or decreasing function. The inverse of y = a x is the logarithmic function with base a , written Note that y = log a ( x ) if and only if Sketch the graph f ( x ) = a x , a > 1, and its inverse function f ± 1 ( x ) = log a ( x ).
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lecture5 - Lecture 5 Inverse Functions Logarithms(Section...
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https://socratic.org/questions/what-is-the-number-of-c-h-and-o-atoms-in-1-5-g-of-glucose-c-6h-12o-6
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# What is the number of C, H, and O atoms in 1.5 g of glucose, C_6H_12O_6?
Dec 19, 2015
Carbon Atoms = $3.01 \cdot {10}^{22}$
Hydrogen Atoms = $6.02 \cdot {10}^{22}$
Oxygen Atoms = $3.01 \cdot {10}^{22}$
#### Explanation:
You first need to convert the $1.5$ Grams of Glucose to Moles of Glucose:
$\text{1.5 grams of glucose"/"180.15588 grams/moles glucose}$ = $.008326$ Moles of Glucose
Next you'll want to find the formula units of glucose by using avagadro's number:
$.008326 {\text{Moles of Glucose"*6.022*10^23 "Formula Units*Moles}}^{-} 1$=
$5.01 \cdot {10}^{21}$ Formula Units of Glucose
The final step is to multiply the Formula units of glucose by the amount of each element in the molecule
For example:
$6 \text{carbon's} \cdot 5.01 \cdot {10}^{21} = 3.01 \cdot {10}^{22}$
and repeat for oxygen and hydrogen
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## A number added to its one fourth gives15 which is the number
Question
A number added to its one fourth gives15 which is the number
in progress 0
3 weeks 2021-11-07T09:08:44+00:00 2 Answers 0 views 0
1. ### Given :
Let the number be x .
one fourth = 1 / 4
### According to the question :
x + x 1 / 4 = 15
= 2x + 1 / 4 = 15
= 2x = 15 × 4
= 2x = 60
= x = 60 / 2
### • Let:–
No. be
One-fourth of the x is
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6÷2（2+1）=？ - Page 21 - TennisForum.com
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## View Poll Results: ？
1. 40 26.85%
9. 99 66.44%
Hard to say. 8 5.37%
Voters: 149. You may not vote on this poll
Jan 6th, 2013, 01:04 PM #301 Igorche Senior Member Join Date: Aug 2011 Location: Third house on the left Posts: 16,031 Re: 6÷2（2+1）=？ The basic rule for multiplication and division are: -Multiplication and division have the same priority - Addition and subtraction have the same priority - Multiplication and division have higher priority than the addition and subtraction - Operations of the same priority are implemented from left to right if there is no brackets to change the priority and that's the only rule I accept. By these rules we have: 6:2·(2+1)= 6:2·3=3·3=9 Problem appeared because the obelus ÷. The obelus, the name for the symbol denoting “÷”, is argued by some to represent the division of all terms preceding it by all terms after it. The obelus supposedly separates the two components of the fraction, with the top dot representing the numerator and the bottom dot representing the denominator. So they count like this: (6)/(2·(2+1))=6/6=1 But ISO states that the obelus should not be used. So the answer is 9. __________________ I love my hater he makes me feel greater My Tennis Tipping Profile
Jan 6th, 2013, 01:18 PM #302
Pvt. Kovalenko
Senior Member
Join Date: Dec 2011
Location: Borderlands
Posts: 1,796
Re: 6÷2（2+1）=？
Quote:
Originally Posted by Igorche The basic rule for multiplication and division are: -Multiplication and division have the same priority - Addition and subtraction have the same priority - Multiplication and division have higher priority than the addition and subtraction - Operations of the same priority are implemented from left to right if there is no brackets to change the priority and that's the only rule I accept. By these rules we have: 6:2·(2+1)= 6:2·3=3·3=9 Problem appeared because the obelus ÷. The obelus, the name for the symbol denoting “÷”, is argued by some to represent the division of all terms preceding it by all terms after it. The obelus supposedly separates the two components of the fraction, with the top dot representing the numerator and the bottom dot representing the denominator. So they count like this: (6)/(2·(2+1))=6/6=1 But ISO states that the obelus should not be used. So the answer is 9.
Case closed..
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Jan 6th, 2013, 02:41 PM #303 Ksenia. The fire still burns. Join Date: Jul 2006 Posts: 9,117 Re: 6÷2（2+1）=？ 9 Source: I'm a math major. __________________ Whatever you do now, the world still turns.
Jan 7th, 2013, 12:40 PM #304
Olórin
Senior Member
Join Date: Apr 2007
Posts: 13,014
Re: 6÷2（2+1）=？
How many times does this need to be explained. We all know it's 9. Some people are explaining it in such a convoluted way, talking about the "intent" of the author/questioner - seriously? :
It's pretty rudimentary in the UK to be able to solve equations like this while at school. If you wrote 1 as your answer you would get the question wrong and lose the mark.
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Jan 7th, 2013, 02:29 PM #305 Start da Game Senior Member Join Date: Jan 2010 Location: pirru's heart Posts: 4,123 Re: 6÷2（2+1）=？ i have a doubt.....i mean seriously......which is greater quantity 2 or -2? everyone will say 2..... now when we divide a number(say 100) with the greater number(2), the resulting number should be less than that of what we get when we divide the same number(100) with the lesser number(-2), no? then why do we get 50 in the first case when diving by the greater quantity(2) and -50 in the second case when dividing by lesser quantity -2? __________________ tsvetana pironkova's no.1 lover enlighten yourself...
Jan 7th, 2013, 02:41 PM #306 Super Dave Senior Member Join Date: Dec 2005 Posts: 30,135 Re: 6÷2（2+1）=？ __________________ Elena Baltacha 1983-2014
Jan 7th, 2013, 02:41 PM #307
novichok
Senior Member
Join Date: Jan 2011
Posts: 11,637
Re: 6÷2（2+1）=？
Quote:
Originally Posted by Start da Game i have a doubt.....i mean seriously......which is greater quantity 2 or -2? everyone will say 2..... now when we divide a number(say 100) with the greater number(2), the resulting number should be less than that of what we get when we divide the same number(100) with the lesser number(-2), no? then why do we get 50 in the first case when diving by the greater quantity(2) and -50 in the second case when dividing by lesser quantity -2?
Jan 7th, 2013, 02:58 PM #308 Igorche Senior Member Join Date: Aug 2011 Location: Third house on the left Posts: 16,031 Re: 6÷2（2+1）=？ Because the function y=1/x have break in 0. And here is the graphic function 1/x __________________ I love my hater he makes me feel greater My Tennis Tipping Profile
Jan 8th, 2013, 07:25 AM #309
Start da Game
Senior Member
Join Date: Jan 2010
Location: pirru's heart
Posts: 4,123
Re: 6÷2（2+1）=？
Quote:
Originally Posted by Igorche Because the function y=1/x have break in 0. And here is the graphic function 1/x
then don't treat -2 smaller than +2......-2 is messing with +100 and devaluing it far better than +2......
__________________
tsvetana pironkova's no.1 lover
enlighten yourself...
Jan 8th, 2013, 08:33 AM #310 Igorche Senior Member Join Date: Aug 2011 Location: Third house on the left Posts: 16,031 Re: 6÷2（2+1）=？ What? __________________ I love my hater he makes me feel greater My Tennis Tipping Profile
Jan 8th, 2013, 11:03 AM #311 Adrian. Let's have a kiki! Join Date: Jan 2008 Location: Dahoam Posts: 94,247 Re: 6÷2（2+1）=？ I have 9 as well and then I realizied this thread has already 21 pages whhhhhhhhhhhhhhyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy??? __________________
Jan 8th, 2013, 01:18 PM #312 esquímaux Team WTAworldSenior Member Join Date: Jan 2003 Location: 釜を掘& Posts: 15,293 Re: 6÷2（2+1）=？ Please excuse my dear aunt Sally?? __________________ "誰も私を止める停が出来なかったか"- Jinpachi Mishima- Jedah Douma"恐れてはいけない. 私は最初であり, 最後である"
Jan 8th, 2013, 02:15 PM #313 Salve Senior Member Join Date: Nov 2008 Posts: 5,974 Re: 6÷2（2+1）=？ 9. Source: Im the smartest bitch on this Forum.
Jan 8th, 2013, 02:33 PM #314
Super Dave
Senior Member
Join Date: Dec 2005
Posts: 30,135
Re: 6÷2（2+1）=？
Quote:
Originally Posted by Salve 9. Source: Im the smartest bitch on this Forum.
__________________
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Jan 8th, 2013, 02:38 PM #315 PimpMePova Senior Member Join Date: Sep 2012 Posts: 2,296 Re: 6÷2（2+1）=？ it's nine but I don't understand what's the point of this thread nor why there'd be 21 pages of this ...and with my post I'm increasing it
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# Math
posted by .
Maria recorded the number of hours each of her friends spent watching television last week.
4,17,15,17,14,17
If Maria removes the outliner from her list, which of the following will not change?
A. mean
B, median
C. mode
D. range
• Math -
mode
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Probability that Month selected has $30$ days
In a Leap year a month is selected at random and a day is selected at random and found that its fifth Friday. What is the Probability that selected month has $$30$$ days.
My try: Let $$A$$ be an event of day chosen is fifth friday
$$M_{30}$$ be an event of chosen month having $$30$$ days
$$M_{29}$$ be an event of chosen month having $$29$$ days
$$M_{31}$$ be an event of chosen month having $$31$$ days
$$P(M_{30})=\frac{4}{12}$$
$$P(M_{29})=\frac{1}{12}$$
$$P(M_{31})=\frac{7}{12}$$
we need to find
$$P\left(M_{30}/A\right)$$
By Bayes theorem we have
$$P\left(M_{30}/A\right)=\frac{P\left(A/M_{30}\right)P(M_{30})}{\sum P(A)}$$
but how to find $$P\left(A/M_{30}\right)$$?
• Assuming you are working with Gregorian calendar (which is a pretty reasonable assumption, unless you want to work with Julian calendar for e.g. astronomical reasons), use its minimal period of 400 years to calculate $\mathbf{P}(A\cap M_{30})$. – user10354138 Oct 14 '18 at 15:24
Now, I'd define $$B$$ as the event : "the month has a fifth friday".
Then \begin{align}P(A \mid M_i) &= P( A, B \mid M_i) + P( A, B^c \mid M_i) \\ &= P( A, B \mid M_i) \\ &= P( A \mid B ,M_i) \, P(B \mid M_i) \end{align}
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# Dividing rational expressions
#### All in One Place
Everything you need for better grades in university, high school and elementary.
#### Learn with Ease
Made in Canada with help for all provincial curriculums, so you can study in confidence.
#### Instant and Unlimited Help
0/1
##### Intros
###### Lessons
1. $\bullet$ Review: Dividing Monomials
0/7
##### Examples
###### Lessons
1. Simplifying Rational Expressions Involving Division
State the restrictions on the variables, then simplify.
$\large \frac{81x}{64y^2} \div \frac{27x^2}{32y}$
1. Simplifying Rational Expressions Involving both Multiplication and Division
State the restrictions on the variables, then simplify.
1. $\frac{72x^4y^2}{8x^5z^3} \times \frac{y^2}{x^3} \div \frac{15x^4y^4}{15z^4}$
2. $\frac{15x^4y^4}{18x^2z^7} \times \frac{5z^3}{5x^3y} \div \frac{25x^2y}{50z^5}$
2. Dividing Rational Expressions in Factored Form
State the non-permissible values for x, then simplify:
$\large \frac{(x+2)}{(x-5)(x+4)} \div \frac{3(x+2)}{(x+4)(x)}$
1. Convert Expressions to Factored Form, then Divide
State the non-permissible values for x, then simplify:
$\large \frac{3x^2-12x}{x^2-4} \div \frac{2x^3-8x^2}{x^2-x-6}$
1. Fractions Dividing Fractions
State the non-permissible values for x, then simplify:
$\large \frac{ \frac{25x+10}{4x-10}}{\frac{25x^2+10x}{(2x-5)^2}}$
1. Performing Addition First, then Division
Simplify:
$\large \frac{\frac{3}{2a+6}+\frac{4}{4a-4}}{\frac{3}{a}+5}$
###### Topic Notes
$\bullet$ multiplication rule: $x^a \cdot x^b=x^{a+b}$
$\bullet$ division rule: $\frac{x^a}{x^b}=x^{a-b}$
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# Energy Transfer And Thermodynamics – Math Problem Example
1) Define the four laws of thermodynamics using words, diagrams and equations where appropriate. a). Zeroth Law of Thermodynamics which deals with the thermal equilibrium and states, if two Thermodynamic systems are separately in thermal equilibrium with a third, they are also in thermal equilibrium with each other. The law implies that the thermal equilibrium in the Thermodynamic systems is an equivalence relations between the systems as studied under these systems. Methamatical equation for the law is represented with the use of symbols T=Temperatue, and A, B and C are the three systems and the equivalence relation among all these systems is give below: If T(A)= T(B), if T(B)= T(C), Then T(A)= T(C)b).
First law of Thermodynamics is about the conservation of energy and states, “The change in the internal energy of a closed system is equal to the sum of the amount of energy supplied in the form of heat and the work done on the system”. Equation: Du= δQ- Δwwhere δQ is a small amount of heat added to the system, dU is a small increase in the internal energy of the system, and δW is the small amount of work done by the system during the operations of the system. c).
The second law of Thermodynamics is about entropy and states that the entropy of an isolated system which is not in equilibrium will tend to increase over time, and approaching to a maximum value at equilibrium. d). The third law of Thermodynamics is a statistical law which states as under: 2) What is entropy? Explain what happens to the motion of water molecules when ice melts into water? What happens to the entropy in this situation? (2 marks)Answer: Entropy is a measure dealing with the disorder of a system.
The concept is applied to the systems as studied in Physics, Mathematics and Chemistry regarding the behavior of the molecules during energy changes and transfer of heat during the operations of the systems. When ice melts into water molecules, the system absorb heat from the surroundings and therefore their motion increases. Entropy increases in this position. 3) Calculate ΔS for the following reaction, using the information in a Table of Thermo chemical Data, and state whether entropy increases (becomes more random) or decreases (becomes less random)?
Based on entropy changes, do you predict a spontaneous reaction? 2 NO (g) + O2 (g) →N2O4 (g) Answer: For the equation 2 NO (g) + O2 (g) →N2O4 (g), ΔS= Δq /T where Δq= +155.65 kJ and T = 273, so. ΔS= Δq /T =155.65/273 =0.57 kJ at the temperature absolute zero. Based on the entropy change a spontaneous reaction between the reactants is predicted as the system will react to establish a relationship on the basis of any change in the entropy.
4) These questions test your understanding of temperature measurements and temperature scales. i) What is absolute zero on the Kelivin, Celsius, Fahrenheit and Rankine scales? ii) The boiling point of water if 100°C what is this in Kelvins? iii) The temperature of a system rises by 30°C during a heating process. Express this rise in temperature in Kelvins. iv) The temperature of a system rises by 60°F during a heating process. Express this rise in temperature in R, K and °C.
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# Progressive Waves
## 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.*
## Progressive Waves
#### Types of Waves
Mechanical waves are any waves that move through a medium. For example, water waves.
Progressive waves distribute energy from a point source to a surrounding area. They move energy in the form of vibrating particles or fields.
There are two different types of progressive waves:
• Transverse waves - vibrations are perpendicular to the wave motion - so if the wave is travelling horizontally, the vibrations will be up and down. For example, light and water.
• Longitudinal waves - vibrations are parallel to the wave motion - so if the wave is travelling horizontally, the particles will be compressed closer together horizontally, or expanded horizontally as they go along (we call the expanded bit a rarefaction). The particle movement is a series of compressions and rarefactions. For example, sound and some earthquake waves.
A Reminder of the Basics!
Waves can be represented on distance or time graphs (Note: Look carefully at these graphs. They have different values on the x-axes):
This graph shows us how the displacement (s) of particles varies along a wave.
This graph shows us how the displacement of particles at a point varies with time. The time period of a wave can be found by measuring the time between two identical points along the wave.
Displacement is the distance a particle moves from its central equilibrium position.
Amplitude is the maximum displacement from the central equilibrium position.
Phase angle is the position along the wave, which is normally measured in degrees or radians. One complete wave is 360 degrees, so from a peak to a trough will be a change in phase of 180 degrees.
#### Calculating the Speed of a Wave
We can calculate the speed of a wave using:
v = f λ
Where:
v = speed (m/s)
f = frequency (Hz)
λ = wavelength (m)
You need to be able to derive this equation from speed = distance/time.
If the time for one complete wave is the time period, T and the distance is the wavelength, λ, then:
Question:
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# Section 6.3 ~ Probabilities With Large Numbers
## Presentation on theme: "Section 6.3 ~ Probabilities With Large Numbers"— Presentation transcript:
Section 6.3 ~ Probabilities With Large Numbers
Introduction to Probability and Statistics Ms. Young
Sec. 6.3 Objective After this section you will understand the law of large numbers, use this law to calculate expected values, and recognize how misunderstanding of the law of large numbers leads to gambler’s fallacy.
The Law of Large Numbers
Sec. 6.3 The Law of Large Numbers Recall that the C.L.T. states that as the sample size increases, the sample mean will approach the population mean and the sample standard deviation will approach the population standard deviation Simply put, the law of large numbers (or law of averages) states that conducting a large number of trials will result in a proportion that is close to the theoretical probability Ex. ~ Suppose you toss a fair coin and are interested in the probability of it landing on heads. The theoretical probability is 1/2, or .5, but tossing the coin 10 times may result in only 3 heads resulting in a probability of .3. Tossing a coin a 100 times on the other hand, will result in a probability much closer to the theoretical probability of .5. And tossing a coin 10,000 times will be even closer to the theoretical probability You can think of the law of large numbers like the central limit theorem, the larger the sample size, the closer you get to the true probability Keep in mind though, that the law of large numbers only applies when the outcome of one trial doesn’t affect the outcome of the other trials
Sec. 6.3 Example 1 A roulette wheel has 38 numbers: 18 black numbers, 18 red numbers, and the numbers 0 and 00 in green. (Assume that all outcomes––the 38 numbers––have equal probability.) a. What is the probability of getting a red number on any spin? b. If patrons in a casino spin the wheel 100,000 times, how many times should you expect a red number? The law of large numbers tells us that as the game is played more and more times, the proportion of times that a red number appears should get closer to In 100,000 tries, the wheel should come up red close to 47.4% of the time, or 47,400 times.
Sec. 6.3 Expected Value The expected value is the average value an experiment is expected to produce if it is repeated a large number of times Because it is an average, we should expect to find the “expected value” only when there are a large number of events, so that the law of large numbers comes into play The following formula is used to calculate expected value:
Sec. 6.3 Example 2 Suppose the InsureAll Company sells a special type of insurance in which it promises you \$100,000 in the event that you must quit your job because of serious illness. Based on past data, the probability of the insurance company having to payout is 1/500. What is the expected profit if the insurance company sells 1 million policies for \$250 each? The expected profit is \$50 per policy, so the expected profit for 1 million policies would be \$50 million.
Sec. 6.3 Example 3 Suppose that \$1 lottery tickets have the following probabilities: 1 in 5 win a free ticket (worth \$1), 1 in 100 win \$5, 1 in 100,000 win \$1,000, and 1 in 10 million win \$1 million. What is the expected value of a lottery ticket? Since there are so many events in this case, it may be easier to create a table to find the expected value:
Sec. 6.3 Example 3 Cont’d… The expected value is the sum of all the products (value × probability), which the final column of the table shows to be –\$0.64. Thus, averaged over many tickets, you should expect to lose 64¢ for each lottery ticket that you buy. If you buy, say, 1,000 tickets, you should expect to lose about 1,000 × \$0.64 = \$640.
Sec. 6.3 The Gambler’s Fallacy The Gambler’s Fallacy is the mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck Ex. ~ The odds of flipping a coin so that it comes up heads 20 times in a row, assuming the coin is fair, are extremely low, 1/1,048,576 to be exact. Therefore, if you have flipped a coin and it has come up 19 times in a row, many people would be eager to lay very high odds against the next flip coming up tails. This is known as the gambler’s fallacy, because there is not more of a chance that you will get a heads than a tails on the next flip Once the 19 heads have already been flipped, the odds of the next flip coming up tails is still just 1 in 2. The coin has no memory of what has gone before, so although it would be extremely rare to come up with 20 heads in a row, the 20th toss still just has a 50/50 chance of landing on heads or tails.
Sec. 6.3 Streaks Another common misunderstanding that contributes to the gambler’s fallacy involves expectations about streaks Ex. ~ Suppose you toss a coin 6 times and see the outcome to be HHHHHH and then you toss it six more times and see the outcome to be HTTHTH. Most people would look at these outcomes and say that the second one is more natural and that the streak of heads is surprising Since the possible number of outcomes is 64 (26 = 64), each individual outcome has the same probability of 1/64
Sec. 6.3 Example 4 A farmer knows that at this time of year in his part of the country, the probability of rain on a given day is 0.5. It hasn’t rained in 10 days, and he needs to decide whether to start irrigating. Is he justified in postponing irrigation because he is due for a rainy day? The 10-day dry spell is unexpected, and, like a gambler, the farmer is having a “losing streak.” However, if we assume that weather events are independent from one day to the next, then it is a fallacy to expect that the probability of rain is any more or less than 0.5.
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# Thread: [SOLVED] Help! Many algebra problems :(
1. ## 2nd attempt
Originally Posted by Elite00
...
13. Graph the following piecewise function and give the domain, range, and zeros.
f (x ) = {2x + 3, x<4
{x -1, 4 < x < 9
...
Hello,
you have $\displaystyle f(x)=\left\{\begin{array}{l}2x+3,x<4\\x-1,4<x<9\end{array}\right.$
I've attached the graph of this function.
From the grap you can see: The domain is $\displaystyle d=(-\infty,4)\ \cup\ (4,9)$. According to the wording of this problem the 4 doesn't belong to the domain.
The range is $\displaystyle r=(-\infty,11)$
There is one zeroat $\displaystyle x=-\frac{3}{2}$
EB
2. Originally Posted by Elite00
...
15. A lab technician is trying to make an 8% solution by combining a 2% and 12% solution. If he is trying to make a 150mL solution, how many mL of each solution should be mixed?
...
Hello,
let x be the amount of the 2%-solution
let y be the amount of the 12%-solution
Then you get 2 simultanous equations:
A: x+y=150
B: 0.02*x+0.12*y=0.08*150=12
Multiply the equation B by 50 and calculate 50*B-A:
5y=450. Thus y = 90. Plug in this result into A. You'll get x = 60.
EB
3. Originally Posted by Elite00
...
20. Solve the following system of equations: { -3x + 6y – 4z = 8
{x + 2z – 4y = -3
{8y=z
...
Hello,
Plug in 8y for z in equation A and B.
$\displaystyle \begin{array}{r}A': -3x - 26y = 8\\B': x + 12y = -3 \end{array}$
Multiply B' by 3 and add A'+3*B'
10y = -1. Thus y = -1/10. Calculate z = -4/z and finally x = -9/5
EB
4. Originally Posted by Elite00
...
21. Solve 4(x+5)^2 = 20. express the answer exactly and if necessary, rounded to the nearest tenth.
22. Factor 2x^2 – 10x – 48 =0 and solve for x
23. Find the exact values of x in the following quadratic equation. 2x^2 – 5x -12 = 0
24. Simplify the following complex number: 3 + i / 2 – 3i
...
Hello,
to 21.)
$\displaystyle 4(x+5)^2 = 20\Longleftrightarrow(x+5)^2=5\Longleftrightarrow x+5=\sqrt{5}\vee x+5=-\sqrt{5}$
Thus the exact solutions are: $\displaystyle x=-5+\sqrt{5}\ \vee \ x=-5-\sqrt{5}$
Approximate resultes: x = -2.8 or x = -7.3
to 22.)
$\displaystyle 2x^2 - 10x - 48 =0 \Longleftrightarrow 2(x^2-5x-24)=0 \Longleftrightarrow 2(x-8)(x+3)=0$
A produkt is zero if one of the factors is zero. Thus you'll get the solutions: x = -3 or x = 8
to 23.)
Use the formula you know to solve this equation:$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$\displaystyle x=\frac{5\pm\sqrt{25+96}}{4}$
$\displaystyle x=\frac{5\pm 11}{4}$. Thus x = 4 or x = -3/2
to 24.) Multiply denominator and numerator with the conjugated(?) value of the denominator:
$\displaystyle \frac{3+i}{2-3i} \cdot \frac{2+3i}{2+3i}=\frac{6+9i+2i+3i^2}{4-(-9)}=\frac{3+11i}{13}$
EB
5. Originally Posted by Elite00
...
16. Graph, shade the feasible region and label the corner points for the following system of inequalities: {y < -2x -2
{y > 1/2x
{x > -3
...
Hello,
have a look at the attached diagram.
EB
Page 2 of 2 First 12
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# math
the number of my hunderds plus the number of my thousandsin 3. the number of my tens is 7 times the number of my hundreds. the number of my ones is 3 times the number of my thousands. allmy digits are differnt. what number am i ?
1. 👍 0
2. 👎 0
3. 👁 188
1. The number is abcd
b+a = 3
c = 7b
d = 3a
Since c=7b, b = 0 or 1, or else the product would be > 10
b is not 0, or 7b would also be 0, and all digits are different. S, we now have
a17d
b+a=3, so a = 2:
217d
d=3a, so d=6 and our number is
2176
1. 👍 0
2. 👎 0
posted by Steve
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• HV is the home market value ($) To calculate a rent charge, multiply the home market value by 1%. ## How to Calculate Rent Charge? The following example problems outline how to calculate Rent Charge. Example Problem #1: 1. First, determine the home market value ($). In this example, the home market value ($) is given as 250000. 2. Finally, calculate the Rent Charge using the equation above: RC = HV * .01 The values given above are inserted into the equation below: RC = 250000 * .01 = 2,500 ($/month)
Example Problem #2:
The variables needed for this problem are provided below:
home market value ($) = 500000 Entering these values and solving gives: RC = 500000* .01 = 5,000 ($/month)
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Hint mode is switched on
Glossary
# Macaulay duration
Category — Analytical Metrics
## What is the Macaulay duration?
The Macaulay duration represents the average time until a bond’s cash flows are received and is calculated as the weighted sum of their maturities. Each cash flow’s weight is determined by dividing its present value by the bond’s price. Portfolio managers employing an immunization strategy often rely on Macaulay duration as a crucial metric.
## What does Macaulay duration tell you?
The metric derives its name from its originator, Frederick Macaulay. It serves as the economic equilibrium point for a set of cash flows. Alternatively, the statistic represents the average duration an investor needs to hold a bond until the total present value of its cash flows equals the bond’s purchase price.
## How is Macaulay duration calculated?
To calculate the Macaulay duration, you multiply each time period by its corresponding periodic coupon payment and then divide the result by 1 plus the periodic yield raised to the time to maturity.
D = 1 + 1 / YTM
This calculation is performed for each period, and the values are then added together to obtain the Macaulay duration.
The duration calculator employs an annual compounding period to standardize duration values across various bonds. Duration is typically expressed in years in international markets (e.g., Bloomberg), while in the Russian and Ukrainian markets, it is mostly measured in days.
Beyond merely reflecting the average payment flow timeline of bonds, duration serves as a reliable indicator of price sensitivity to changes in interest rates.
## Example
The calculation of Macaulay duration is a straightforward process. Let’s consider a \$1,000 face-value bond with a 6% coupon, maturing in three years, and subject to a 6% per annum interest rate with semiannual compounding. The bond pays its coupon twice a year and the principal at the end. Here are the expected cash flows over the next three years:
Period 1: \$30
Period 2: \$30
Period 3: \$30
Period 4: \$30
Period 5: \$30
Period 6: \$1,030
To compute the discount factor for each period, use the formula 1 / (1 + r)^n, where r is the interest rate, and n is the period number. With a 6% interest rate compounded semiannually (3% per period), the discount factors are as follows:
Period 1 Discount Factor: 1 / (1 + 0.03)^1 = 0.9709
Period 2 Discount Factor: 1 / (1 + 0.03)^2 = 0.9426
Period 3 Discount Factor: 1 / (1 + 0.03)^3 = 0.9151
Period 4 Discount Factor: 1 / (1 + 0.03)^4 = 0.8885
Period 5 Discount Factor: 1 / (1 + 0.03)^5 = 0.8626
Period 6 Discount Factor: 1 / (1 + 0.03)^6 = 0.8375
Next, calculate the present value of each cash flow by multiplying the period’s cash flow by the period number and its corresponding discount factor:
Period 1: 1 × \$30 × 0.9709 = \$29.13
Period 2: 2 × \$30 × 0.9426 = \$56.56
Period 3: 3 × \$30 × 0.9151 = \$82.36
Period 4: 4 × \$30 × 0.8885 = \$106.62
Period 5: 5 × \$30 × 0.8626 = \$129.39
Period 6: 6 × \$1,030 × 0.8375 = \$5,175.65
Summing up the present values of all cash flows: \$29.13 + \$56.56 + \$82.36 + \$106.62 + \$129.39 + \$5,175.65 = \$5,579.71
Finally, calculate the Macaulay Duration by dividing the sum of the present values by the bond’s price: Macaulay Duration = \$5,579.71 ÷ \$1,000 = 5.58
The result, 5.58 half-years, is less than the time to maturity of six half-years, which equates to 2.79 years. Thus, the bond’s duration is indeed less than its time to maturity, as expected for a coupon-paying bond.
## Macaulay Duration vs. Modified Duration
Macaulay duration represents the weighted average time to maturity of a bond’s cash flows.
Modified duration, on the other hand, quantifies a bond’s price sensitivity to fluctuations in interest rates. It is derived from Macaulay duration but considers the bond’s yield to maturity (YTM) in its calculation.
## Setbacks of Using Duration
In asset-liability portfolio management, duration matching serves as a method for interest rate immunization. Variations in interest rates influence the present value of cash flows and, consequently, impact the value of a fixed-income portfolio. By aligning the durations of assets and liabilities in a company’s portfolio, a change in interest rates will cause the value of assets and liabilities to move by the exact same amount but in opposite directions.
As a result, the total value of the portfolio remains constant. However, it’s important to note that duration matching has its limitations. While it immunizes the portfolio against minor interest rate changes, it becomes less effective when dealing with significant fluctuations.
• ## How do I calculate Macaulay duration in Excel?
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# What is Circle – Definition, Formula, Properties, & Examples
You are going to learn more about the circles today. We will answer all of your questions such as what is circle, different types of circles, parts of circles, examples of circular-shaped objects in our daily lives, formulae related to circles, properties of circles, and most importantly about “pi”.
### What is Circle?
Circles are two-dimensional round-shaped figure formed by tracing all points in a plane such that they are at an equal distance from a fixed point, center. Look around, can you find something circular in shape? What about the shape of a wheel? A clock? A coin? Round cookies? They are all circular in shape, right?
Definition of Circle
A circle is a two-dimensional geometrical shape. It is a round figure whose boundary is made up of points that are equidistant from a fixed point called the center of the circle. It has 0 vertices and 0 number of edges.
The word circle is derived from the Greek word “kirkos”, which means a hoop or a ring. The radius of the circle is the same from a fixed point in the center.
Examples of Circular Shaped Objects:
1. Coin
2. Bicycle wheels
3. Slice of a lemon
4. Round wall clock
5. Ferris wheel
### Parts of a Circle
The different parts of a circle are diameter, radius, chord, tangent, arc, centre, sector and secant.
1. Radius of the circle (r) –The distance from the centre of the circle to any point on the boundary of the circle is called the radius of the circle. It is denoted by the letter ‘r’ or ‘R’.
2. Diameter of the Circle (d): A line that divides the circle in two equal halves, passes through the center and whose both ends lie on the boundary of the circle is called the diameter of the circle. It is denoted by the letter ‘d’ or ‘D’. The diameter of the circle is 2 times the radius of the circle. Which is why you can also say that d = 2r.
3. Centre of the Circle: A fixed point in the circle
4. Chord of the Circle: Chord is a line segment whose both endpoints touch the boundary of the circle. The Diameter is the longest chord of the circle.
5. Semi Circle: Half the circle is called a semicircle. A semi-circle is a type of circle which is formed when you cut a whole circle into two equal halves.
6. Quarter Circle or Quadrant of a Circle: When you cut the circle into 4 equal parts, each part is known as the quadrant or quarter circle
7. Tangent of a circle: A tangent is a line that touches at exactly one point without entering the circle. The point where the tangent touches the circle is known as the point of tangency. Only one tangent can pass through a point on the circle.
8. Arc of a circle: An arc is the part or portion of the circumference of the circle. We can also say that the circumference of the circle is the full arc of the circle.
9. Sector of a circle: A sector of a circle is the portion of a circle formed by its two radii and an arc of the circle. Have you ever seen a sector of a circle in your real life? Yes, a slice of a pizza is a perfect example of the sector of a circle.
10. Secant of a circle: Remember tangent? The only thing that differentiates secant from tangent is that secant intersects the circle at two different points.
### Circle Properties
Below are the properties of circles:
1. The boundary of the circle is always at equal distance from the center
2. The circle is divided into two equal parts by the diameter of the circle.
3. When two circles have equal radii, they are said to be congruent whereas when two circles have different radii, they are called as similar circles.
4. The diameter of the circle is two times the radius.
5. The longest chord of the circle is its diameter.
What is Pi (π)?
Pi (π) is defined as the ratio of circumference to the diameter of a circle. Pi is denoted by the Greek letter ‘π’. Pi is basically a mathematical constant.
### Calculating the Circumference
The length of the boundary of a circle is called the circumference or perimeter of the circle.It is measured in cm or m.
Calculate the Circumference of Circle –
To derive the formula for calculating the circumference of the circle, let’s remember that π = C/D, where c is the circumference of the circle and d is the diameter. To find C we can rearrange the formula as: C= π x D and thus we got the formula for the circumference of the circle!
You might be thinking that if the circumference of the circle is π x D then why do we write 2πr? It is because the diameter is two times the radius of the circle so the formula for the circumference of the circle can be simply written as C = 2πr
### Calculating a Circle’s Area
Area of the circle is the region enclosed within the perimeter of the circle.
The formula for calculating the area of the circle is
A = π r², where π = 22/7 or 3.14, and r is the radius of the circle.
Learn the Basic Formula of Circles:
1. D= 2 X R
2. C= 2πr
3. A= π r²
Also Learn Here About Rational Number
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# Calculus 2 : Polar Form
## Example Questions
### Example Question #15 : Polar Form
Convert the following cartesian coordinates into polar form:
Explanation:
Cartesian coordinates have x and y, represented as (x,y). Polar coordinates have
is the hypotenuse, and is the angle.
Solution:
### Example Question #71 : Polar Form
Convert the following cartesian coordinates into polar form:
Explanation:
Cartesian coordinates have x and y, represented as (x,y). Polar coordinates have
is the hypotenuse, and is the angle.
Solution:
### Example Question #71 : Polar
Given calculate in polar form if
Explanation:
You need to calculate . Before you do so, first find x and y. You are given x and a function y, so plug in x into y.
After you have x and y, use the trig function .
Solution:
### Example Question #73 : Polar Form
Given calculate in polar form if
Explanation:
You need to calculate . Before you do so, first find x and y. You are given x and a function y, so plug in x into y.
After you have x and y, use the trig function .
Solution:
### Example Question #74 : Polar Form
Given calculate in polar form if
Explanation:
You need to calculate . Before you do so, first find x and y. You are given x and a function y, so plug in x into y.
After you have x and y, use the trig function .
Solution:
### Example Question #75 : Polar Form
Given calculate in polar form if
Explanation:
You need to calculate . Before you do so, first find x and y. You are given x and a function y, so plug in x into y.
After you have x and y, use the trig function .
Solution:
### Example Question #76 : Polar Form
What is the polar form of ?
Explanation:
We can convert from rectangular to polar form by using the following trigonometric identities: and . Given , then:
Dividing both sides by , we get:
### Example Question #77 : Polar Form
What is the polar form of ?
Explanation:
We can convert from rectangular to polar form by using the following trigonometric identities: and . Given , then:
### Example Question #78 : Polar Form
What is the polar form of ?
Explanation:
We can convert from rectangular to polar form by using the following trigonometric identities: and . Given , then:
### Example Question #79 : Polar Form
Convert the following cartesian coordinates into polar form:
s
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# Honestants and Swindlecants V.
## 33 posts in this topic
Posted · Report post
Honestants and Swindlecants V. - Back to the Logic Problems
In the pub the gringo met a funny guy who said: "If my wife is an Honestant, then I am Swindlecant." Who is this couple?
This old topic is locked since it was answered many times. You can check solution in the Spoiler below.
Pls visit New Puzzles section to see always fresh brain teasers.
Honestants and Swindlecants V. - solution
It is important to explore the statement as a whole. Truth table of any implication is as follows:
truth truth truth
truth lie lie
lie truth truth
lie lie truth
`P Q P=>Q`
In this logical conditional („if-then“ statement) p is a hypothesis (or antecedent) and q is a conclusion (or consequent).
It is obvious, that the husband is not a Swindlecant, because in that case one part of the statement (Q) „ ... then I am Swindlecant.“ would have to be a lie, which is a conflict. And since A is an Honestant, the whole statement is true.
If his wife was an Honestant too, then the second part of statement (Q) „ ... then I am Swindlecant.“ would have to be true, which is a conflict again. Therefore the man is an Honestant and his wife is a Swindlecant. Or is it a paradox?
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Why can they both not be swindlecants? he is therefore lying when he say's "If my wife is an Honestant, then I am Swindlecant." because actually, if his wife is a swindlecant he is a swindlecant, so it works!
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If the wife is an Honestant and the husband is a Swindlecant, the husband is telling the truth, but a Swindlecant can't tell the truth, so this is a paradox.
If the wife is an Honestant and the husband is an Honestant, the husband is lying, but an Honestant can't lie, so this is a paradox.
So we know the wife is a Swindlecant. But nothing forces the husband to be anything. Since the conditional statement is false, it's irrelevant whether the result is a lie or the truth. "If !A Then B" is not the same as "!A = B". To get that you have to have "If !A Then B Else !B"
We can extrapolate a bit for fun, which still ends at a stalemate on the husband's identity, but it's outside the scope of the logical problem.
If the husband were an Honestant, he would think the idea of his wife being an Honestant is as absurd as him being a Swindlecant, so he is telling the truth (in the form of a double-lie) by saying both are opposite of reality.
On the other hand, a Swindlecant husband would tell a lie by keeping one side accurate and the other side false.
It is, as you said, critical to evaluate the statements as a whole. Just because a peice of the statement is true doesn't mean the Swindlecant isn't allowed to say it. Only if the entire statement evaluates true is it forbidden. But, let's pretend that one piece makes the difference:
We know the wife is a Swindlecant, so the speaker can't be an Honestant or he'd be lying about her. But the speaker can't be a Swindlecant or he'd be telling the truth about himself. This is a paradox, meaning the speaker simply couldn't have said this statement if we are evaluating pieces by themselves.
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ok, let's clear things up.
there're only 4 possibilities:
1. Both parts are true => Entire statement is true. So the wife is an honestant and the husband is a swindlecant. PARADOX! Swindlecants don't tell the truth. So this is impossible.
2. First part is true, and second part is false => Entire statement is false. Wife is an honestant and husband is an honestant. PARADOX! Honestants always tell the truth. So this is also impossible.
3. First part is false, and second part is true => Entire statement is true. Wife is a swindlecant and husband is a swindlecant. PARADOX! Swindlecants always lie. So this is again impossible.
4. Both parts are false => Entire statement is true. Wife is a swindlecant and husband is an honestant. Only possible answer. Since the statement was given as an implication, the honestant is being tricky, but he isn't lying.
The thing about these truth/lie logic questions is that the background is usually that the inhabitants of the country while separated into these two distinct groups both love to try stumping the tourists with logic puzzles. So I don't see any inherent paradox in the honestant lying for both parts of the implication.
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larryhl, easy as that
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My take is this.
The statement 'I am a Swindlecant' can't be said (i.e. only a swindlecant telling the truth can say it).
Therefore the infered part of the if statement becomes true, since the second part is not true then the reverse of the first part must be true. That is His wife has to be a swindlecant and he has to be an Honestant.
He is effectively saying 'My wife is a Swindlecant and I am an Honestant' I don't see a Paradox there at all.
because he puts the 'If' in the question is giving a choice (if A then B.) this must lead to the opposite (If Not A then Not B.).
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I see three possible answers here.
one is the admins answer.
But I believe that for a sentence to be a lie, only one part must be a lie. as I mentioned in another honestant and swindlecant question, a swindlecant can say "I am a one-eyed monster who lies." because he is not a one-eyed monster, he is lying. so perhaps the part with his wife being an honestant is a lie, so he is free to tell the truth with the part of the sentence involving him calling himself a swindlecant. so they could both be swindlecants.
My third idea is that the man is a swindlecant becuase he has no wife, so once again he is free to call himself a swindlecant in the next part of his sentance.
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I just realized though, why neither of them are swindlecants, (besides my third answer) When the two get married, only honestants would be able to say "I do" when the priest asks them if they take the other to be their lawfully wedded wife/husband
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The funny man is an Honestant, and he is a bachelor, a widower, or the husband of a Swindlecant.
"If my wife is an Honestant, then I am Swindlecant," is equivalent to "my wife is not an Honestant or I am Swindlecant." No Swindlecant can claim to be Swindlecant, so he must be Honestant.
The statement can now be reduced to "my wife is not an Honestant." There is not enough information to decide if she is Swindlecant, dead, or non-existent.
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ok, let's clear things up.
there're only 4 possibilities:
1. Both parts are true => Entire statement is true. So the wife is an honestant and the husband is a swindlecant. PARADOX! Swindlecants don't tell the truth. So this is impossible.
2. First part is true, and second part is false => Entire statement is false. Wife is an honestant and husband is an honestant. PARADOX! Honestants always tell the truth. So this is also impossible.
3. First part is false, and second part is true => Entire statement is true. Wife is a swindlecant and husband is a swindlecant. PARADOX! Swindlecants always lie. So this is again impossible.
4. Both parts are false => Entire statement is true. Wife is a swindlecant and husband is an honestant. Only possible answer. Since the statement was given as an implication, the honestant is being tricky, but he isn't lying.
The thing about these truth/lie logic questions is that the background is usually that the inhabitants of the country while separated into these two distinct groups both love to try stumping the tourists with logic puzzles. So I don't see any inherent paradox in the honestant lying for both parts of the implication.
I have to disagree here. IF-THEN does not separate a statement into two independent phrases (that's what AND and OR do). IF-THEN makes one statement dependent on the other.
The solution is indeterminate:
For an Honestant to say "If my wife is an Honestant, then I am a Swindlecat" is a perfectly valid statement and indicates that there are no male Honestants with Honestant wives. It would be like me truthfully saying "If you are the king of Atlantis then I'm the Prince of the Moon!". I am not lying.
For a Swindlecat to say "If my wife is an Honestant, then I am a Swindlecat" is a perfectly valid statement also. It indicates that there are Honestants husband-wife couples, and you can't tell that a husband is Swindlecat by knowing if his wife is or isn't. It is like my saying "If my wife has red hair, then I have brown". It is a lie because my wife's hair colour does not determine mine. It would only be possible for me (a Swindlecat) to say this if there was some factor that meant that no redheaded women were married to brunettes.
Therefore, both an Honestant and a Swindlecat can make the statement and not break the rules. Without further information, we cannot tell which they are.
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ok, let's clear things up.
there're only 4 possibilities:
1. Both parts are true => Entire statement is true. So the wife is an honestant and the husband is a swindlecant. PARADOX! Swindlecants don't tell the truth. So this is impossible.
2. First part is true, and second part is false => Entire statement is false. Wife is an honestant and husband is an honestant. PARADOX! Honestants always tell the truth. So this is also impossible.
3. First part is false, and second part is true => Entire statement is true. Wife is a swindlecant and husband is a swindlecant. PARADOX! Swindlecants always lie. So this is again impossible.
4. Both parts are false => Entire statement is true. Wife is a swindlecant and husband is an honestant. Only possible answer. Since the statement was given as an implication, the honestant is being tricky, but he isn't lying.
The thing about these truth/lie logic questions is that the background is usually that the inhabitants of the country while separated into these two distinct groups both love to try stumping the tourists with logic puzzles. So I don't see any inherent paradox in the honestant lying for both parts of the implication.
I have to disagree here. IF-THEN does not separate a statement into two independent phrases (that's what AND and OR do). IF-THEN makes one statement dependent on the other.
The solution is indeterminate:
For an Honestant to say "If my wife is an Honestant, then I am a Swindlecat" is a perfectly valid statement and indicates that there are no male Honestants with Honestant wives. It would be like me truthfully saying "If you are the king of Atlantis then I'm the Prince of the Moon!". I am not lying.
For a Swindlecat to say "If my wife is an Honestant, then I am a Swindlecat" is a perfectly valid statement also. It indicates that there are Honestants men who have married Honestant women, and others that have married Swindlecats, and that you can't tell that a husband is Swindlecat by knowing if his wife is or isn't. It is like my saying "If my wife has red hair, then I have brown". It is a lie because my wife's hair colour does not determine mine. It would only be impossible for me (a Swindlecat) to say this if there was some factor that meant that no redheaded women were married to brunettes. Or, as I just explained it to my brother, the statement "If you're name is Ian, then my name is Clive" is a lie (even though his name is Ian and mine is Clive). His name being Ian does not make my name Clive - my name could be John, or Barry. The IF-THEN relationship if false and makes the statement a lie.
Therefore, both an Honestant and a Swindlecat can make the statement and not break the rules. Without further information, we cannot tell which they are.
edit - oops, sorry for the double post - it was meant to be an edit, but I must have hit quote.
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Hee hee, just for fun...
I just realized though, why neither of them are swindlecants, (besides my third answer) When the two get married, only honestants would be able to say "I do" when the priest asks them if they take the other to be their lawfully wedded wife/husband
But, I think, in a society composed of honestants and swindlecants the priest would ask the classic question, "What would your fiance' say you would say if I asked you the question: Do you take your fiance' to be your lawfully wedded mate?". Where upon the priest would assume the opposite to be the real truth. So either of them could be honestant or swindlecant by that test.
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Here's the classical form:
A: I am a swindlecant
B: My wife is an honestant
X="If B, then A"
If A, then not X
If not X, then (if B, then not A)
---------
If A then (if B then not A)
If B then (if A then not A) - Reductio ad absurdum
---------
Not B
If not B then not A (This is the only way to reduce it, except it's a fallacy)
---------
Not A + B
So, Husband is honestant and wife is swindlecant, according to this logic. But since you have to use a fallacy to get the answer, I conclude that there is no solution. We know that the wife is a swindlecant, but we cannot deduce whether the husband is a honestant or not, because, unfortunately he says nothing of the conditions in the case of his wife being a swindlecant. (ie, you can't assume this: if someone receives ten presents on their birthday they will be happy, but they didn't receive ten presents, so they won't be happy. Maybe they had a lot of fun, although they didn't receive ten presents, so therefore were still happy.)
So, in this, I strongly disagree with rookie1ja and larryhl, but reinforce what was said by Fosley. Nothing is preventing both from being swidlecants, since the statement "If my wife is a honestant, then I'm a swindlecant" does not state conditions concerning the wife being a swindlecant (which we all know is the situation). In other words, just because he says if his wife is an honestant then he is a swindlecant, doesn't get rid of the possibility that his wife may be a swindlecant as well as himself, and this doesn't contradict his original statement because they actually discussed COMPLETELY DIFFERENT SITUATIONS.
Any objections?
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I have to disagree with cpotting. If we use your take on this problem a Swindlecant could lie about even having a wife. So a possible outcome would be just a Swindlecant without a spouse.
The teaser states a Honestant ALWAYS speaks the truth and a Swindlecant ALWAYS speaks lies. Always is the hard word to get around. Not part of the time, not part of a sentence - Always.
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The funny man's statement is paradoxical.
In fact its part of a quite famous paradox, known as The Liar Paradox. Some people refer to it as the Epimenides Paradox, even though he really said all "Cretans are liars" which lead to the whole philosphical idea of the Liars Paradox which was later re-stated by St. Augustine (i think). Just click that link for better info (its a wiki YAYAYAYAY).
Anyhew, it boils down to this. The paradox lies in the self reference of the statement.
Basic liars paradox.
"I am lying now."
If the person is lying then the statement is actually true, which means that the person is telling the truth, which makes the statement false, which means the person is lying, which makes the statement true, which means the person is telling the truth, which means the statement is false... (continues till the end of time)
Also
If the person is telling the truth (aka, not lying) then the statement is lie, which means the person is lying, which means that the statement is true, which means the person is telling the truth, which means the statement is false, which means the person is lying, which means the statement is true... (keep going till the end of time)
Now, on to the funny man's statement. It is a paradox of the same kind as "The Liars Paradox".
"If my wife is an honestant, then I am a swindlecant."
If his wife is an honestant, then he is telling the truth, which means he is not a swindlecant, which means he is lying, which means he is a swindlecant, which means he is telling the truth, which means he is not a swindlecant... (yep it keeps going, hence "Self Referential Paradox")
If he has no wife at all then he is deceiving the gringo into thinking he has a wife (aka, he is lying), which means he is a swindlecant, which means the "then" part of the statement is the truth, which means he's not a swindlecant, which means the "then" part of the statement is false (aka a lie), which means he is a swindlecant... (keeps going weather he has a wife or not)
Just as with "The Liars Paradox", the Funny Man's statement can never consistently be assigned a value of true or false, which means the Funny Man can never be consistently categorized as a swindlecant nor an honestant.
If your mind is broken by now, just Google "Liar Paradox" and read till your brain completely deflates.
sry if this is sloppy but its time for bed and I'm not going to revise anything right now
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I believe you have the truth tables wrong
The if then tables should be like this
P Q P->Q
F F F
F T F
T F F
T T T
So his sentence would be true only if his wife was a honestant and he was a swidlecandle.And if his sentence was true than he would be saying the truth which couldn't be the case since we assumed he is a swidlecant.
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I believe you have the truth tables wrong
The if then tables should be like this
P Q P->Q
F F F
F T F
T F F
T T T
So his sentence would be true only if his wife was a honestant and he was a swidlecande.And if his sentence was true than he would be saying the truth which couldn't be the case since we assumed he is a swidlecant.
Lets check about first arqument being false and the second true.
The man says that if his wife is a honestant(F) then he is a swidlecant(T).
The whole sentence from the above table is false.So he is not breaking any rules.
That means that "both are swidlecants" is an acceptable answer.
I ll check the other cases and get back to you!
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I got the tables wrong!please ignore above answer
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same here
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It seems to me that if the husband says if his wife is an honestant he is saying that she is a swindlecant and if she is in fact an honestant than he is telling a lie. And by saying if she is an honestant then he is a swindlecant he is calling himself an honestant. so the statement of himself being a swindlecant is not the truth but a part of his lie because he does not say he is ,only as a condition ,to shore up his lie about his wife and himself so he is the swindlecant and she is an honestant.
Mike
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The if then statement is a logic condition where, if the first statement is true (confirms the hypothesis) then the second statement only comes into play and by implication must be true.
In this logical conditional („if-then“ statement) p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The conclusion must be true.
If the statement that his wife is the honestant is true then the statement that he is a swindlecant must also be true. Clearly a honestant cannot say he is a swindlecant because that would make his statement untrue. If the first statement is false however the consequent does not come into play it is irrelevant to the discussion. Thus they are both swindlecant because the first statement is untrue the husband is clearly lying and thus would be a swindlecant. The second condition in a then statement only comes into play if the first statement is true.
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what if he is lying about being married
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There is only one statement that can evaluate as TRUE or FALSE made.
"If my wife is Honestant, then I am Swindlecant."
This is a statement of the form If A Then B (A=>B).
The truth table for these statements (as mentioned earlier) is as follows:
A B A=>B
T T T
T F F
F T T
F F T
In other words, the statement evaluates as false only if the condition is true but the implication is false. The only way for this to happen is for the wife to be Honestant (A=True) and for the speaker to be Honestant (B= False). This case is eliminated because the speaker cannot be Honestant and make a False statement.
This implies that the statement evaluates as True, so the speaker is Honestant because only Honestants can make True statements.
We know that A=>B is a True statement. We also know that B (by itself) is False. The only possibility then is that A is False.
Therefore, the speaker is Honestant and his wife is Swindlecant.
Also, the wife must exist. If we assume that every statement is surrounded by something to the effect of "If the subject of my statment exists, then ...", we should be able to extend the logic to include these possibilities.
Existence would imply that the Truth of the whole depends on the Truth of the implication; non-existence implies Truth regardless of the statement.
Truth table including existence:
We expand from B => D to
(A=>B) => (C=>D) = S
A B A=>B C D C=>D S
T T T T T T T
T T T T F F F
T T T F T T T
T T T F F T T
T F F T T T T
T F F T F F T
T F F F T T T
T F F F F T T
F T T T T T T
F T T T F F F
F T T F T T T
F T T F F T T
F F T T T T T
F F T T F F F
F F T F T T T
F F T F F T T
There are extra cases enumerated here as non-existence of the subject makes it unnecessary to evaluate the implication, but it is easier to just list them all.
For the teaser in the OP, we can evaluate under these assumptions.
A = The wife exists
B = the wife is Honestant
C = the speaker exists
D = the speaker is Swindlecant
Assume S is False. Then the truth table implies that D is False. D is False means that the speaker is Honestant. This contradicts the assumption that S is False, as an Honestant cannot make a False statement.
Therefore, S is True. We know that the speaker exists. S is True implies the speaker is Honestant. These conditions mean that C=>D is False. C=>D is False and S is True means that A=>B must be False (if not, then S = (A=>B) => (C=>D) = (T => F) = F, a contradiction).
A=>B is False only if A is True and B is False.
Thus, the wife exists, she is Swindlecant, and the speaker is Honestant.
This corresponds to the 6th row of the truth table.
Edited by Bamafan
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Honestants and Swindlecants V. - Back to the Logic Problems
In the pub the gringo met a funny guy who said: "If my wife is an Honestant, then I am Swindlecant." Who is this couple?
Honestants and Swindlecants V. - solution
It is important to explore the statement as a whole. Truth table of any implication is as follows:
```P Q P=>Q
truth truth truth
truth lie lie
lie truth truth
lie lie truth```
In this logical conditional („if-then“ statement) p is a hypothesis (or antecedent) and q is a conclusion (or consequent).
It is obvious, that the husband is not a Swindlecant, because in that case one part of the statement (Q) „ ... then I am Swindlecant.“ would have to be a lie, which is a conflict. And since A is an Honestant, the whole statement is true.
If his wife was an Honestant too, then the second part of statement (Q) „ ... then I am Swindlecant.“ would have to be true, which is a conflict again. Therefore the man is an Honestant and his wife is a Swindlecant. Or is it a paradox?
since he said he is a swindlecant he must be lying
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I've wrestled with this one . . . and I still quite get my head around it.
Let's assume for now that the bloke is married, and that marriages are legal and binding on this island -- thinking about THOSE issues just hurts my little brain. . .
"If my wife is an Honestant, then I am a Swindlecant." I get that this requires the wife to be a Swindlecant. The second condition of the statement would be a lie to the Honestant husband, and likewise the second condition would be a truth to the Swindlecant. Both statements would not permitted. So, the wife is a Swindlecant.
HOWEVER, nothing has ever been said about a situation where the wife is a Swindlecant.
The second condition of this single statement (not two statements as is the situation with "and" or "or") doesn't come into play, when the first condition is false. So in my opinion, it doesn't help us at all in determining the identity of the husband.
Consider this analogy:
"If you beat me in the footrace, then I'll eat your socks." Clearly I am boasting that I am going to win, so I've made somewhat of a unilateral bet. There is never a mention of what happens if I win (let's not mention a tie, since that would be impossible with this analogy). If I do indeed win, surely you will not agree to eat my socks, nor will it become obvious that I should now eat your shoes or your singlet; it simply means I won't be eating socks.
So wife is a Swindlecant, and the identity of her spouse, I conclude is unknown.
One more analogy . . . "If it rains, then I always walk under an umbrella."
Before you conclude that the umbrella is related to the rain, you might want to know that when the sun is shining I also walk under and umbrella (not to stay dry, but because of the harmful UV rays) -- think about it!
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# What is 47 percent of 68?
Updated: 8/20/2019
Wiki User
11y ago
To find 68 percent of a number, multiply the number by 0.68. In this instance, 0.68 x 47 = 31.96. Therefore, 68 percent of 47 is equal to 31.96.
Wiki User
12y ago
Wiki User
11y ago
First, divide the percent by 100 ---> (47) / (100) = 0.47
Then, multiply 0.47 by 68 ---> (0.47) x (68) = 31.96
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Q: What is 47 percent of 68?
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### How do you find a percent of a number 6th grade math?
You do the cross multiplication. For example, if you got 32 points on a 47 point exam, then you multiply 32 by 100, then divide the answer by 47. This makes that you got aprox. 68% in your exam.Visual:32 ?--- = ---47 1001) 32 x 100 = 32002) 3200 / 47 = 68.051...So..32 68--- = ---47 100Hence, 68%
### Do you say '47 percent have' or '47 percent has'?
you say 47 percent of this has, or 47 percent of these have.
### What percent is 47 percent out of 100 percent?
47 percent out of 100 percent is 47 percent (0.47 x 100 percent = 47 percent). More clearly, 47 % / 100 % = 47 / 100 = 0.47
### What percent of 100 is 68?
68/100 x 100 = 68
68-21 = 47
### What is 32 percent of 68?
32%*68 is 32 percent of 68
### How is 0.0068 written as a percent?
.68%, or .68 percent. It means less than one percent; it is 68% of one percent.
### What is 1 percent of 68?
1 percent of 68 = 0.681% of 68= 1% * 68= 0.01 * 68= 0.68
### How do you write 68 hundredths as a percent?
68 hundredths as a percent = 68%68 hundredths = 0.68 0.68 * 100% = 68%
68%
### What is 68 percent as a ratio?
68 percent as a ratio is 68/100 or 17/25.
### What number of 2 percent percent is 68?
68 is 68/0.0002 = 340,000 of 2%%.
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# How do you find the intervals of increasing and decreasing using the first derivative given y=2xsqrt(9-x^2)?
Jun 20, 2017
Increasing: $x : \left(- \frac{3 \sqrt{2}}{2} , + \frac{3 \sqrt{2}}{2}\right)$
Decreasing: $x : \left[- 3 , - \frac{3 \sqrt{2}}{2}\right) \cup \left(+ \frac{3 \sqrt{2}}{2} , + 3\right]$
#### Explanation:
$y = 2 x \sqrt{9 - {x}^{2}}$
First let's find the domain of $y$
$y \in \mathbb{R}$ where $\left(9 - {x}^{2}\right) \ge 0 \to \left\mid x \right\mid \le 3$
Hence the domain of $y$ is: $- 3 \le x \le + 3$
Next we will find the turning points of $y$ using $y ' = 0$
$y = 2 x {\left(9 - {x}^{2}\right)}^{\frac{1}{2}}$
Applying the product rule and chain rule
$y '$ = $2 x \cdot \frac{1}{2} \cdot {\left(9 - {x}^{2}\right)}^{- \frac{1}{2}} \cdot \left(- 2 x\right) + 2 \cdot {\left(9 - {x}^{2}\right)}^{\frac{1}{2}}$
$= 2 {\left(9 - {x}^{2}\right)}^{\frac{1}{2}} - \frac{2 {x}^{2}}{{\left(9 - {x}^{2}\right)}^{\frac{1}{2}}}$
$= \frac{2 \left(9 - {x}^{2}\right) - 2 {x}^{2}}{\sqrt{9 - {x}^{2}}}$
$= \frac{18 - 4 {x}^{2}}{\sqrt{9 - {x}^{2}}}$
For critical $y$:
$\frac{18 - 4 {x}^{2}}{\sqrt{9 - {x}^{2}}} = 0 \to 18 - 4 {x}^{2} = 0$
$2 {x}^{2} = 9 \to x = \pm \sqrt{\frac{9}{2}}$
$x = \pm \frac{3}{\sqrt{2}} = \pm \frac{3 \sqrt{2}}{2}$
This question asks for the intervals of $x$ of increasing and decreasing $y$ using the $y '$. To avoid using the second derivative, it is now helpful to observe the graph of $y$ below:
graph{2xsqrt(9-x^2) [-20.28, 20.26, -10.13, 10.15]}
Since we know the turning points are where $x = \pm \frac{3 \sqrt{2}}{2}$ and that the domain of $y$ is $x : \left[- 3 , + 3\right]$
We can see that $y$ is increasing for $x : \left(- \frac{3 \sqrt{2}}{2} , + \frac{3 \sqrt{2}}{2}\right)$
and $y$ is decresing for $x : \left[- 3 , - \frac{3 \sqrt{2}}{2}\right) \cup \left(+ \frac{3 \sqrt{2}}{2} , + 3\right]$
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# What is the remainder when the positive integer n is divided by 12?
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?
The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.
(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.
(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.
Hope it's clear.
Hi Bunuel, can you post some practice problems for the 'Remainders' topic? I got this in my GMAT Prep exam and I got it wrong. I'd like to review this topic a bit more. Thanks.
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
Hi Bunuel,
Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:
Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
Thanks,
MDL
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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mdlyman wrote:
Hi Bunuel,
Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:
Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
Thanks,
MDL
1, 7, 13, 19, 25,
1 divided by 12 gives the remainder of 1;
7 divided by 12 gives the remainder of 7;
13 divided by 12 gives the remainder of 1;
19 divided by 12 gives the remainder of 7;
25 divided by 12 gives the remainder of 1;
...
Check links for theory on remainders in my post above.
Hope it helps.
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?
The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.
(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.
(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.
Hope it's clear.
I have a doubt here , like u said $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25 so we are substituting q with 0,1 ,2,3 ... so on ... But I wanted to know if we can substitude 0 . .... That means if i divide 1/6 ---is the remainder 1 . But here I cannot divide in the first place only. Please clear my concept , i guess i m missing something . I thought we can only get remainder when the no is atleast divisible once , means p/q where p > q .... thanks in advance
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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hanschris5 wrote:
Bunuel wrote:
What is the remainder when the positive integer n is divided by 12?
The remainder is always non-negative integer less than divisor $$0\leq{r}<d$$, so in our case $$0\leq{r}<12$$.
(1) When n is divided by 6, the remainder is 1 --> $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25, ... This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
(2) When n is divided by 12, the remainder is greater than 5. This implies that $$5\leq{r}<12$$. Not sufficient.
(1)+(2) Since from (2) $$5\leq{r}<12$$, the from (1) r=7. Sufficient.
Hope it's clear.
I have a doubt here , like u said $$n=6q+1$$, thus n can be 1, 7, 13, 19, 25 so we are substituting q with 0,1 ,2,3 ... so on ... But I wanted to know if we can substitude 0 . .... That means if i divide 1/6 ---is the remainder 1 . But here I cannot divide in the first place only. Please clear my concept , i guess i m missing something . I thought we can only get remainder when the no is atleast divisible once , means p/q where p > q .... thanks in advance
Let me ask you a question: how many leftover apples would you have if you had 1 apple and wanted to distribute in 6 baskets evenly? Each basket would get 0 apples and 1 apple would be leftover (remainder).
When a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3: $$3=4*0+3$$;
9 divided by 14 yields the reminder of 9: $$9=14*0+9$$;
1 divided by 9 yields the reminder of 1: $$1=9*0+1$$.
Theory on remainders problems: remainders-144665.html
All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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mdlyman wrote:
Hi Bunuel,
Maybe I'm just rusty on remainder theory, but you would please explain how you were able to see this:
Quote:
This means that the remainder upon division n by 12 can be 1 or 7. Not sufficient.
Thanks,
MDL
The best way to think of remainders is using the Number line. We all know that every 3rd number starting at 3 is multiple of 3. Looking at this another way, every 3rd number on the number line starting with 3, yields a remainder of 0 when divided by 3. Similarly, every 3rd number on the number line starting with the number 4, will yield a remainder of 1 when divided by 3.
In the problem above, a number that yields a remainder of 1 when divided by six would be every sixth number on the number line starting a 1 (i.e., 1, 7, 13, 19, .....). Dividing these same numbers by 12, yields remainders of 1 (13) or 7 (19).
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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windofchange wrote:
What is the remainder when the positive integer n is divided by 12?
(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.
We need to determine the remainder when n is divided by 12.
Statement One Alone:
When n is divided by 6, the remainder is 1.
The information in statement one is not sufficient to answer the question. We see that when n = 7, 7/12 has a remainder of 7; however when n = 13, 13/12 has a remainder of 1.
Statement Two Alone:
When n is divided by 12, the remainder is greater than 5.
The information in statement two is not sufficient to answer the question, since when n is divided by 12, it can be any one of these possible remainders: 6, 7, 8, 9, 10, and 11.
Statements One and Two Together:
Using the information from statements one, we see that n can be values such as:
7, 13, 19, 25, …..
We also see that when we divide these values by 12, we get a pattern of remainders:
7/12 has a remainder of 7
13/12 has a remainder of 1
19/12 has a reminder of 7
25/12 has a remainder of 1
Since we have found a pattern, we do not have to test any further numbers. Furthermore, since statement two tells us that the remainder when N is divided by 12 is greater than 5, the only possible remainder is 7.
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
Hi,
Please, I have problem understanding why c. In my opinio, the answer is E because not only 7 meets the criteria but also 31 !
can you tell me where am i wrong ?
Thank you !
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
hichem wrote:
Hi,
Please, I have problem understanding why c. In my opinio, the answer is E because not only 7 meets the criteria but also 31 !
can you tell me where am i wrong ?
Thank you !
The question asks to find the value of r not n. What is the remainder when the positive integer n is divided by 12? That being said, n can take infinitely many values: 7, 19, 31, 43, ... not just the two you mention.
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
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Statement 1 : When n is divided by 6, the remainder is 1.
Let's say N = 6k +1
When N is ODD , Values of N are 7,19,31,43...
The remainder when N is divided by 12 is 7.
When N is even, Values of N are 1,13,24,37..
The remainder when N is divided by 12 is 1.
So we can conclude the when N is divided by 12 , the remainder could be 1 or 7.
Hence Statement 1 alone is insufficient.
Statement 2:When n is divided by 12, the remainder is greater than 5.
So the remainder could be 6,7,8,9,10,11
So Statement 2 alone is insufficient.
When you combine both statements, we can conclude that the Remainder is 7
option C is the right answer.
Thanks,
Clifin J Francis
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
Statement 1 is clearly insufficient, coz N could be 1,7,13,19,25,31 and each one of those possible values of N gives different value for the remainder when divided by 12
But i donot understand statement 2.
This is what i get. if N/12 and the remainder is greater than 5, this means N could be 6,7,8,9,10,11 but N could also be 18,19,20,21,22,23.
If N 18, then 18/12, the remainder is 6,which is greater than 5
If N 19, then 19/12, the remainder is 7,which is greater than 5
etc....
Conclusion/
From statement 1 N could be 1,7,13,19,25,31
From statement 2 N could be 6,7,8,9,10,11 but N could also be 18,19,20,21,22,23
Both statement together: N could 7 or 19 and that is why i have E as answer choice.
Can anyone tell me where i go wrong in my approach?
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Re: What is the remainder when the positive integer n is divided by 12? [#permalink]
windofchange wrote:
What is the remainder when the positive integer n is divided by 12?
(1) When n is divided by 6, the remainder is 1.
(2) When n is divided by 12, the remainder is greater than 5.
avigutman sir I am not able to approach the 2 statements
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# Factors And Multiples
Here we will learn about factors and multiples, including their definitions, listing factors and multiples, and problem solving with factors and multiples.
There are also factors and multiples worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
## What are factors and multiples?
Factors and multiples are two different types of numbers.
Factors are numbers that will divide into an integer (a whole number) with no remainder. Another name for a factor is a divisor.
Multiples are the result of multiplying a number by an integer.
### Factors
There are a finite number of factors of a number.
For example, the factors of 18 are 1,2,3,6,9, and 18.
To find all of the factors of any integer, we write out all of the factor pairs in order.
Step-by-step guide: Factors
The highest common factor (HCF), or greatest common factor, is the largest number that is a factor of two or more numbers.
For example, the highest common factor of the numbers 6,8 and 10 is 2.
Step-by-step guide: Highest common factor
Prime numbers have only two factors, themselves and 1.
Any positive integer that is not a prime number is a composite number. Composite numbers have at least 2 factor pairs.
Step-by-step guide: Prime numbers
### Multiples
There are an infinite number of multiples of a number.
For example, the first 5 multiples of 18 are 18,36,54,72, and 90, but we can continue this list indefinitely.
To calculate a multiple of a number n, we have to multiply it by an integer.
Step-by-step guide: Multiples
The lowest common multiple (LCM), or least common multiple, is the smallest number that is a multiple of two or more numbers.
For example, the lowest common multiple of the numbers 8 and 10 is 40.
Step-by-step guide: Lowest common multiple
We can use factors and multiples to solve problems involving probability, area, substitution, solving quadratics, and equivalent fractions.
## How to list factors
In order to list all of the factor pairs of a number n:
1. State the pair \bf{1 \times n} .
2. Write the next smallest factor of \bf{n} and calculate its factor pair.
3. Repeat until the next factor pair is the same as the previous pair.
4. Write out the list of factors for \bf{n} .
### Related lessons onfactors, multiples and primes
Factors and multiples is part of our series of lessons to support revision on factors, multiples and primes. You may find it helpful to start with the main factors, multiples and primes lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
## Factors examples
### Example 1: listing factors (odd number)
List the factors of 15.
1. State the pair \bf{1 \times n} .
As n=15, you have the first factor pair 1\times{15}.
2Write the next smallest factor of \bf{n} and calculate its factor pair.
As 15 is an odd number, 15\div{2} is not an integer and so 2 is not a factor of 15.
15\div{3}=5 and so the next factor pair is 3\times{5}.
3Repeat until the next factor pair is the same as the previous pair.
So far you have,
\begin{aligned} &1\times{15}\\\\ &3\times{5} \end{aligned}
As 15 is odd, you cannot divide 15 by any even number and get an integer and so 4 is not a factor of 15.
The next factor to try is 5.
As factors are commutative, 3\times{5}=5\times{3} which is the same as the previous factor pair.
We have now found all of the factor pairs,
\begin{aligned} &1\times{15}\\\\ &3\times{5} \end{aligned}
4Write out the list of factors for \bf{n} .
Reading down the first column of factors, and up the second column, the factors of 15 are 1, 3, 5, and 15.
### Example 2: listing factors (square number)
List the factors of 16.
State the pair \bf{1 \times n} .
Write the next smallest factor of \bf{n} and calculate its factor pair.
Repeat until the next factor pair is the same as the previous pair.
Write out the list of factors for \bf{n} .
### Example 3: listing factors (common factors)
The factors of 21 are 1,3,7, and 21. By finding the factors of 6, determine the common factors of 6 and 21.
State the pair \bf{1 \times n} .
Write the next smallest factor of \bf{n} and calculate its factor pair.
Repeat until the next factor pair is the same as the previous pair.
Write out the list of factors for \bf{n} .
## How to calculate multiples
In order to calculate multiples of a number n:
1. State the first multiple of \bf{n} .
2. Calculate the second multiple of \bf{n} .
3. Continue until you have calculated the number of multiples needed.
4. Write the solution.
## Multiples examples
### Example 4: listing multiples (two digit number)
List the first 5 multiples of 12.
State the first multiple of \bf{n} .
Calculate the second multiple of \bf{n} .
Continue until you have calculated the number of multiples needed.
Write the solution.
### Example 5: calculate a specific multiple
What is the 13th multiple of 6?
State the first multiple of \bf{n} .
Calculate the second multiple of \bf{n} .
Continue until you have calculated the number of multiples needed.
Write the solution.
### Example 6: common multiples
Given that the first 5 multiples of 12 are 12,24,36,48, and 60, find a common multiple of 8 and 12.
State the first multiple of \bf{n} .
Calculate the second multiple of \bf{n} .
Continue until you have calculated the number of multiples needed.
Write the solution.
### Common misconceptions
• Factors and multiples
Factors and multiples are easily mixed up. Remember multiples are the multiplication table, whereas factors are the numbers that go into another number without a remainder.
• Remember \bf{1} and the number itself for factors
All numbers are a factor of themselves and 1 is a factor of every number.
For example, the factors of 6 are 1,2,3,6 and so 6 is a factor of itself.
• Remember the number itself for multiples
All numbers are a multiple of themselves.
For example, the multiples of 6 are 6,12,18,24 and so on and so 6 is a multiple of itself.
### Practice factors and multiples questions
1. List the factors of 24.
2 and 12, 3 and 8, 4 and 6
24, 48, 72, 96, and 120
2\times{2}\times{2}\times{2}
1, 2, 3, 4, 6, 8, 12 and 24
The factor pairs of 24 are,
\begin{aligned} &1\times{24}\\\\ &2\times{12}\\\\ &3\times{8}\\\\ &4\times{6} \end{aligned}
So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
2. List the factors of 49.
1 and 49
49, 98, 147, 196, 245
1, 7, and 49
7\times{7}
The factor pairs of 49 are,
\begin{aligned} &1\times{49}\\\\ &7\times{7} \end{aligned}
So the factors of 49 are 1, 7, and 49.
3. The factors of 8 are 1,2,4 and 8. By finding the factors of 30, determine the common factors of 8 and 30.
30, 60, 90, 120, 150
240
1 and 2
2\times{3}\times{5}
The factor pairs of 30 are,
\begin{aligned} &1\times{30}\\\\ &2\times{15}\\\\ &3\times{10}\\\\ &5\times{6} \end{aligned}
So the factors of 30 are \colorbox{yellow}{1}, \colorbox{yellow}{2}, \ 3, \ 5, \ 6, \ 10, \ 15, and 30.
The factors of 8 are \colorbox{yellow}{1}, \colorbox{yellow}{2}, \ 4, and 8.
The common factors of 8 and 30 are 1 and 2.
4. List the first 5 multiples of 30.
1, 2, 3, 5, 6
30\times{5}=150
30, 60, 90, 120, 150
1, 2, 3, 5, 6
\begin{aligned} &30\times{1}=30 \\\\ &30\times{2}=60 \\\\ &30\times{3}=90 \\\\ &30\times{4}=120 \\\\ &30\times{5}=150 \end{aligned}
So, the first 5 multiples of 30 are 30, 60, 90, 120, and 150.
5. What is the 4th multiple of 15?
15, 30, 45, 60
60
3.75
15
15\times{4}=60
6. Determine the first 2 common multiples of 3 and 4.
4, 8, 12, 16, 20, 24
12, 24
1
24, 36
The first 8 multiples of 3 are 3, \ 6, \ 9, \colorbox{yellow}{12}, \ 15, \ 18, \ 21 and \colorbox{yellow}{24}.
The first 8 multiples of 4 are 4, \ 8, \colorbox{yellow}{12}, \ 16, \ 20, \colorbox{yellow}{24}, \ 28 and 32.
The first common factor of 4 and 3 is 12 and the second is 24.
### Factors and multiples GCSE questions
1. Here is a list of numbers,
1, \ 2, \ 3, \ 4, \ 6, \ 10, \ 12, \ 16, \ 24, \ 25 .
(a) Write down the multiples of 4.
(b) Which numbers have a factor of 3?
(c) A common multiple of two numbers is 18. The numbers also have a common factor of 3. Write down the two numbers.
(d) Which number has exactly 5 factors?
(5 marks)
(a) 4, 12, 16, 24
(1)
(b) 3, 6,12, 24
(1)
(c) 3 and 6
(2)
(d) 16
(1)
2. Bus A and Bus B leave the depot at 7:40am.
It takes 40 minutes for Bus A to return to the depot.
It takes 30 minutes for Bus B to return to the depot.
What time will both buses be back at the depot?
(3 marks)
Multiples of 30 and 40 listed.
(1)
120 minutes = 2 hours
(1)
9:40am
(1)
3. (a) The length and width of a rectangle are both integers. How many possible rectangles can be drawn with an area of 24cm^{2}?
(b) An isosceles triangle also has side lengths that are integers. How many triangles can be drawn with a perimeter of 8cm?
(4 marks)
(a)
Factor pairs of 24 are 1 and 24, 2 and 12, 3 and 8, 4 and 6 .
(1)
4 rectangles
(1)
(b)
\cfrac{8-[2, 4, 6]}{2}
(1)
1 \ (2cm, \ 3cm, \ 3cm only)
(1)
## Learning checklist
You have now learned how to:
• Use and understand the terms factors and multiples
• Recognise and use factor pairs and commutativity in mental calculations
• Identify factors including all factor pairs of a given number and common factors of two numbers
• Solve problems involving multiplying and dividing including knowledge of factors and multiples
## Still stuck?
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths tuition programme.
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# Problem: A compound has a molar mass of 180.18 g/mol. Given the following percent composition, calculate the molecular formula. 39.99% C, 6.73% H, 53.28% O
###### FREE Expert Solution
We are asked to determine the molecular formula of a compound with a molar mass of 180.18 g/mol and a percentage composition of 39.99% C, 6.73% H, 53.28% O.
This means we need to do the following steps:
Step 1: Calculate the mass and moles of C, H, and O in the compound.
Step 2: Determine the lowest whole number ratio of C, H, and O to get the empirical formula.
Step 3: Get the ratio of the molar mass and empirical mass to determine the molecular formula.
Step 1: The compound is composed of C, H, O and we’re given the mass percent of C (39.99% C), H (6.73% H) and O (53.28% O)Check if the mass percentages of a compound add up to 100%.
This means:
Recall that mass percent is given by:
Assuming we have 100 g of the compound, this means we have 39.99 g C, 6.73 g H, and 53.28 g O
Now, we need to get the moles of each element in the compound.
The atomic masses are 12.01 g/mol C, 1.00 g/mol H, 15.99 g/mol O.
82% (383 ratings)
###### Problem Details
A compound has a molar mass of 180.18 g/mol. Given the following percent composition, calculate the molecular formula.
39.99% C, 6.73% H, 53.28% O
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## 3430
3,430 (three thousand four hundred thirty) is an even four-digits composite number following 3429 and preceding 3431. In scientific notation, it is written as 3.43 × 103. The sum of its digits is 10. It has a total of 5 prime factors and 16 positive divisors. There are 1,176 positive integers (up to 3430) that are relatively prime to 3430.
## Basic properties
• Is Prime? No
• Number parity Even
• Number length 4
• Sum of Digits 10
• Digital Root 1
## Name
Short name 3 thousand 430 three thousand four hundred thirty
## Notation
Scientific notation 3.43 × 103 3.43 × 103
## Prime Factorization of 3430
Prime Factorization 2 × 5 × 73
Composite number
Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 5 Total number of prime factors rad(n) 70 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 3,430 is 2 × 5 × 73. Since it has a total of 5 prime factors, 3,430 is a composite number.
## Divisors of 3430
1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 343, 490, 686, 1715, 3430
16 divisors
Even divisors 8 8 4 4
Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 7200 Sum of all the positive divisors of n s(n) 3770 Sum of the proper positive divisors of n A(n) 450 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 58.5662 Returns the nth root of the product of n divisors H(n) 7.62222 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 3,430 can be divided by 16 positive divisors (out of which 8 are even, and 8 are odd). The sum of these divisors (counting 3,430) is 7,200, the average is 450.
## Other Arithmetic Functions (n = 3430)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 1176 Total number of positive integers not greater than n that are coprime to n λ(n) 588 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 486 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 1,176 positive integers (less than 3,430) that are coprime with 3,430. And there are approximately 486 prime numbers less than or equal to 3,430.
## Divisibility of 3430
m n mod m 2 3 4 5 6 7 8 9 0 1 2 0 4 0 6 1
The number 3,430 is divisible by 2, 5 and 7.
• Arithmetic
• Abundant
• Polite
• Frugal
## Base conversion (3430)
Base System Value
2 Binary 110101100110
3 Ternary 11201001
4 Quaternary 311212
5 Quinary 102210
6 Senary 23514
8 Octal 6546
10 Decimal 3430
12 Duodecimal 1b9a
20 Vigesimal 8ba
36 Base36 2na
## Basic calculations (n = 3430)
### Multiplication
n×i
n×2 6860 10290 13720 17150
### Division
ni
n⁄2 1715 1143.33 857.5 686
### Exponentiation
ni
n2 11764900 40353607000 138412872010000 474756150994300000
### Nth Root
i√n
2√n 58.5662 15.081 7.65286 5.094
## 3430 as geometric shapes
### Circle
Diameter 6860 21551.3 3.69605e+07
### Sphere
Volume 1.69033e+11 1.47842e+08 21551.3
### Square
Length = n
Perimeter 13720 1.17649e+07 4850.75
### Cube
Length = n
Surface area 7.05894e+07 4.03536e+10 5940.93
### Equilateral Triangle
Length = n
Perimeter 10290 5.09435e+06 2970.47
### Triangular Pyramid
Length = n
Surface area 2.03774e+07 4.75572e+09 2800.58
## Cryptographic Hash Functions
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# Change a percentage into a fraction (with examples and a table)
When making comparisons or connections between amounts, we frequently use percents and fractions. As an illustration, we rank students by comparing their grades and overall performance using percentages and percents. Percentages are numerical representations of fractions of 100.
The term "percent" is more easily memorized than the fraction "percentage" because it describes a proportion of a whole. What percentage of the whole something it makes up Half can be written as 1/2, and a quarter as 1/4, for instance. Converting from percent to fraction can be done in the same way.
Finding out what percentage of something is important in everyday life. Examining an illustration can help you grasp the concept of converting percentages to fractions. What fraction of the student body is female if there are 865 students at a school and 389 of them are female? You can solve this by dividing the two numbers. As a percentage, it is 389 out of 865, or 389/865.0 0.44971 The percentage form, which can be used to simplify the calculation: 44.9711% Let's dive deeper into the percent-to-fraction conversion process now.
## A Percentage of What?
The term "percentage" literally means "out of one hundred." The Latin word "per centum," from which we get the word "per cent," means "one hundred percent." That's the number you use when you want to represent a percentage of 100. The symbol for this concept is a percent sign (%). Regarding this, 64% can be expressed as 64/100 as a fraction To use another illustration, let's say that 80 percent of the students at a given school are female.
## The Meaning of Fractions
A fraction is a quantified division of a whole. It is a visual representation of the fractional size of a whole. To write a fraction like 1/2, the numbers above the line are called the numerator, and the numbers below the line are called the denominator.
Parts are represented by the numerator, and the total number of parts that compose the whole is indicated by the non-zero denominator. If, for some reason, you only need three of the apple's four pieces, you can express this as a fraction written as 3/4, where the numerator, 3, represents the number of apple pieces used and the denominator, 4, indicates that there are four equal apple pieces.
## Fraction to Percentage Conversion
Using the formula on this page, converting between percentages and fractions is a breeze. For those who don't know, "percent" literally means "out of one hundred." We must first convert a percentage to a decimal, and then the decimal to a fraction.
Verify: Convert a fraction to a percentage using an online calculator.
### Ways of Breaking Down Percentage into Fractions
Method One
One must first divide the specified percentage by 100. There will be a decimal result from this.
Second, determine how many digits are present after the decimal point. D stands for the digit in this context.
Number 2 in decimal notation Two digits follow the decimal point in 56, so we can write d = 2.
Third, determine the value of the constant f. Integer form will be calculated from the decimal using the following formula:
The fourth step is to multiply and divide the decimal number by the factor.
The GCD of the fraction is the fifth step.
Simplify the fraction by dividing the numerator and denominator by the GCD value, as in Step 6:
Method Two
First, we take the decimal number obtained by dividing the given percentage by 100: i e , % of One Hundred
Step 2: Multiply each digit after the decimal point by 10 if the percentage is not a whole number.
Thirdly, we simplify the fraction.
## Conversion of Percentages with Mixed Numbers
We already know how to convert between simple percentages and fractions. In this section, we will examine the process by which a percent value expressed as a mixed number can be converted into a fractional value. Please proceed as outlined below:
• Change the percent into a proper fraction
• Remove the percent sign by multiplying 1/100 by 100.
• To simplify a fraction, reduce it to its lowest terms.
Examples:
1. Figure out 11's equivalent fraction 1 / 2 %
11 1/2% is the aggregate given percentage.
Changing this into the correct fraction
11 1/2 % = 23/2 %
Divide by a hundredth to get rid of the % sign:
23/2 x 1/100 = 23/200
Since 23/200 is a problem that cannot be further reduced, it is equivalent to 200.
1. How to fractionize 2.5%
Given,
2 1/2 %
Changing a percentage in mixed form to its proper form.
⇒ 2 1/2 % = 5/2 %
Subtract 1% by multiplying by 1/100 now
⇒ 5/2 x 1/100
⇒ 5/200
⇒ 1/40
Percent Fraction
## What are the Issues, and How Can We Fix Them?
First Example: Change 11% with respect to a fraction
Solution:
First, convert the percentage to a decimal by dividing it by 100.
Second, since the specified percentage is a whole number, proceed to Step 3.
Third, we can't reduce 11/100 any further because it's already a fraction of 100.
11 out of 100 is the correct response.
Second Illustration: dividing 75% by 10
Solution:
First, we'll get the decimal by dividing the percentage by 100.
Proceed to Step 3 if the provided percentage is a whole number (Step 2).
Third, reduce the fraction 75/100 to its simplest form.
The correct response is "4/5"
Third Illustration: 62 Converted 5% to a decimal
Solution:
First, enter 62% into your calculator. 5/100
Step 2: Since the given percentage is a fraction rather than a whole number, multiply both the numerator and denominator by 10 to get the correct answer.
Step 3:
As a fourth-step, simplify the fraction:
The correct response is 5/8.
## The Conversion from Percent to Decimal
Here are the steps to take to convert a percentage to its corresponding decimal value:
• Subtract the percent sign and multiply the result by 100.
• Next, remove any constants that are present in both the numerator and denominator.
• Reduce to the lowest decimal form to see how it compares
### Examples
By dividing 10% by 100%, we obtain;
10/100 = 1/10 = 0 1
Thus, 0 The decimal value of 10% is 1.
1. Convert 77 How to convert 5% to decimal form
Subtracting 77 The result of dividing 5 by 100 is;
77.5% = 77 5/100 = 7 75/10 = 0 775
Thus, 0 In decimal notation, 77 converts to 775. 5%
## Converting a Decimal to a Fraction
Follow these steps to convert a decimal value to its corresponding fractional form:
• Using multiplication and division by 10, find the decimal equivalent. n , where n is the number of significant digits after the decimal point
• Check the numerator and denominator for their HCF.
• Subtract the GCF from both the numerator and denominator to get the answer.
• When the numerator and denominator have no more factors in common, the fraction is written in its simplest form.
Examples:
1. Convert 1 Five converted to a fraction
One of the things we can do is write. 5 as 1 5/1
Now that we have only one digit after the decimal point, we can simplify by multiplying the numerator and denominator by 10 to obtain;
⇒ (1 5/1) x (10/10)
⇒ 15/10
Product of 15 and 10 (GCF) = 5
Therefore,
⇒ (15÷5)/(10÷5)
⇒ 3/2
1. Convert 2 To convert 25 to a fraction
⇒ 2 25/1
Perform a 10x and 100 calculation. 2 = 100; rounding up to the nearest whole number is allowed to a maximum of two places.
⇒ (2 25/1) x (100/100)
⇒ 225/100
A GCF of 225 and 100 is 25
Therefore,
⇒ (225÷25)/(100÷25)
⇒ 9/4
Download BYJU'S-The Learning App and subscribe to BYJU'S to access interactive and educational videos.
Three percent is equivalent to a fraction of one tenth.
The process of converting percent to fraction is straightforward. To get rid of the '%' symbol, divide the given percentage by 100 first. Simplify the fraction to its lowest terms.
8.25% as a fraction equals 2/25
8% = 8/100 = 2/25
The equivalent of 70% is a score of 7/10.
70% = 70/100 = 7/10
34 is the same as 75%
¾ = 0 75 = 0 75 x 100 = 75%
Convert Chinese Yuan to United States Dollar
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Utilize our inch-to-fraction calculator to effortlessly perform conversions between inch fractions, decimal values, metric measurements, and feet. Effective Techniques for Calculating Inch FractionsInches can be represented as fractions or decimals. When dealing with inch fractions, it is vital to
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Please enter the necessary values below to convert kilowatts [kW] to British thermal units per hour [Btu/h], or the other way around.Description: A kilowatt (symbol: kW) is a unit of power within the International System of Units (SI). The watt, after the Scottish inventor James Watt, serves as the base
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HomeFINANCESeize Time: the Vector of Life!
# Seize Time: the Vector of Life!
## What is vector or scalar time?
Many physical magnitudes are absolutely defined by a telephone number and a unit, these same magnitudes are called scalars. Eg volume, temperature, time, mass, etc.
### How long is a vector?
In physics and mathematics, a vector is a segment of a straight line, endowed with a notation, that is, oriented within a two- or three-dimensional Euclidean plane. Or what is the same: a vector is a factor in a vector space.
### What is a vector in daily life?
Vectors allow you to calculate the speed of moving objects and also their accelerations. In addition, they are used to obtain other types of measurements, such as the sides of a building. Design roads and highways.
## What magnitudes are vectors?
Vector magnitude: it is the magnitude that is absolutely determined by a number, a unit, a direction and a value. The phone number and the unit are called a module. For example, the magnitude of the speed of the wind is 3 km/h.
### Why is time not a vector magnitude?
Time has no direction or notation, for this very reason it is a scalar and not a vector.
### What is a scalar and vector magnitude examples?
The scalar magnitude is the quantity that we can measure of a certain property that does not depend on its direction or location in space.
Scalar and vector magnitude.
Vectors scalars
examples Displacement, speed, weight, forces. Length, speed, mass, density, temperature.
## What are the vector magnitudes?
Vector magnitudes are those that are characterized by a quantity (intensity or module), a direction and a meaning. In a Euclidean space, of no more than three dimensions, a vector is represented by an oriented segment.
#### What are the 7 vector magnitudes?:
The seven that you drink as essential units, from which all the others are derived. They are length, time, mass, intensity of electric current, temperature, amount of substance, and light intensity.
#### What are the vector and scalar magnitudes?:
A scalar magnitude is one that is fully determined with a telephone number and its own relevant units, and a vector magnitude is one that, in addition to a numerical value and its own units (module), we must specify its direction and value.
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This is math, not art, but you should pretend you’re Rembrandt when you hit a geometry question on the PSAT/NMSQT. Why? Because geometry concerns shapes and lines, and if you can see them, you can understand them better. Follow these steps:
• Examine the diagram, if the question has one. The little drawings on the exam are a good starting place. Look at the information they provide and notice every number and variable (45°, 2 feet, x, and so forth).
The drawings on the PSAT/NMSQT are as accurate as possible, but they can deceive you. Something may look like a right angle, for instance, but not actually be one. Check the question; it may state that ABD is a right angle or include a little square drawn in the angle.
But don’t assume anything! Rely on the information given and your knowledge of math, not on an estimate.
Some diagrams are labeled not drawn to scale. If you see that phrase, be extra careful. The drawings don’t give you an idea of relative length or size.
• If no diagram appears, sketch one in the test booklet. Don’t take time for museum-ready quality. Just be sure that you have everything in the right place.
• Read the information supplied by the question, calculate if necessary, and add everything you can to the diagram. If the question tells you that one angle is twice as large as another and the smaller angle is labeled 30°, label the bigger angle 60°.
• Search for basic shapes hidden inside more complicated diagrams. You may see a triangle with one side extended like a flagpole, for example. So? It’s still a triangle! Everything you know about triangles still applies.
Plus, the “flagpole” may supply extra information, such as the measurement of an angle outside the triangle. Because straight lines always equal 180°, you can sometimes figure out the angle inside the triangle by looking at the angle outside the triangle.
• Reread the question and identify what the test-makers want to know. Are you looking for the length of side b or trying to find out what number can’t possibly represent the length of side b? If you answer the wrong question, your math skills won’t matter.
As you read the question, underline key words that indicate what you have to figure out. In the preceding bullet, for example, you might underline “can’t” and “length of side b.” The underlining focuses your attention on your goal.
• Use the formula box only as a reminder. Because they have kind hearts and aren’t actually trying to torture you, the question-writers provide a little box of information at the beginning of each math section. The box tells you the number of degrees in a circle, straight line, and triangle. It also supplies formulas for area and volume, the Pythagorean Theorem, and the measurements of special right triangles.
• If you’re nervous, peek at the box to check on these basics. However, you may be pressed for time, and turning back to find formulas can eat up precious seconds. Your goal is to know this information before you receive the test booklet.
Mathematicians love specialized terms almost as much as grammarians do. The good news is that you don’t need to know many terms to solve PSAT/NMSQT geometry problems. A few basic words will get you through. So even though your math teacher throws around words such as scalene triangle, you don’t have to memorize the fact that a scalene triangle has three unequal sides.
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# Eureka Math Grade 5 Module 4 Mid Module Assessment Answer Key
## Engage NY Eureka Math 5th Grade Module 4 Mid Module Assessment Answer Key
### Eureka Math Grade 5 Module 4 Mid Module Assessment Task Answer Key
Multiply or divide. Draw a model to explain your thinking.
a. $$\frac{1}{2}$$ × 6
Explanation:
Given, $$\frac{1}{2}$$ × 6
= $$\frac{6}{2}$$
= 3.
b. $$\frac{1}{2}$$ × 7
7 ÷ 2 = 3 1/2
Explanation:
c. $$\frac{3}{4}$$ × 12
Explanation:
3/4 × 12 = 9
d. $$\frac{2}{5}$$ × 30
Explanation:
2/5 × 30 = 12
e. $$\frac{1}{3}$$ of 2 feet = _8_ inches
2 × 12 inches = 24 inches
Explanation:
1 feet = 12 inches
2 feet = 24 inches
Hence 1/3 × 24 inches = 8 in.
f. $$\frac{1}{6}$$ of 3 yards = _________feet”
Answer: 1 1/2 feet
Explanation:
1/6 ×3 yards = 1 1/2 feet
g. (3 + $$\frac{1}{2}$$) × 14
(3 ×14) + ( 1/2 ×14)
42 + 7
49
Explanation:
(3 ×14) + ( 1/2 ×14)
42 + 7
49
h. 4$$\frac{2}{3}$$ × 13
60 2/3
Explanation:
( 4×13) + (2/3 ×13)
52 + (2×13)/3
52 + (26/3)
52 + 8 2/3
60 2/3
Question 2.
If the whole bar is 3 units long, what is the length of the shaded part of the bar? Write a multiplication equation for the diagram, and then solve.
Answer: $$\frac{9}{4}$$ or 2.25 units
Explanation:
Given, the whole bar is 3 units long,
The bar has 4 total parts, The shaded region has 3 parts.
So, the ratio of shaded parts to total parts will be $$\frac{3}{4}$$
In order to determine length of the shaded part, just multiply the ratio of the shaded parts to total parts by the total length of the bar.
Then, $$\frac{3}{4}$$ × 3 = $$\frac{9}{4}$$ or 2.25 units.
Question 3.
Circle the expression(s) that are equal to $$\frac{3}{5}$$ × 6. Explain why the others are not equal using words, pictures, or numbers.
a. 3 × (6 ÷ 5)
b. 3 ÷ (5 × 6)
c. (3 × 6) ÷ 5
d. 3 × $$\frac{6}{5}$$
Answer:Â a, c, d are correct answers
Explanation:
All the correct options have the result of $$\frac{18}{5}$$
Option b. 3 ÷ (5 × 6) gives the output of $$\frac{3}{30}$$
Question 4.
Write the following as expressions.
a. One-third the sum of 6 and 3.
1/3 × ( 6 +3 )
Explanation:
The expression for the following question is 1/3 × ( 6 +3 ).
b. Four times the quotient of 3 and 4.
4 × (3÷4)
Explanation:
The expression for the following question is 4 × (3÷4).
c. One-fourth the difference between $$\frac{2}{3}$$ and $$\frac{1}{2}$$.
1/4 × ( 2/3 – 1/2)
Explanation:
The expression for the following question is 1/4 × ( 2/3 – 1/2).
Question 5.
Mr. Schaum used 10 buckets to collect rainfall in various locations on his property. The following line plot shows the amount of rain collected in each bucket in gallons. Write an expression that includes multiplication to show how to find the total amount of water collected in gallons. Then, solve your expression.
5/8 + (4 × 1 2/8) + (2 × 1 5/8)+ 2 1/8 + (2 × 2 3/8)
5/8 + 4 + (4 ×2)/8 + 2 + (2×5)/8 + 2 1/8 + 4 + (2×3)/8
12 + 5/8 + 8/8 + 10/8 + 1/8 + 6/8
12 + 5/8 + 8/8 + 10/8 + 1/8 + 6/8
13 + 22/8
15 6/8
15 3/4
Explanation:
As asked in the question I have solved the expression and the total amount of water collected in gallons are 15 3/4.
Question 6.
Mrs. Williams uses the following recipe for crispy rice treats. She decides to make 2/3 of the recipe.
2 cups melted butter
24 oz marshmallows
13 cups rice crispy cereal
a. How much of each ingredient will she need? Write an expression that includes multiplication. Solve by multiplying.
Butter: 2/3 × 2 cups = (2×2)/3 = 4/3 = 1 1/3 cups
Marshmallows: 2/3 × 24 oz = (2×24)/3 = 48/3 = 16 oz
Cereal: 2/3 × 13 cups = (2×13)/3 = 26/3 = 8 2/3 cups
Explanation:
I have written an expression that includes multiplication is she will need 1 1/3 cups of butter, 16 ounces of marshmallows, and 8 2/3 cups of rice crispy cereal.
b. How many fluid ounces of butter will she use? (Use your measurement conversion chart, if you wish.)
1 cup = 8 ounces
1 1/3 × 8 = (1×8) + (1/3 ×8)
= 8 + 8/3
= 8 + 2 2/3
= 10 2/3
Explanation:
In this, by using the measurement conversion chart she will use 10 2/3 fluid ounces of butter.
c. When the crispy rice treats have cooled, Mrs. Williams cuts them into 30 equal pieces. She gives two-fifths of the treats to her son and takes the rest to school. How many treats will Mrs. Williams take to school? Use any method to solve.
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## What are the 3 basic characteristics of an inelastic collision?
What are their basic characteristics?
• Linear momentum is conserved.
• Total energy is conserved.
• K E is not conserved.
• Some or all forces involved may be non-conservative.
## How do you identify an inelastic collision?
If objects stick together, then a collision is perfectly inelastic. When objects don’t stick together, we can figure out the type of collision by finding the initial kinetic energy and comparing it with the final kinetic energy. If the kinetic energy is the same, then the collision is elastic.
## What are the characteristic of each type of collision?
Answer. A collision is the event in which two or more bodies exert forces on each other in about a relatively short time. Inelastic collisionCollision which conserves momentum but not kinetic energy. Elastic collision:-Collision in which there is no net loss in kinetic energy in the system as a result of the collision
## What are 3 types of collisions?
Collisions are of three types:
• perfectly elastic collision.
• inelastic collision.
• perfectly inelastic collision.
## What are the 4 types of collisions?
If two objects (a car and a truck, for example) collide, momentum will always be conserved. There are three different kinds of collisions, however, elastic, inelastic, and completely inelastic. Just to restate, momentum is conserved in all three kinds of collisions.
## What happens in an inelastic collision?
An inelastic collision is a collision in which there is a loss of kinetic energy. While momentum of the system is conserved in an inelastic collision, kinetic energy is not. This is because some kinetic energy had been transferred to something else. Such collisions are simply called inelastic collisions.
## What happens in a completely inelastic collision?
A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. In a perfectly inelastic collision, i.e., a zero coefficient of restitution, the colliding particles stick together. In such a collision, kinetic energy is lost by bonding the two bodies together.
## What is an example of a perfectly inelastic collision?
Another common example of a perfectly inelastic collision is known as the “ballistic pendulum,” where you suspend an object such as a wooden block from a rope to be a target.
## What is the formula for perfectly inelastic collision?
Inelastic Collision Formula
V= Final velocity. M1= mass of the first object in kgs. M2= mas of the second object in kgs. V1= initial velocity of the first object in m/s.
## What is the difference between an inelastic and perfectly inelastic collision?
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost.
## What is difference between elastic and inelastic collision?
An inelastic collision can be defined as a type of collision where this is a loss of kinetic energy.
Differences between elastic and inelastic collisions.
Elastic Collision Inelastic Collision
The total kinetic energy is conserved. The total kinetic energy of the bodies at the beginning and the end of the collision is different.
Momentum does not change. Momentum changes.
## What is elastic and inelastic collision give example?
A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision.
## What are the two types of collision?
There are two general types of collisions in physics: elastic and inelastic. An inelastic collisions occurs when two objects collide and do not bounce away from each other.
## Is an inelastic collision a closed system?
Inelastic collision.
In an inelastic collision, the collision changes the total kinetic energy in a closed system. If you can observe appreciable energy losses due to nonconservative forces (such as friction), kinetic energy isn’t conserved.
## What happens when two objects collide?
When two objects collide, each object pushes the other. Newton’s third law states that when one object exerts a force on another, the second object exerts an equal but opposite force on the first object. These forces are sometimes called action force andreaction force or force pairs.
## Are elastic collisions open or closed?
For all collisions in a closed system, momentum is conserved. In some collisions in a closed system, kinetic energy is conserved. When both momentum and kinetic energy are conserved, the collision is called an elastic collision.
## Is velocity conserved in a collision?
Figure 8.7 A one-dimensional inelastic collision between two objects. Momentum is conserved, but kinetic energy is not conserved. for inelastic collisions, where v′ is the final velocity for both objects as they are stuck together, either in motion or at rest.
## How do you find the speed of an elastic collision?
If two particles are involved in an elastic collision, the velocity of the second particle after collision can be expressed as: v2f=2⋅m1(m2+m1)v1i+(m2−m1)(m2+m1)v2i v 2 f = 2 ⋅ m 1 ( m 2 + m 1 ) v 1 i + ( m 2 − m 1 ) ( m 2 + m 1 ) v 2 i .
## Do objects stick together in an elastic collision?
– An elastic collision is one in which no energy is lost. – A partially inelastic collision is one in which some energy is lost, but the objects do not stick together. – The greatest portion of energy is lost in the perfectly inelastic collision, when the objects stick.
## Where does energy go in inelastic collision?
While the total energy of a system is always conserved, the kinetic energy carried by the moving objects is not always conserved. In an inelastic collision, energy is lost to the environment, transferred into other forms such as heat.
## What does it mean if a good is perfectly inelastic?
Perfectly inelastic demand means that quantity demanded remains the same when price increases or decreases. Consumers are completely unresponsive to changes in price.
## What makes a product inelastic?
Inelastic means that a 1 percent change in the price of a good or service has less than a 1 percent change in the quantity demanded or supplied. If the price increase had no impact whatsoever on the quantity demanded, the medication would be considered perfectly inelastic.
## Is toothpaste elastic or inelastic?
Well, toothpaste is an essential necessity to keep teeth clean. If the price fluctuated a little on toothpaste, most consumers would still be likely to purchase it because of its usefulness. Therefore, toothpaste is essential and inelastic.
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# Calculate the mach number after an oblique shockwave
• JD88
In summary, when the angle of the shockwave exceeds the maximum allowable theta, the shockwave becomes detached and the downstream Mach number is equal to the upstream Mach number.
JD88
I am trying to calculate the mach number after an oblique shockwave. The freestream mach number is 2.0 and the angle theta is 25 deg. But on the Theta-Beta-Mach Diagram the line Theta=25deg does not intersect the Mach=2 line. So this means that the angle has exceeded the maximum allowable theta and the shockwave has come detached, right? If it were not detached I would just look up Beta off of the diagram and use it to calculate the normal mach number normal to the shockwave and go from there.
So how would I go about determining the mach number after the oblique shockwave when the angle has exceed the max and the wave is detached?
When an oblique shockwave is detached, the downstream Mach number will be the same as the upstream mach number. In this case, the downstream Mach number after the detached shockwave is 2.0.
## 1. What is a mach number and why is it important to calculate after an oblique shockwave?
A mach number is the ratio of an object's speed to the speed of sound in the surrounding medium. It is important to calculate after an oblique shockwave because it helps determine the level of compression and heating that occurs during the shockwave, which can have significant impacts on the object and its surroundings.
## 2. How is the mach number calculated after an oblique shockwave?
The mach number after an oblique shockwave can be calculated using the equation M2 = [(M1^2 * sin^2θ) + 2] / [(2 * M1^2 * sin^2θ) - (M1^2 - 1)], where M1 is the mach number before the shockwave and θ is the angle of the shockwave relative to the object.
## 3. What factors can affect the mach number after an oblique shockwave?
The mach number after an oblique shockwave can be affected by the angle of the shockwave relative to the object, the mach number before the shockwave, and the properties of the surrounding medium such as temperature and density.
## 4. Can the mach number after an oblique shockwave be greater than the mach number before the shockwave?
Yes, it is possible for the mach number after an oblique shockwave to be greater than the mach number before the shockwave. This occurs when the shockwave is at a shallow angle and the object is traveling at a high mach number, resulting in an increase in the mach number after the shockwave.
## 5. How is the mach number after an oblique shockwave used in real-world applications?
The mach number after an oblique shockwave is used in aerospace engineering to design and analyze the behavior of supersonic and hypersonic vehicles. It is also used in the study of high-speed aerodynamics and in understanding the effects of shockwaves on structures and materials.
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Specific Objectives
By the end of the topic the learner should be able to:
(a) Calculate length of an arc and a chord;
(b) Calculate lengths of tangents and intersecting chords;
(c) State and use properties of chords and tangents;
(d) Construct tangent to a circle,
(e) Construct direct and transverse common tangents to two circles;
(f) Relate angles in alternate segment;
(g) Construct circumscribed, inscribed and escribed circles;
(h) Locate centroid and orthocentre of a triangle;
(i) Apply knowledge of circles, tangents and chords to real life situations.
Content
(a) Arcs, chords and tangents
(b) Lengths of tangents and intersecting chords
ecolebooks.com
(c) Properties of chords and tangents
(d) Construction of tangents to a circle
(e) Direct and transverse common tangents to two circles
(f) Angles in alternate segment
(g) Circumscribed, inscribed and escribed circles
(h) Centroid and orthocentre
(i) Application of knowledge of tangents and chords to real life situations.
Length of an Arc
The Arc length marked red is given by
Example
Find the length of an arc subtended by an angle of at the centre of the circle of radius 14 cm.
Solution
Length of an arc =
=
Example
The length of an arc of a circle is 11.0 cm.Find the radius of the circle if an arc subtended an angle ofat the centre .
Solution
Arc length =
Therefore 11 =
Example
Find the angle subtended at the centre of a circle by an arc of 20 cm, if the circumference of the circle is 60 cm.
Solution
=
But 2
Therefore,
Chords
Chord of a circle: A line segment which joins two points on a circle. Diameter: a chord which passes through the center of the circle. Radius: the distance from the center of the circle to the circumference of the circle
Perpendicular bisector of a code
A perpendicular drawn from the centre of the circle to a chord bisects the chord.
Note;
• Perperndicular drawn from the centre of the circle to chord bisects the cord ( divides it into two equal parts)
• A straight line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
The radius of a circle centre O is 13 cm.Find the perpendicular distance from O to the chord, if AB is 24 cm.
Solution
OC bisects chord AB at C
Therefore, AC =12 cm
In O
Therefore
, OM = = 5 cm
Parallel chords
Any chord passing through the midpoints of all parallel chords of a circle is a diameter
Example
In the figure below CD and AB are parallel chords of a circle and 2 cm apart. If CD = 8 cm and AB= 10 cm, find the radius of the circle
Solution
• Draw the perpendicular bisector of the chords to cut them at K and L .
• Join OD and OC
• In triangle ODL,
• DL = 4 cm and KC =5 cm
• Let OK = X cm
• Therefore (
In triangle OCK;
•
• Therefore (
•
•
• 4x = 5
• X =
Using the equation
=
=
= 5.154 cm
Intersecting chords
In general
Example
In the example above AB and CD are two chords that intersect in a circle at Given that AE = 4 cm, CE =5 cm and DE = 3 cm, find AB.
Solution
Let EB = x cm
4
Since AB = AE + EB
AB = 4 + 3.75
= 7.75 cm
Equal chords.
• Angles subtended at the centre of a circle by equal chords are equals
• If chords are equal they are equidistant from the centre of the circle
Secant
A chord that is produced outside a circle is called a secant
Example
Find the value of AT in the figure below. AR = 4 cm, RD = 5 cm and TC = 9 cm.
Solution
AC x AT
(x + 9) x = (5 + 4) 4
(x + 12) (x- 3) = 0
Therefore, x = – 12 or x = 3
Tangent and secant
Tangent
A line which touches a circle at exactly one point is called a tangent line and the point where it touches the circle is called the point of contact
Secant
A line which intersects the circle in two distinct points is called a secant line (usually referred to as a secant).The figures below A shows a secant while B shows a tangent .
A B
Construction of a tangent
• Draw a circle of any radius and centre O.
• Join O to any point P on the circumference
• Produce OP to a point P outside the circle
• Construct a perpendicular line SP through point P
• The line is a tangent to the circle at P as shown below.
Note;
• The radius and tangent are perpendicular at the point of contact.
• Through any point on a circle , only one tangent can be drawn
• A perpendicular to a tangent at the point of contact passes thought the centre of the circle.
Example
In the figure below PT = 15 cm and PO = 17 cm, calculate the length of PQ.
Solution
OT = 8 cm
Properties of tangents to a circle from an external point
If two tangents are drawn to a circle from an external point
• They are equal
• They subtend equal angles at the centre
• The line joining the centre of the circle to the external point bisects the angle between the tangents
s
Example
The figure below represents a circle centre O and radius 5 cm. The tangents PT is 12 cm long. Find: a.) OP b.) Angle TP
Solution
• Join O to P
• <
= 0.9231
Therefore,
Hence <
Two tangent to a circle
Direct (exterior) common tangents Transverse or interior common tangents
Tangent Problem
The common-tangent problem is named for the single tangent segment that’s tangent to two circles. Your goal is to find the length of the tangent. These problems are a bit involved, but they should cause you little difficulty if you use the straightforward three-step solution method that follows.
The following example involves a common external tangent (where the tangent lies on the same side of both circles). You might also see a common-tangent problem that involves a common internal tangent (where the tangent lies between the circles). No worries: The solution technique is the same for both.
Given the radius of circle A is 4 cm and the radius of circle Z is 14 cm and the distance between the two circles is 8 cm.
Here’s how to solve it:
1.)Draw the segment connecting the centers of the two circles and draw the two radii to the points of tangency (if these segments haven’t already been drawn for you).
Draw line AZ and radii AB and ZY.
The following figure shows this step. Note that the given distance of 8 cm between the circles is the distance between the outsides of the circles along the segment that connects their centers.
2.) From the center of the smaller circle, draw a segment parallel to the common tangent till it hits the radius of the larger circle (or the extension of the radius in a common-internal-tangent problem).
You end up with a right triangle and a rectangle; one of the rectangle’s sides is the common tangent. The above figure illustrates this step.
3.)You now have a right triangle and a rectangle and can finish the problem with the Pythagorean Theorem and the simple fact that opposite sides of a rectangle are congruent.
The triangle’s hypotenuse is made up of the radius of circle A, the segment between the circles, and the radius of circle Z. Their lengths add up to 4 + 8 + 14 = 26. You can see that the width of the rectangle equals the radius of circle A, which is 4; because opposite sides of a rectangle are congruent, you can then tell that one of the triangle’s legs is the radius of circle Z minus 4, or 14 – 4 = 10.
You now know two sides of the triangle, and if you find the third side, that’ll give you the length of the common tangent.
You get the third side with the Pythagorean Theorem:
(Of course, if you recognize that the right triangle is in the 5 : 12 : 13 family, you can multiply 12 by 2 to get 24 instead of using the Pythagorean Theorem.)Because opposite sides of a rectangle are congruent, BY is also 24, and you’re done.
Now look back at the last figure and note where the right angles are and how the right triangle and the rectangle are situated; then make sure you heed the following tip and warning.
Note the location of the hypotenuse. In a common-tangent problem, the segment connecting the centers of the circles is always the hypotenuse of a right triangle. The common tangent is always the side of a rectangle, not a hypotenuse.
In a common-tangent problem, the segment connecting the centers of the circles is never one side of a right angle. Don’t make this common mistake.
HOW TO construct a common exterior tangent line to two circles
In this lesson you will learn how to construct a common exterior tangent line to two circles in a plane such that no one is located inside the other using a ruler and a compass.
Problem 1
For two given circles in a plane such that no one is located inside the other, to construct the common exterior tangent line using a ruler and a compass.
Solution
We are given two circles in a plane such that no one is located inside the other (Figure 1a).
We need to construct the common exterior tangent line to the circles using a ruler and a compass.
First, let us analyze the problem and make a sketch (Figures 1a and 1b). Let AB be the common tangent line to the circles we are searching for.
Let us connect the tangent point A of the first circle with its center P and the tangent point B of the second circle with its center Q (Figure 1a and 1b).
Then the radii PA and QB are both perpendicular to the tangent line AB (lesson A tangent line to a circle is perpendicular to the radius drawn to the tangent point under the topic Circles and their properties ). Hence, theradii PA and QB are parallel.
Figure 1a. To the Problem 1
Figure 1b. To the solution of the Problem 1
Figure 1c. To the construction step 3
Next, let us draw the straight line segment CQ parallel to AB through the point Q till the intersection with the radius PA at the point C (Figure 1b). Then the straight line CQ is parallel to AB. Hence, the quadrilateral CABQ is a parallelogram (moreover, it is a rectangle) and has the opposite sides QB and CA congruent. The point C divides the radius PA in two segments of the length (CA) and (PC). It is clear from this analysis that the straight line QC is the tangent line to the circle of the radius with the center at the point P (shown in red in Figure 1b).
It implies that the procedure of constructing the common exterior tangent line to two circles should be as follows:
1) draw the auxiliary circle of the radius at the center of the larger circle (shown in red in Figure 1b);
2) construct the tangent line to this auxiliary circle from the center of the smaller circle (shown in red in Figure 1b). In this way you will get the tangent point C on the auxiliary circle of the radius
3) draw the straight line from the point P to the point C and continue it in the same direction till the intersection with the larger circle (shown in blue in Figure 1b). The intersection point A is the tangent point of the common tangent line and the larger circle. Figure 1c reminds you how to perform this step.
4) draw the straight line QB parallel to PA till the intersection with the smaller circle (shown in blue in Figure 1b).
The intersection point B is the tangent point of the common tangent line and the smaller circle;
5) the required common tangent line is uniquely defined by its two points A and B.
Note that all these operations 1) – 4) can be done using a ruler and a compass. The problem is solved.
Problem 2
Find the length of the common exterior tangent segment to two given circles in a plane, if they have the radii and and the distance between their centers is d.
No one of the two circles is located inside the other.
Solution
Let us use the Figure 1b from the solution to the previous Problem 1.
This Figure is relevant to the Problem 2. It is copied and reproduced
in the Figure 2 on the right for your convenience.
figure 2
It is clear from the solution of the Problem 1 above that the common
exterior tangent segment |AB| is congruent to the side |CQ| of the
From the other side, the segment CQ is the leg of the right-angled
triangle DELTAPCQ. This triangle has the hypotenuse’s measure d and
the other leg’s measure . Therefore, the length of the common
exterior tangent segment |AB| is equal to
|AB| =
Note that the solvability condition for this problem is d >.
It coincides with the condition that no one of the two circles lies inside the other.
Example 1
Find the length of the common exterior tangent segment to two given circles in a plane, if their radii are 6 cm and 3 cm and the distance between their centers
is 5 cm.
Solution
Use the formula (1) derived in the solution of the Problem 2.
According to this formula, the length of the common exterior tangent segment to the two given circles is equal to
= =
= 4 cm
The length of the common exterior tangent segment to the two given circles is 4 cm
Contact of circles
Two circle are said to touch each other at a point if they have a common tangent at that point.
Point T is shown by the red dot.
Internal tangent externally tangent
Note;
• The centers of the two circles and their point of contact lie on a straight line
• When two circles touch each other internally, the distance between the centers is equal to the difference of the radii i.e. PQ= TP-TA
• When two circles touch each other externally, the distance between the centers is equal to the sum of the radii i.e. OR =TO +TR
.
Alternate Segment theorem
The angle which the chord makes with the tangent is equal to the angle subtended by the same chord in the alternate segment of the circle.
Angle a = Angle b
Note
The blue line represents the angle which the chord CD makes with the tangent PQ which is equal to the angle b which is subtended by the chord in the alternate segment of the circle.
Illustrations
• Angle s = Angle t
• Angle a = Ange b
Tangent – secant segment length theorem
If a tangent segment and secant segment are drawn to a circle from an external point, then the square of the length of the tangent equals the product of the length of the secant with the length of its external segment.
Example
In the figure above ,TW=10 cm and XW = 4 cm. find TV
Solution
=
TV =
Circles and triangles
Inscribed circle
• Construct any triangle ABC.
• Construct the bisectors of the three angles
• The bisectors will meet at point I
• Construct a perpendicular from O to meet one of the sides at M
• With the centre I and radius IM draw a circle
• The circle will touch the three sides of the triangle ABC
• Such a circle is called an inscribed circle or in circle.
• The centre of an inscribed circle is called the incentre
Circumscribed circle
• Construct any triangle ABC.
• Construct perpendicular bisectors of AB , BC, and AC to meet at point O.
• With O as the centre and using OB as radius, draw a circle
• The circle will pass through the vertices A , B and C as shown in the figure below
Escribed circle
• Construct any triangle ABC.
• Extend line BA and BC
• Construct the perpendicular bisectors of the two external angles produced
• Let the perpendicular bisectors meet at O
• With O as the centre draw the circle which will touch all the external sides of the triangle
Note;
Centre O is called the ex-centre
AO and CO are called external bisectors.
End of topic
Did you understand everything?If not ask a teacher, friends or anybody and make sure you understand before going to sleep!
Past KCSE Questions on the topic.
1. The figure below represents a circle a diameter 28 cm with a sector subtending an angle of 750 at the centre.
Find the area of the shaded segment to 4 significant figures
(a)
2. The figure below represents a rectangle PQRS inscribed in a circle centre 0 and radius 17 cm. PQ = 16 cm.
Calculate
1. The length PS of the rectangle
2. The angle POS
3. The area of the shaded region
3. In the figure below, BT is a tangent to the circle at B. AXCT and BXD are
straight lines. AX = 6 cm, CT = 8 cm, BX = 4.8 cm and XD = 5 cm.
Find the length of
(a) XC
(b) BT
4. The figure below shows two circles each of radius 7 cm, with centers at X and Y. The circles touch each other at point Q.
Given that 0 and lines AB, XQY and DC are parallel, calculate the area of:
a) Minor sector XAQD (Take π 22/7)
b) The trapezium XABY
5. The figure below shows a circle, centre, O of radius 7 cm. TP and TQ are tangents to the circle at points P and Q respectively. OT =25 cm.
Calculate the length of the chord PQ
6. The figure below shows a circle centre O and a point Q which is outside the circle
Using a ruler and a pair of compasses, only locate a point on the circle such that angle OPQ = 90o
7. In the figure below, PQR is an equilateral triangle of side 6 cm. Arcs QR, PR and PQ arcs of circles with centers at P, Q and R respectively.
Calculate the area of the shaded region to 4 significant figures
8. In the figure below AB is a diameter of the circle. Chord PQ intersects AB at N. A tangent to the circle at B meets PQ produced at R.
Given that PN = 14 cm, NB = 4 cm and BR = 7.5 cm, calculate the length of:
(a) NR
(b) AN
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0
# How do you write 1 minute 38.29 seconds in word form?
Updated: 9/16/2023
Wiki User
14y ago
one minute and thirtyeight,twentynine hundredth of seconds
Wiki User
14y ago
Earn +20 pts
Q: How do you write 1 minute 38.29 seconds in word form?
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### How do you write the word form of 1 minute and 38.29 seconds?
The word form of 1 minute and 38.29 seconds is: one minute thirty-eight and twenty-nine hundredths seconds.
### How do you write 1 minute and 38.29 seconds in word form?
One minute, thirty-eight and twenty-nine hundredths seconds.
### How do you write 1 minute 38.29 seconds into word form?
One minute and thirty-eight and twenty-nine hundredths seconds.
### How would you write the word form of 1 minute and 38.29 seconds?
1 minute, thirty-eight and twenty-nine hundredths seconds.
### How do you write 1 minute an 38.29 seconds in decimal form?
1.6381666... minutes
### Write the word form for 1 min 38.29 seconds?
One minute and thirty-eight and twenty-nine hundredths seconds.
### How do you write the word form for 1minute 38.29 sec?
One minute, thirty-eight and twenty-nine hundredths seconds.
### How do you write the word form for 1minute and 38.29 sec?
One minute, thirty-eight and twenty-nine hundredths seconds.
### How to write the word form of 1 min 38.29secs?
One minute, thirty-eight and twenty-nine hundredths seconds.
### How do you write the word form of 1 min 38.29secs?
One minute, thirty-eight and twenty-nine hundredths seconds.
### What fraction is 9 seconds in a minute?
One minute is 60 seconds. 9 seconds is 9/60 of a minute. In simplest form it is 3/20 of a minute.
### What is the word form for 1 minute 38.29 sec?
The word form for 1 minute 38.29 seconds is: one minute and thirty-eight and twenty-nine hundredths seconds.
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Chapters Links Problems Enroll for Updates Help
CHAPTER 5 PROBLEMS
Introduction Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 All problems
Home > Problems > Chapter 5
4. In question 3, we assumed that we already have a global alignment of a set of sequences so that a scoring matrix could be made from the alignment. Although we may know that a set of sequences has the same function, and thus should align, the sequences may vary so much that it is difficult to align them globally. In this case, we have to resort to a statistical analysis to find conserved patterns. The following problem goes through the first few steps required to find the best alignment by a statistical method. Students will need to study first the example of the expectation maximization algorithm in the text. Analyze the following ten DNA sequences by the expectation maximization algorithm. Assume that the background base frequencies are each 0.25 and that the middle three positions are a motif. The size of the motif is a guess that is based on a molecular model. The alignment of the sequences is also a guess. ```seq1 C CAG A seq2 G TTA A seq3 G TAC C seq4 T TAT T seq5 C AGA T seq6 T TTT G seq7 A TAC T seq8 C TAT G seq9 A GCT C seq10 G TAG A ``` To start the PSSM, make a table with three columns (position in motif) and four rows (1 for each base). Calculate the observed frequency of each base at each of the three middle positions in the alignment. Using the frequencies in the column tables, and the background frequencies, calculate the odds likelihood of finding the motif at each of the possible locations in sequence 5. Calculate the probability of finding the motif at each position in sequence 5. Calculate what change will be made to the base count in each column of the motif table as a result of matching the motif to the first position in sequence 5. This is usually a fractional number of one base. What other steps are taken to update or maximize the table values?
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# Need help in graph problem - Solved with approach defined - Google Kickstart Round H, Diagonal Puzzle
This is not necessarily a graph problem, but it can be solved using graphs. I was doing some practice on past google kickstart and found this problem.
Now, in the analysis section of this problem, one of the solutions is creating the graph where each node is a diagonal and each cell represents edge which connects them. I am not able to visualize this analogy and thus can’t able to proceed. It seems to be a really nice trick. Can you help me to understand this solution?
It would be nice if you can mention such problems which can be solved using such analogy.
==============================SOLUTION APPROACH ======================
I have been going over many articles for this problem. There are two approaches that can solve this given problem.
1. In order to arrive at this, it required huge pattern recognition and mathematics visually intuition. The Solution lies in that fact that “You are guaranteed that it is possible to make all the squares black.
It basically meaning, there is a mathematical relationship that exist over diagonals that while flipping never ended up revolving each other. (thus infinite loop if we simply do greedy - like when we got “W” simply rotate one of the diagonal and check for other “W”)
If we closely work over many examples then we wil find that it (this relationship among diagonals) depends over each other through each grid cell. (of course, one grid cell can be part of two diagonals, ). Example:
000
000
000
Now, If its “W” that meaning we have to flip one diagonal only (out of the two) to make it black. While, If its “B” already meaning, we wether never flip both diagonal or flip both diagonal(to retain its colour). In order to make number of flips minimum, we have to start from the diagnal which must have the highest impact.
That will be the middle diagonal and sub middle diagonal. We will then have 4 choices wether we flip one or both or nothing and look for White cells, in both these diagonal. If there is white, then we flip another diagonal which is perpendicular to current (these two longest diagonal). Again, there will be solution exist by doing this because as given in problem, You are guaranteed that it is possible to make all the squares black. In other words, intuitively things will get unfold once, you automate this technique as it will never ended up revolving and its will result in least as we are starting from diagnal which is having the highest impact(meaning containing highest number of grid cells hence the intersection point).
NOTE: I never think, that I can able to come up with this approach while competing in real time. That’s, why i need to learn this 2nd technique to encounter such type of question in future.
2. In this, you have to find a pattern. If you closely look at the examples given you will see, (as discussed in 1st approach above ) that for each cell two diagonal will be connected using this cell. And the fate of this cell(Colour of this cell) depends upon the flipping status of these two diagonals.
So the pattern here is:
If Current cell is “black”- we either flip both or none
If Current cell is “white”- we flip one of the two
Look at this in another way. If Cell is “black”, we want to behave these two diagonals in same way while if cell is “white” we want to behave these two diagonals in different way.
Any pattern yet?
Actually this is something similar to what we do in bipartite graphs. In bipartite graph, if current node is coloured lets say, “Colour 1” then all of its neighbour should have to be in different colour. Only the difference here, is that here the diagonal which is acting here as node while these grid cell is acting as edges. If this grid cell is “white” (meaning edge) then it means we want to have two of its endpoing to have different colour while if Cell is “black” (meaning that edge) we wants two endpoint to behave in similar fashion.
So, We create a graph in which nodes are the diagonals, and edges are the grid cell. A diagonal is connected with others by these cells. So total edges in this graph will be n*n. And we will do dfs staring from any random node(or diagonal). While dfs, we will look at the edge colour. if its “black” then we will colour node (other endpoint in consideration) in same colour, lets say “Colour 1” while if edge or cell is white then we will colour other endpoint diagonal or node with different colour lets say “colour 2”.
We will also make a count of total number of colour 1 and colour 2 that we did. And which ever is minium we will add that to the result.
One implementation problem, that I encouter is how to translate this diagonal in node. That can look into the property as below.
00 01 02
10 11 12
20 21 22
Take above diagonal, some of the indices are always 2. So if its below left to top right diagonal, then its indices sum is equal.
While,
00 01 02
10 11 12
20 21 22
While above diagonals, can be distinguies by the difference in the indices, as it will be same.
For Images, and Code.
Credit: https://zhuanlan.zhihu.com/p/94047090
1 Like
is it necessary to use google-chrome to login into https://codingcompetitions.withgoogle.com/ ? Because i can’t login, and it doesn’t show any kind of error.
Not necessary. I’m able to view it with FireFox!
Visible without chrome
@niks_vat Though I am not sure I think this is it.
Every cell has two diagonals passing through it. So, we can somehow create a graph with diagonals as nodes and connect those nodes with the other diagonal of every cell in that diagonal.
Hope this will be of some help to you
Thanks @satwik_bhv1. So it seems like we need to look at each cell(now edge) if its color white, meaning one of the diagonal crossing over it needs to be flipped(white->black). But if its colour is black, meaning whether no diagonal crossing over it needs to flip or both needs to be flipped. So now after this, how to proceed?
But there are no colored vertices in our graph! We only have white and black edges. And if we arbitrarily choose a white edge to toggle, how do we decide which vertex (diagonal) to flip?
I meant cell.
Yeah, that’s fine. But how do we choose which diagonal to flip?
Example:
BWB
BBW
BBB
Here choosing if we start out dfs by choosing one vertex, we’ll get count as 4 and for the other vertex it’ll be done with count = 1. I’m not able tackle this issue…
we start with a white cell and run two dfs on either diagonal as a starting vertex. minimum of both is the answer.
@aneee004 your problem only arises at the starting vertex and after that in the dfs you won’t get that issue because you already chosen a diagonal of that cell if its white no problem if its black call the dfs on other diagonal as a vertex
B W B
B B W
B B B
1. lets say, we chosen, (0,1) W lets, we did dfs on left diagonal. We will get
B B B
WBW
B B B
1. Now, again lets we choose, (1,0) W, and did dfs on above digonal we will get
B W B
B B W
B B B
Now, we arrive at same intial configuration, so didn’t we ended up getting in infinite loop. or am I missing something?
we will choose the other diagonal. we maintain visited at each cell for both left and right diagonals
Ok, so if we encounter any cell which is white in dfs, we need to check whether we already did dfs(visited) on this or not, despite whatever the configration of the matrix currently is?
yes
i think it’s better not to change the contents of the matrix. because we have the initial colour and number of flips. anyhow it all comes under the implementation part.
I am not getting the intuition that, lets we have arrive a vertex(W) (Currently not visited before) and did dfs on it, which will essentially change the configuration of the whole matrix. Now, lets say after doing some more dfs, we arrive at same vertex again(it got White again due to some other flipping of diagonal). But since, current configuration is different, we might get the result by doing just one more flip(as now configuration is changed)
That’s right. So we just try both paths and return the minimum?
yeah
Do you agree that we can flip the colour of a current cell only in two ways either left or right. retain the colour in two ways both or none.
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# Calculating Pond Volume
For ponds that are significantly square or rectangular, use this formula:
Length x Width x Average Depth x 7.5 = Gallons
Example:
A rectangular pond is 90' long by 40' wide with an average depth of 4'.
90 x 40 x 4 x 7.5 = 108,000 gallons of water
------------------------------------------------------------------------------------------------------------
For ponds that are significantly circular, use this formula:
Diameter x Diameter x Average Depth x 6 = Gallons
Example:
A circular pond is 125' across with an average depth of 5'.
125 x 125 x 5 x 6= 468,750 gallons of water
------------------------------------------------------------------------------------------------------------
For ponds that are significantly elliptical, use this formula:
Length x Width x Average Depth x 6 = Gallons
Example:
An elliptical pond 200' long by 80' wide with an average depth of 6'
200 x 80 x 6 x 6 = 576,000 Gallons of Water
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Home » Posts tagged 'Theorem'
Tag Archives: Theorem
Theorem of parallel axis
State the theorem of parallel axes for the moment of inertia.
Maulik
Gauss Theorem – Electrostatics
“WHAT IS THE MAGNITUDE OF ELECTRIC FIELD IN THE GAUSSIAN SURFACE OF A CUBE,AT ITS FACE ,AT ITS CENTRE, AT ITS CORNERS OR AT ANY OTHER POINT INSIDE THE CUBE. ALSO TELL ME THE THE WAY TO KNOW IT.”
Ans:
According to Gauss Theorem, the total electric flux through a closed surface,
The Gaussian surface is an imaginary surface. So, for calculating the electric field at a point using Gauss theorem, we have to imagine a Gaussian surface symmetric with the given charge distribution.
I have assumed that there is a point charge Q at the centre of the cube.
At its face (at a point on the face on the line radially outwards from the point charge at the centre of the cube and perpendicular to the face)
The distance is equal to half the length of side of the cube (Say L).
Therefore the electric field,
On the corner,
Calculate the distance from the centre of the cube to its corner and replace (L/2) in the above equation with that distance.
A question from Kinetic energy
A lorry and a car moving with the same KE are brought to rest having same retarding force .which one of them should come at rest at shorter distance.
Ans:
According to Work Kinetic Energy theorem
Both should stop travelling the same distance
(This was a quick answer without a deep thought. If the readers have a diffrent opinion, please comment)
A Question from Electric Flux and Gauss Theorem
How can you prove that dS in φ =E.dS is dS Cos θ if the area is tilted at an angle ? I need the mathematical Steps.
Ans;
Dear Zeenath,
Electric Flux is defined as the total no of field lines passing normal to the surface. So while calculating, we need to consider the area of the surface normal to the field lines only. That is why we take the dot product of E and ds, where ds is the area vector (not just the area: Area vector is a vector whose magnitude is equal to the area and dirtected normal to the surface.)
Then by definition of dot product,
dφ =E.dS = EdS cos θ
which gives the component of E and the component of area vector in the direction of E (When area vector is in the direction of E, the actual component of area is perpendicular to E)
Hope the matter is clear.
Mathew Abraham
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## Thursday, July 2, 2015
### Five Things I Learned at PCMI: Thursday 7/2/15
1. Jo Boaler at Stanford University a website called youcubed.com that provides resources and online courses for teachers, parents, and students with a focus on growth mindset in the mathematics classroom. The site focuses on practices that promote depth over speed, equitable group work, and math efficacy. There is also a free set of videos and lessons for the first week of school that are designed to promote growth mindset, introduce the 8 mathematical practices, and set the tone for the year. Take a look here: http://youcubed.org/
2. There are multiple ways to complete congruence proofs with transformations. You could use transformations of the entire plane to map one figure onto another, starting with one point. Next, assume that the other points do not line up, and use properties of your figure and of transformations to explain why they do. Of course, this proof can take many forms, but there is another method too. You can express the properties of a figure by using circles to preserve distance and rays to preserve angles to create a locus of points where the vertices of the figure could exist. If there is only one set of vertices, then the figures are congruent.
3. Although there is some debate, the term "constant of proportionally" should generally be reserved for relationships between a domain and a range, and has the form y = k x. On the other hand, "scale factor" refers to a relationship between two geometric figures. It should normally be presented as one number, not a ratio. However, it can be useful to use ratios to compare lengths within one geometric figure. For example, we could say that two triangles with sides 4:6:8 and 6:9:12 have a 3/2 or 1.5 scale factor. You can thank Bill McCullen for that.
4. I can construct the incircle of a triangle using Euclid: The Game, level 15. And yes, I already knew the properties of an incircle and incenters, but the construction takes an extra trick.
5. The mountains in Utah are amazing. We took a 6 mile hike that turned into an 8 mile hike to Middle Mountain in Wasatch Mountain State Park. The trail was heavily washed out and it was a bit steeper than anticipated, but the view from the top and the experience was worth it.
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# solve strategy for word problems
I would instruct the students to write that information down first and then put the equal sign. In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Step 6. Part c): Part c) is the algebraic equation that is needed to solve for x. Start with a fresh slate and begin to think positive thoughts like the student in the cartoon below. You may need to read the problem two or more times. Since then you’ve increased your math vocabulary as you learned about more algebraic procedures, and you’ve had more practice translating from words into algebra. If ever a student leaves out a question on an exam, you can be sure it would be a word problem. It may help to first restate the problem in one sentence, with all the important information. b) X = c) Equation d) Find x. e) Answer part a). Before students look for keywords and try to figure out what to do, they need to slow... 2. Find the number. All the words in the sentence after the verb were transcribed into an algebraic expression and placed on the right hand side of the equation. $3\enspace\Rightarrow$ three, $+\enspace\Rightarrow$ more than, $2b\enspace\Rightarrow$ twice the number of bananas. Translate into an equation. If my sister and I buy our mother a present, how much will each of us pay? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? This method works as long as the situation is familiar to you and the math is not too complicated. She had planned a lesson on word problems, so I could see the issue first-hand. Strategies for Solving Word Problems 1. When grading their homework or marking an exam paper, I would assign five marks for a word problem. Part of the reason for this is the student often has difficulty in deciding what steps to take to analyze and understand what the problem is about. Question ID 142694, 142722, 142735, 142761. The left hand side of the equation comes from all the words in the sentence that appear before the verb. Answer: a) What is the second even consecutive number? If we take control and believe we can be successful, we will be able to master word problems. of carrots for her horses. Nga’s car insurance premium increased by $\text{\60}$, which was $\text{8%}$ of the original cost. What was the original cost of the premium? They opened their textbooks to page 47 and she read a word problem out loud. Practice mindfulness with your attitude about word problems, Apply a general problem-solving strategy to solve word problems. When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Make sure you understand all the words and ideas. Translate into an equation. Part e): Using the value for x that they found in part d), students then used that information to answer the question asked in part a). Our website is made possible by displaying online advertisements to our visitors. The word problems applied math to everyday situations. This strategy will help you become successful with word problems. What is this plan for solving math word problems? Now we’ll develop a strategy you can use to solve any word problem. Included in this section would be a list of items and one of them would be equal to x. Divide both sides by $0.08$. a)? No matter what level of math, I have found the following method to be very successful when solving word problems. What is the second number? If she has 100 horses, how many pounds does each horse get?” “OK class, she has 40 pounds that she will give to each horse. CC licensed content, Specific attribution, $\color{red}{2}\cdot18=\color{red}{2}\cdot\frac{1}{2}p$, Let $b=\text{number of bananas}$, $11\enspace\Rightarrow$ The number of apples, $11\color{red}{-3}=2b+3\color{red}{-3}$, $\frac{8}{\color{red}{2}}=\frac{2b}{\color{red}{2}}$, Let $c=\text{the original cost}$. If students just gave me the correct answer without following the Five Step Plan, they would only receive one point for their answer. Students were required to write part e) in a full sentence. Choose a variable to represent that quantity. Here is a chart I would put on the board when teaching this strategy to my students. We’ll demonstrate the strategy as we solve the following problem. Usually this could be found in the sentence containing the question mark. Differentiate Word Problems. Solve the equation using good algebra techniques. If the question was stated as a command, for example, ‘Find the number.’ That would become the question to be written in part a). When it comes to word problems, a positive attitude is a big step toward success. If there are words you don’t understand, look them up in a dictionary or on the Internet. Read the positive thoughts and say them out loud. Here is a chart I would put on the board when teaching this strategy to my students. Please consider supporting us by disabling your ad blocker. The number of apples was three more than twice the number of bananas. Name what you are looking for. Read the Entire Word Problem. Check the answer in the problem and make sure it makes sense. Often finding the value of x is not the answer to the word problem. For a review of how to translate algebraic statements into words, watch the following video. How much should I tip the server at a restaurant? These word problem graphic organizers/mats help walk students through the thinking and analyzing process that is automatic for us. One of most dreaded assignments students have in math is solving word problems. Yash brought apples and bananas to a picnic. In the same manner that we teach students to comprehend texts, we should also teach them how to breakdown and analy… The world is full of word problems. Write Algebraic Expressions from Statements: Form ax+b and a(x+b). I call it the Five Step Plan. The students sat at round tables, and she stood at the front. No matter what level of math – pre- algebra, algebra I, algebra II, pre-calculus, calculus, trigonometry, or statistics, using the Five Step Plan helps students to discover exactly what information is given and what they need to find in order to answer a word problem. Yash brought $11$ apples to the picnic. Step 7. Part d): Students would then use the equation that they constructed in part c) and solve the equation for x. b) Let x = the number c) 6x = 4x + 4 d) 2x = 4. Do NOT follow this link or you will be banned from the site. The next period, I was scheduled to visit Ms. Hartwell’s class. Step 1. Answer the question with a complete sentence. Have you ever had thoughts like the student in the cartoon below? In the next example, we will apply our Problem-Solving Strategy to applications of percent. Add or Subtract, Divide or Multiply? What was the original price of the shirt? Example: A number multiplied by six is four more than four times the number. We have also translated English sentences into algebraic equations and solved some word problems.
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# How long would it take to triple an investment at 10 compounded annually?
Contents
## How long will it take a money to triple itself at 10% compound interest?
In what time does a sum of money triple? A sum of money triples itself in 15 years 6 months.
## How long does it take for an investment to triple compound interest?
hence to the nearest year, it will it take 18 years for an investment to triple, if it is continuously compounded at 6% per year.
## How long will it take for money to double itself at 10% compounded monthly?
In reality, a 10% investment will take 7.3 years to double ((1.107.3 = 2). The Rule of 72 is reasonably accurate for low rates of return. The chart below compares the numbers given by the Rule of 72 and the actual number of years it takes an investment to double.
## How long in years will it take your money to triple at an annual percentage rate of 6% compounded annually?
It will approximately take 18 years 10 months.
## How long will it take for an amount to become triple of itself at 20% per annum simple interest?
Answer Expert Verified
= 10 years . A sum becomes 3 times. Rate = 20% p.a.
## How long will it take your money to triple at a rate of 9%?
For example, with a 9% rate of return, the simple calculation returns a time to double of eight years. If you use the logarithmic formula, the answer is 8.04 years—a negligible difference. In contrast, if you have a 2% rate of return, your Rule of 72 calculation returns a time to double of 36 years.
## How long would it take to double your money in an account that paid 6% per year?
To use the Rule of 72 in order to determine the approximate length of time it will take for your money to double, simply divide 72 by the annual interest rate. For example, if the interest rate earned is 6%, it will take 12 years (72 divided by 6) for your money to double.
## How long will it take for \$7000 to double at the rate of 8?
The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.
THIS IS INTERESTING: Question: How does investment in human resource give higher return in future?
## How long in years and months will it take for an investment to double at 13% compounded monthly?
1 Expert Answer
13 = 5.33 years and ln(2)/. 15 = 4.62 years.
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2220hw6
# 2220hw6 - Math 2220 Section 5.3 Problem Set 6 Spring 2010 3...
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Unformatted text preview: Math 2220 Section 5.3 : Problem Set 6 Spring 2010 3 ex 6. Sketch the region of integration for the integral 0 1 2 dy dx , then reverse the order of integration and evaluate both integrals, the original one and the one with the order reversed. π /2 cos x 8. Do the same things for 1 sin x dy dx . 0 −x x2 −2 0 10. For the integral −2 (x − y ) dy dx reverse the order of integration to obtain a 1 x 2 2−x 0 sum of integrals, and evaluate the resulting integrals. 12. Reverse the order of integration in a single integral, and evaluate this integral. ππ sin x 16. Evaluate dx dy x 0y 2 1 y/2 0 0 sin x dy dx + 1 sin x dy dx to obtain 18. Evaluate 0 e−x dx dy 2 Section 5.4 : 2. Evaluate [0,1]×[0,2]×[0,3] (x2 + y 2 + z 2 ) dV 12. Integrate the function f (x, y, z ) = y over the region bounded by the plane x + y + z = 2 , the cylinder x2 + z 2 = 1 , and y = 0 . 14. Integrate f (x, y, z ) = z over the region in the first octant bounded by the cylinder y 2 + z 2 = 9 and the planes y = x , x = 0 , and z = 0 . 16. Integrate f (x, y, z ) = 3x over the region in the first octant bounded by z = x2 + y 2 , x = 0 , y = 0 , and z = 4 . 20. Find the volume of the region inside both the cylinders x2 + y 2 = a2 and x2 + z 2 = a2 . 1 1 0 0 36−4x2 −4y 2 5x2 x2 22. Change the order of integration of equivalent iterated integrals. 2 0 1 2 f (x, y, z ) dz dx dy to give five other √ 36−9x2 24. Consider the iterated integral 0 0 2 dz dy dx . (a) Describe the region of integration in R3 . (b) Set up an equivalent triple integral (c) Set up an equivalent triple integral 2 dz dx dy . Do not evaluate your answer. 2 dy dz dx . Do not evaluate your answer. ...
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# Types Of Quadrilaterals And Its Properties Pdf
File Name: types of quadrilaterals and its properties .zip
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Published: 30.04.2021
Power Thinking PowerPoint Document 3. And best of all they all well, most! Beginners to the answers allow you might encounter a scale match up the length, … Irregular Quadrilateral 7.
A polygon made using four lines, straight or slanted, and at some angle to each other come under the family of a quadrilateral. This structure is one of the primary lessons that beginners learn about as part of an early geometry subject. Teachers or instructors employ quadrilateral charts to familiarize young learners with the concept of a quadrilateral. Often I am asked to prepare quadrilateral charts to offer practice material to Geometry beginners. So, here are some very engaging ones made for your use. The study of quadrilaterals can tell us that the world is almost unimaginable without this polygon family.
A quadrilateral is a polygon in Euclidean plane geometry with four edges sides and four vertices corners. Other names for quadrilateral include quadrangle in analogy to triangle , tetragon in analogy to pentagon , 5-sided polygon, and hexagon , 6-sided polygon , and 4-gon in analogy to k -gons for arbitrary values of k. The word "quadrilateral" is derived from the Latin words quadri , a variant of four, and latus , meaning "side". Quadrilaterals are either simple not self-intersecting , or complex self-intersecting, or crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple and planar quadrilateral ABCD add up to degrees of arc , that is [2].
One special kind of polygons is called a parallelogram. It is a quadrilateral where both pairs of opposite sides are parallel. If we have a parallelogram where all sides are congruent then we have what is called a rhombus. The properties of parallelograms can be applied on rhombi. If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid.
## Free Printable Quadrilateral Classification, Properties & theorem flow Chart [PDF]
Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device! In this chapter, you will learn more about different kinds of triangles and quadrilaterals, and their properties. You will explore shapes that are congruent and shapes that are similar. You will also use your knowledge of the properties of 2D shapes in order to solve geometric problems.
In this tutorial on basic geometry concepts, we cover the types and properties of quadrilaterals: Parallelogram, rectangle, square, rhombus, trapezium. Do opposite sides in a quadrilateral have to be equal in order for it to have diagonals that are perpendicular? It is not correct. The diagonals are perpendicular if and only if all the sides are equal as in the case of a rhombus or a square. The diagonals will be perpendicular also if the pairs of adjacent sides are equal as in the case of a kite. However, in a kite the diagonals are perpendicular but the diagonals are not congruent.
To name different quadrilaterals. • To describe the properties of different types of quadrilaterals. • To work out Match each word to its meaning. Equal. Bisect.
Types of quadrilaterals are discussed here: 1. Parallelogram: A quadrilateral whose opposite sides are parallel and equal is called a parallelogram. Its opposite angles are equal. Here LMNO is a parallelogram.
Year 10 Interactive Maths - Second Edition. A quadrilateral is a closed plane figure bounded by four line segments. For example, the figure ABCD shown here is a quadrilateral. A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.
Therefore, identifying the properties of quadrilaterals is important when trying to distinguish them from other polygons. In this article, you will get an idea about the 5 types of quadrilaterals and get to know about the properties of quadrilaterals. The diagram given below shows a quadrilateral ABCD and the sum of its internal angles.
Его любимым развлечением было подключаться к ее компьютеру, якобы для того, чтобы проверить совместимость оборудования.
#### Types of triangles
Конечно, согласился. Вы же мой шеф. Вы заместитель директора АНБ. Он не мог отказаться. - Ты права, - проворчал Стратмор. - Поэтому я его и попросил.
- Она наклонилась и принялась рыться в сумке. Беккер был на седьмом небе. Кольцо у нее, сказал он. Наконец-то. Он не знал, каким образом она поняла, что ему нужно кольцо, но был слишком уставшим, чтобы терзаться этим вопросом. Его тело расслабилось, он представил себе, как вручает кольцо сияющему заместителю директора АНБ. А потом они со Сьюзан будут лежать в кровати с балдахином в Стоун-Мэнор и наверстывать упущенное время.
Могу я поинтересоваться, кто со мной говорит.
Никто даже не заподозрит, что эти буквы что-то означают. К тому же если пароль стандартный, из шестидесяти четырех знаков, то даже при свете дня никто их не прочтет, а если и прочтет, то не запомнит. - И Танкадо отдал это кольцо совершенно незнакомому человеку за мгновение до смерти? - с недоумением спросила Сьюзан.
О да, конечно, - медленно проговорила женщина, готовая прийти на помощь потенциальному клиенту. - Вам нужна сопровождающая. - Да-да.
Что-то шевельнулось в углу. Сьюзан подняла. На плюшевом диване, закутавшись в махровый халат, грелся на солнце Дэвид и внимательно за ней наблюдал. Она протянула руку, поманив его к. - Без воска? - тихо спросила она, обнимая .
## Tranpamenkei
Properties of Kites. In a kite,. 1. Two disjoint pairs of consecutive sides are congruent by definition. 2. The diagonals are perpendicular. 3. One diagonal is the.
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Home » Banking » Reasoning » Distance And Direction
Practice Questions
1.
From her house, Kimi went 15 km to the North. Then, she turned West and covered 10 km. Then, she turned south and covered 5 km. Finally turning to East, she covered 10 km. In which direction is she from her house? A.East B.West C.North D.South
Movements of Kimi are shown in the given figure.
Clearly, final position is T which is in the north of her house.
2.
From his house, Deepak went 25 km to North then, he turned west and covered 15 km. Then, turned south and covered 10 km. Finally, turning to East, he covered 15 km. In which direction is he from his house? A.North B.South C.West D.East
Show the figure,
Starting from the point A and passing through the points B, C and D, Deepak reaches the point E. Hence, he is in North direction from his house.
3.
Shyam goes to 5 km in the north from his school. Now, turning to the left, he goes to 10 km and again turn to left and goes to 5 km. How far he is from his school and which direction? A.10 km, South from school B.10 km, North from school C.10 km, West from school D.10 km, East from school
Let point A is the starting point I.e., school of Shyam
Point D is the ending point,
From, figure, AB = CD = 5 km
And AD = BC = 10 km
So, Shyam is 10 km far away from his school and in West direction from school.
4.
A direction pole was situated on the road crossing. Due to an accident, the pole turned in such a manner that the pointer which was showing East, started showing South. Sita, a traveller went to the wrong direction thinking it to be west. In what direction actually she was travelling? A.North B.West C.East D.South
The pointer which was showing last started showing South.
After accident, E – S and S – W
W – N and N – E
So, Pointer turned 90 degree clockwise.
5.
A and B start walking in opposite directions. A covers 3 km and B covers 4 km. Then, a turns right and walks 4 km while B turns left and walks 3 km. How far is each from the starting point? A.10 km B.8 km C.5 km D.4 km
Let A and B started from point O. A towards West and B towards East and at last they reached N and Q. Respectively,
ON = √[(NM)2+ (MO)2]
ON = √(42+ 32)
ON = 5 km
And,
OQ = √[(NQ)2+ (NO)2]
OQ = √(32+ 42)
OQ = 5 km
6.
A man walks 6 km south, turns left and walks 4 km, again turns left and walks 5 km. Which direction is he facing now? A.South B.North C.East D.West
Let the man starts from the point A and passing through,
B and C, he reaches D. Clearly, he is now facing North.
7.
Ram cycled 10 km southward from his home, turned right and cycled 6 km, turned right and cycled 10 km, turned left and cycled 15 km. How many kilometres will he have cycled to reach straight home? A.10 km B.21 km C.16 km D.20 km
Figure is drawn as,
Now, to go back home Ram has to cycle = ( ED + DA ) = (15 + 6) = 21 km
8.
One day, Ravi left home and cycled 10 km southwards, turned right and cycled 5 km and turned right and cycled 10 km and turned left and cycled 10 km. How many kilometres will he now have to cycle in a straight line to reach his home? A.10 km B.15 km C.20 km D.25 km
Now, distance Ravi to cycle to reach home is (ED + DA) = (10 + 5) = 15 km
9.
A man travels 4 km towards North, then travels 6 km towards East and further travels 4 km towards North. How far he is from the starting point? A.6 km B.14 km C.8 km D.10 km
Distance between the starting point and end point is AD.
Now, AD2 = AE2 + DE2
AD = √100 = 10 km
10.
A man is facing west. He turns 45 degree in the clockwise direction and then another 180 degree in the same direction and then 270 degree in the anti-clockwise direction. Which direction is he facing now? A.South B.North - West C.West D.South - West
The movement of man is shown in the figure below
From figure, finally, he is facing in the OS direction, which is South West.
httpswwwviagrapascherfrcomachat-sildenafil-citrate said: ( 23.11.2017. 10:12 )
Today I went to the beach front with my kids.
I found a sea shell and gave it to my 4 year old daughter and said You can hear the ocean if you put this to your ear. She
placed the shell to her ear and screamed. There was a hermit crab inside and it pinched her
ear. She never wants to go back LoL I know this is totally off topic but
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aftab qazi said: ( 18.03.2017. 09:12 )
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Very interresting
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namrata singh said: ( 13.10.2016. 14:40 )
good question
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# Java Program to Convert Kilogram to Ounce and Ounce to Kilogram
In the previous article, we have discussed about Java Program to Convert Kilogram to Metric Ton and Metric Ton to Kilogram
In this article we will see how to convert Kilogram to Ounce and Ounce to Kilogram by using Java programming language.
## Java Program to Convert Kilogram to Ounce and Ounce to Kilogram
Before jumping into the program let’s know the relationship between Kilogram and Ounce and how we can convert Kilogram to Ounce and vice versa.
Generally Kilogram and Ounce are used as unit in case of mass/weight measurement.
1 Kilogram = 35.274 Ounce
1 Ounce = 0.0283495 Kilogram
Formula to convert Kilogram to Ounce.
Ounce = Kilogram * 35.274
Formula to convert Ounce to Kilogram.
Kilogram = Ounce * 0.0283495 (OR) Ounce / 35.274
Let’s see different ways to convert Kilogram to Ounce and Ounce to Kilogram.
### Method-1: Java Program to Convert Kilogram to Ounce and Ounce to Kilogram By Using Static Input Value
Approach:
• Declare Kilogram and Ounce value.
• Then convert Kilogram to Ounce and Ounce to Kilogram by using the formula.
• Print result.
Program:
public class Main
{
public static void main(String args[])
{
//value of square kilogram declared
double kilogram = 1;
//value of ounce declared
double ounce = 1;
//converting kilogram to ounce
double o = kilogram * 35.274;
//converting ounce to kilogram
double kg = ounce * 0.0283495;
//printing result
System.out.println("Value of "+kilogram+" kilogram in ounce: "+ o);
System.out.println("Value of "+ounce+" ounce in kilogram: "+ kg);
}
}
Output:
Value of 1.0 kilogram in ounce: 35.274
Value of 1.0 ounce in kilogram: 0.0283495
### Method-2: Java Program to Convert Kilogram to Ounce and Ounce to Kilogram By Using User Input Value
Approach:
• Take user input of Kilogram and Ounce value.
• Then convert Kilogram to Ounce and Ounce to Kilogram by using the formula.
• Print result.
Program:
import java.util.*;
public class Main
{
public static void main(String args[])
{
//Scanner class object created
Scanner sc=new Scanner(System.in);
//Taking the value input of double variable kilogram
System.out.println("Enter value of kilogram: ");
double kilogram = sc.nextDouble();
//Taking the value input of double variable ounce
System.out.println("Enter value of ounce: ");
double ounce = sc.nextDouble();
//converting kilogram to ounce
double o = kilogram * 35.274;
//converting ounce to kilogram
double kg = ounce * 0.0283495;
//printing result
System.out.println("Value of "+kilogram+" kilogram in ounce: "+ o);
System.out.println("Value of "+ounce+" ounce in kilogram: "+ kg);
}
}
Output:
Enter value of kilogram:
2
Enter value of ounce:
20
Value of 2.0 kilogram in ounce: 70.548
Value of 20.0 ounce in kilogram: 0.56699
### Method-3: Java Program to Convert Kilogram to Ounce and Ounce to Kilogram By Using User Defined Method
Approach:
• Take user input of Kilogram and Ounce value.
• Call a user defined method by passing Kilogram and Ounce value as parameter.
• Inside method convert Kilogram to Ounce and Ounce to Kilogram by using the formula.
• Print result.
Program:
import java.util.*;
public class Main
{
public static void main(String args[])
{
//Scanner class object created
Scanner sc=new Scanner(System.in);
//Taking the value input of double variable kilogram
System.out.println("Enter value of kilogram: ");
double kilogram = sc.nextDouble();
//Taking the value input of double variable ounce
System.out.println("Enter value of ounce: ");
double ounce = sc.nextDouble();
//calling user defined method convert()
convert(kilogram, ounce);
}
//convert() method to convert kilogram to ounce and vice versa
public static void convert(double kilogram, double ounce)
{
//converting kilogram to ounce
double o = kilogram * 35.274;
//converting ounce to kilogram
double kg = ounce * 0.0283495;
//printing result
System.out.println("Value of "+kilogram+" kilogram in ounce: "+ o);
System.out.println("Value of "+ounce+" ounce in kilogram: "+ kg);
}
}
Output:
Enter value of kilogram:
20
Enter value of ounce:
5
Value of 20.0 kilogram in ounce: 705.48
Value of 5.0 ounce in kilogram: 0.1417475
Beginners and experienced programmers can rely on these Best Java Programs Examples and code various basic and complex logics in the Java programming language with ease.
Related Java Programs:
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Home > Standard Error > Def Standard Error Measurement
# Def Standard Error Measurement
## Contents
It can only be calculated if the mean is a non-zero value. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Standard errors provide simple measures of uncertainty in a value and are often used because: If the standard error of several individual quantities is known then the standard error of some BREAKING DOWN 'Standard Error' The term "standard error" is used to refer to the standard deviation of various sample statistics such as the mean or median. navigate here
We could be 68% sure that the students true score would be between +/- one SEM. Notice that s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} is only an estimate of the true standard error, σ x ¯ = σ n Statistical Notes. The True score is hypothetical and could only be estimated by having the person take the test multiple times and take an average of the scores, i.e., out of 100 times https://en.wikipedia.org/wiki/Standard_error
## Standard Error Of Measurement Vs Standard Error Of Mean
In this scenario, the 2000 voters are a sample from all the actual voters. American Statistical Association. 25 (4): 30–32. The smaller standard deviation for age at first marriage will result in a smaller standard error of the mean.
The standard error can be computed from a knowledge of sample attributes - sample size and sample statistics. n is the size (number of observations) of the sample. Student B has an observed score of 109. Standard Error Of Measurement Example The observed score and its associated SEM can be used to construct a “confidence interval” to any desired degree of certainty.
Perspect Clin Res. 3 (3): 113–116. Standard Error Of Measurement Calculator Standard error functions more as a way to determine the accuracy of the sample or the accuracy of multiple samples by analyzing deviation within the means. The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years. her latest blog The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE.
II. Standard Error Of Measurement Vs Standard Deviation If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean This is not a practical way of estimating the amount of error in the test. SEM SDo Reliability .72 1.58 .79 1.18 3.58 .89 2.79 3.58 .39 True Scores / Estimating Errors / Confidence Interval / Top Confidence Interval The most common use of the
## Standard Error Of Measurement Calculator
ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, David; Barr, Christopher; Çetinkaya-Rundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org ^ T.P. The mean of all possible sample means is equal to the population mean. Standard Error Of Measurement Vs Standard Error Of Mean The standard deviation is used to help determine validity of the data based the number of data points displayed within each level of standard deviation. Standard Error Of Measurement Formula Correction for finite population The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered
These assumptions may be approximately met when the population from which samples are taken is normally distributed, or when the sample size is sufficiently large to rely on the Central Limit check over here The standard error is an estimate of the standard deviation of a statistic. As will be shown, the mean of all possible sample means is equal to the population mean. They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg/dL). Standard Error Of Measurement And Confidence Interval
With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. For the runners, the population mean age is 33.87, and the population standard deviation is 9.27. The table below shows formulas for computing the standard deviation of statistics from simple random samples. his comment is here Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply.
Repeating the sampling procedure as for the Cherry Blossom runners, take 20,000 samples of size n=16 from the age at first marriage population. Standard Error Of Measurement Spss In each of these scenarios, a sample of observations is drawn from a large population. For example, the sample mean is the usual estimator of a population mean.
## See unbiased estimation of standard deviation for further discussion.
Educators should consider the magnitude of SEMs for students across the achievement distribution to ensure that the information they are using to make educational decisions is highly accurate for all students, About the Author Nate Jensen is a Research Scientist at NWEA, where he specializes in the use of student testing data for accountability purposes. Retrieved 17 July 2014. Standard Error Of Measurement Reliability Unfortunately, the only score we actually have is the Observed score(So).
As the reliability increases, the SEMdecreases. The unbiased standard error plots as the ρ=0 diagonal line with log-log slope -½. The mean age was 23.44 years. weblink Because the 5,534 women are the entire population, 23.44 years is the population mean, μ {\displaystyle \mu } , and 3.56 years is the population standard deviation, σ {\displaystyle \sigma }
Think about the following situation. The mean age was 33.88 years. The standard error is computed solely from sample attributes. In this scenario, the 2000 voters are a sample from all the actual voters.
The smaller the spread, the more accurate the dataset is said to be.Standard Error and Population SamplingWhen a population is sampled, the mean, or average, is generally calculated. National Center for Health Statistics (24). The standard error is a measure of variability, not a measure of central tendency. The proportion or the mean is calculated using the sample.
Population parameter Sample statistic N: Number of observations in the population n: Number of observations in the sample Ni: Number of observations in population i ni: Number of observations in sample The standard error of a proportion and the standard error of the mean describe the possible variability of the estimated value based on the sample around the true proportion or true For each sample, the mean age of the 16 runners in the sample can be calculated. Learn how MAP helps you prep Learn how Measures of Academic Progress® (MAP®) users can use preliminary Smarter Balanced data to prepare for proficiency shifts.
This lesson shows how to compute the standard error, based on sample data. Another estimate is the reliability of the test. The confidence interval of 18 to 22 is a quantitative measure of the uncertainty – the possible difference between the true average effect of the drug and the estimate of 20mg/dL. Standard error From Wikipedia, the free encyclopedia Jump to: navigation, search For the computer programming concept, see standard error stream.
Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". American Statistician. doi:10.2307/2682923. If σ is not known, the standard error is estimated using the formula s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} where s is the sample
Note: the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations: the standard error of the mean is a biased estimator
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Math 1554 Fall 2017 Course Information
# Linear Algebra, Math 1554 Fall 2017 Course Information
HW7 The quiz based on this homework will take place on October 25th (Wednesday).
Read section 4.9 of Lay, and handout on PageRank distributed on via email October 19th, and do the following problems: Handout do the exercise on the last page of the handout (page 10 in the scan), 4.9: 1, 3, 11, 13, 17ad (parts bc are not needed because any square matrix with a zero row has a nontrivial null space).
HW6 The quiz based on this homework will take place on October 11th (Wednesday).
Read section 3.3 of Lay, and do the following problems 3.3: 16, 18, 20, 32b.
1: Show that for any square matrix A the number det(A^T A) is nonnegative.
2: If Q is square matrix and Q Q^T=I, then |det(Q)|=1.
3: If det(A)=2, what is det(A^k) for a nonzero integer k?
Do not forget to do MyMathLab homework (due at midnight on Thursday, October 12th).
HW5 The quiz based on this homework will take place on October 4th (Wednesday).
Read sections 2.8, 2.9, 3.1, 3.2 of Lay. Do the following problems 2.8: 11, 21, 22, 32, 36; 2.9: 17, 18, 20, 21, 24; 3.1: 9, 15, 31, 32, 39, 40; 3.2: 20, 27, 28, 32, 34, 35, 36, 40. Do not forget to do MyMathLab homework (due at midnight on Tuesday, October 3rd).
HW4 The quiz based on this homework will take place on September 25th (Monday).
Read sections 2.1, 2.2, 2.3 of Lay. We skip section 2.4 (which is actually quite useful in applications but we have no time for that). Do the following problems 2.1: 15, 16, 19, 20, 21, 28; 2.2: 19, 20, 33; 2.3: 13, 14, 15, 20, 26, 34. Also do #22, 23, 24 from test 1 practice problem list. Do not forget to do MyMathLab homework (due at midnight on Monday, September 25th).
HW3 The quiz based on this homework will take place on September 13th (Wednesday).
Read sections 1.7, 1.8, 1.9 of Lay. We skip sections 1.6 and 1.10 (which contain easy applications of linear algebra). Do the following problems 1.7: 21cd, 22, 26, 28, 36, 37, 38; 1.8: 21ab, 25, 31, 34, 35; 1.9: 23, 24, 26, 29, 30, 35. Do not forget to do MyMathLab homework (due at midnight on Monday, September 11th).
HW2 The quiz based on this homework will take place on September 6th (Wednesday).
Read sections 1.3, 1.4, 1.5 of Lay. Do the following problems 1.3: 21, 23, 24, 25, 32; 1.4: 18, 23, 24, 30, 32. 1.5: 12, 14, 23, 24, 30, 32, 40.
There is no MyMathLab homework this week.
Announcements:
• As announced in class the PLUS sessions are to be held Monday and Wednesdays, 6-7pm in CULC 262 run by Prabhav Chawla, email: pc_1998 at gatech.edu.
HW1 The quiz based on this homework will take place on August 30th (Wednesday).
Read sections 1.1, 1.2 of Lay. Do the following problems 1.1: 18, 23, 24, 26; 1.2: 21, 22(except c and e), 24, 29, 30, 31.
Do not forget to do MyMathLab homework (due at midnight on Monday, August 28th).
• office hours in Skiles 240B: Monday 11:00-11:50am, Thursday noon-12:50pm, or by appointment (to be made via email).
• phone: (404) 385-0053 (please do not leave messages as I do not check voice mail).
• Email: ib at math dot gatech dot edu
This is the best way to contact me. Please include 1554 in the subject header. Please email me from the Georgia Tech address: this would ensure your message won't end up in the spam folder, and besides, I shall not discuss more private matters, such as grades, to someone with non Georgia Tech email address.
• Course homepage: http://www.math.gatech.edu/~ib/1554.html. T-square will be only used for storing the grades.
• Lectures:
• Sections B1-B4 will meet TR 8:00-09:15am, Boggs B9.
• Sections K1-K4 will meet TR 13:30-14:45pm, Clough 152.
Teaching Assistants by section - name, email and office hours in Clough 280:
• B1 Jieun Seong, jseong8 at gatech.edu, Monday 2-3pm in Clough 280.
• B2 Xiaofan Yuan, xyuan at gatech.edu, Thursday 5-6pm in Clough 280.
• B3 Jack Olinde, jolinde at gatech.edu, Monday 2-3pm in Clough 280.
• B4 Haodong Sun, haodongsun at gatech.edu, Wednesday 5-6pm in Clough 280.
• K1 Osama Ghani, osama.ghani at gatech.edu, Monday 1:15-2:15pm in Skiles 230.
• K2 Bhanu Kumar, bkumar3 at gatech.edu, Monday 1:10-2:10pm in Clough 280.
• K3 Timothy Kierzkowski, tkierzkowski3 at gatech.edu, Thursday 2-3pm and Wednesday from 5-6pm.
• K4 Soham Gadgil, sgadgil6 at gatech.edu, Wednesday 11am-noon in Skiles 230.
Content and Course Objectives: Linear Algebra concepts and methods are fundamental in many problems of Sciences and Engineering. The intellectual goal of the course is to teach you to solve specific problems in Linear Algebra, and help you understand the ideas behind the solutions. An education goal to to help you transition from "elementary" math based on drill and rote learning to "higher" math with more abstract ideas and methods. More details on the course content can be found in the Official Syllabus which we shall follow closely.
Prerequisites: The prerequisites are MATH 1113 (Precalculus) with Minimum Grade of D, or SAT Mathematics 600, or Converted ACT Math 600.
Textbook: Linear Algebra and its applications (5th Edition) by David C. Lay, Steven R. Lay, Judi J. McDonald. Henceforth we refer to the book as Lay.
MyMathLab (MML): Georgia Tech has its own MML custom package for Math 1554 which you need to buy. Do not buy MML from other vendors!
Here is how to register for MML: go to http://www.mymathlab.com, click on student registration, use your Georgia Tech email (or at least the one you will check regularly). Our MyMathLab course ID is belegradek53373. When creating Pearson Account please set your user ID to your T-square ID (as e.g. in gburdell3), which will ease converting grades from MML to T-square. To register you need either to pay or enter the access code purchased elsewhere (namely, bookstore or from the MML previous course at Georgia Tech). You can pay online with a credit card or PayPal. If you need time to arrange payment, you can delay paying for up to two weeks and pick temporary access at registration. Thus you can start using MML immediately. All grades earned with temporary access will convert seemlessly once you register.
MyMathLab (MML) fine print:
• We will mainly use MML as the online homework system. MML comes with an electronic version of Lay's textbook, Lay's study guide (as well as Thomas calculus and Thomas solution manual which we won't use but it is used in all our calculus courses if you take any within the next 18 months).
• The most frugal option is probably to buy MML at http://www.mymathlab.com while registering for MML (the latter you have to do anyway).
• If you need to won a hard copy of Lay's text, the best deal is probably to get it at the bookstore (book plus MML access code). The bookstore also has other packages, including a copy of Thomas's text.
• Should you prefer buy/rent used Lay's text, know that the 4th edition is not much different from the 5th one, and in any case you will have an electronic copy of the 5th edition within the MML package.
• This is probably not needed, but just in case the line-by-line registration instruction in pdf format are here.
• If you already have an account on MyMathLab using this combined textbook within the past 18 months, then you do not need to purchase a new code. Login to your account on MyMathLab, select the option to add a new course, and enter the MML course ID: belegradek53373.
• Pearson offers 24/7 support via chat, see here.
Some features of MyMathLab
• Each student gets individual problems, that is, the numbers differ but the problems are similar.
• In MyMathLab homework you have unlimited number of attempts but the problem will change to a similar one after 3 attempts (click in similar exercises to get another problem). Thus nothing stops you from getting 100% on each MyMathLab homework.
• I set up MyMathLab to accept unsimplified answers, namely 4/6 is as good as 2/3.
Homework will come in two varieties: computational (done and graded in MyMathLab) and theoretical (posted at the top of this webpage). Homework will be usually assigned weekly on Monday, and its theoretical portion will not be graded or collected.
Quizzes:
• There will be weekly quizzes in recitations, usually every Wednesday (but rare changes are possible due to holidays, and those changes will be announced on the course webpage.).
• Each quiz will consists of 1-2 problems from the theoretical homework assigned on Monday of the previous week.
• All quizzes are closed-book and closed-notes. No "help sheet" is allowed on quizzes. No electronics (such as calculators, computers, mobile devices, headphones) are allowed on quizzes.
• Three quizzes with the lowest score will be dropped.
• The quizzes will be graded by the Teaching Assistant (TA). All questions about grading quizzes should be first addressed to the TA. (I shall deal with whatever cannot be resolved by the TA).
• There will be no makeup quizzes (except for those who miss the quiz due to Georgia Tech sponsored event, see below). If your excuse for missing a quiz seems valid to me, then the other quizzes will be given higher weight. To arrange for this you need to contact me promptly (by email).
Tests:
• There will be three in-class midterms and one cumulative final.
• All tests are closed-book and closed-notes. A one-page (two-sided if needed) "help sheet" will be allowed on midterms; you may write/type anything there. A four-page "help sheet" (again, two-sided) is allowed on the final.
• No electronics (such as calculators, computers, mobile devices, headphones) are allowed on tests.
• There will be no makeup tests (except for those who miss the test due to Georgia Tech sponsored event, see below). If your reason for missing a test seems valid to me, then the corresponding part of the final will be given a higher weight; to arrange for that you must contact me as soon as possible. With rare exceptions acceptable reasons for missing a test are limited to illness, court appearance, and taking part in Georgia Tech events. In particular, the popular excuse ``alarm clock malfunction'' will not be honored.
• Midterms will be graded by the TA/Instructor team with a detailed grading key provided by the Instructor.
• Tests will be mainly based on theoretical homework.
• Midterm test dates: September 19th (Tuesday, in lecture), October 18th (Wednesday, in recitation), November 21st (in lecture, Tuesday before Thanksgiving).
• The final will take place in the lecture room at the following times:
• Sections B1-B4: December 12th (Tuesday) 2:50-5:40pm.
• Sections K1-K4: December 7th (Thursday) 2:50-5:40pm.
Make ups: students who miss a test/quiz due to Georgia Tech sponsored event (such as travel of student athletes) can get it made up. To do so they must bring me a note from appropriate Georgia Tech authority well before the test/quiz.
• MyMathLab homework is worth 10% of the final grade. Quizzes will count for 10%. Each of the midterms is worth 15%, and the final is worth 35%. No test score will be dropped.
• All grades will be recorded in T-square, but we will not use T-square for any other purpose.
• There will be no "curving" of the grades, that is, performance of your fellow students will have no affect on your grades. There is no set quota for the number of A's, B's etc, in particular, it is possible (though unlikely) that everybody will get an A.
Piazza is an online discussion forum where where you can ask other students (or me if I am available) any questions about the course. You may register here with your Georgia Tech email. The course is Math 1554 (Belegradek). Please be positive and constructive on Piazza. Your identity will be hidden from other students but not from me.
Random thoughts on how to do well in this course:
• Do all homework!
• Attend lectures and recitations; while some rare individuals can do well without going to class, there is a strong evidence that those who attend most lectures and recitations get a better grade.
• Join/form a study group: explaining ideas to others helps clarify them for yourself, not to mention that your peers may have something to teach you too, and most importantly to tell you when you are wrong.
• Always go to review sessions.
• Do not hesitate to ask questions, come to the office hours etc.
Needless to say that cheating will not be tolerated. Please report all cheating to me or your TA, and do so promptly.
• What constitutes cheating on a test or quiz? Examples are use of unauthorized materials, use of electronic devices, and getting outside help including talking, and looking in other students' papers.
• What constitutes cheating on MML homework? Examples are looking up your answers online and asking others to do homework for you. On other other hand, discussing MML homework problems in study groups is perfectly acceptable.
See the Georgia Tech Honor Code for your rights and obligations.
How to get help:
• any concerns should be promptly discussed with me. All feedback on teaching and administrative issues is appreciated.
• Please check out the School of Mathematics free tutoring center: Math Lab which will open one week after classes begin and closes the Thursday before finals. All those in the lab should be able to help you, but those teaching Math 1553 and Math 1554 should have everything on their fingertips.
Georgia Tech Disclaimer: THIS PAGE IS NOT A PUBLICATION OF THE GEORGIA INSTITUTE OF TECHNOLOGY AND THE GEORGIA INSTITUTE OF TECHNOLOGY HAS NOT EDITED OR EXAMINED THE CONTENT. THE AUTHOR(S) OF THE PAGE ARE SOLELY RESPONSIBLE FOR THE CONTENT.... Thank goodness!
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# Tangents
To circle with a radius of 41 cm from the point R guided two tangents. The distance of both points of contact is 16 cm. Calculate the distance from point R and circle centre.
Correct result:
d = 41.8 cm
#### Solution:
$41^2=c_1(c_1+c_2) \ \\ \dfrac{ 16^2}{4}= c_2 c_1 \ \\ \ \\ 41^2=c_1^2 + \dfrac{ 16^2}{4} \ \\ c_1 = \sqrt{ 41^2 - \dfrac{ 16^2}{4} }= 40.212\ cm \ \\ c_2 = \dfrac{ 16^2}{4} / c_1 = 1.592\ cm \ \\ d=c_1+c_2 = 41.8 \ \text{cm}$
We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!
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#### You need to know the following knowledge to solve this word math problem:
We encourage you to watch this tutorial video on this math problem:
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The lengths of the legs of the right triangle ABC are in ratio b = 2: 3. The hypotenuse is 10 cm long. Calculate the lengths of the legs of that triangle.
• Sphere in cone
A sphere is inscribed in the cone (the intersection of their boundaries consists of a circle and one point). The ratio of the surface of the ball and the contents of the base is 4: 3. A plane passing through the axis of a cone cuts the cone in an isoscele
• Dodecagon
Calculate the size of the smaller of the angles determined by lines A1 A4 and A2 A10 in the regular dodecagon A1A2A3. .. A12. Express the result in degrees.
| 818 | 2,922 |
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| 4.46875 | 4 |
CC-MAIN-2020-45
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latest
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en
| 0.859979 |
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