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From the sum $(\text{eq}5)+ (\text{eq}6) +(\text{eq}7)$, we get $$2(x^2 + y^2 + z^2) - 2(xy + yz + zx) = \frac{14}{a^2}$$ hence $$a^2-3b=\frac{7}{a^2}\qquad(\text{eq}8)$$ Subtracting $(\text{eq}8)$ from $(\text{eq}4)$, we get \begin{align*} &(2a^2-3b) - (a^2-3b) = 8-\frac{7}{a^2}\\[4pt] \implies\;&a^2 = 8-\frac{7}{a^2}... | {
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# Math Help - Logic
1. ## Logic
Express the propositions $p\implies q$ and $q\wedge p$ using only $\downarrow$ and $\sim.$
$p\downarrow q,$ where $p$ and $q$ are propositions, it's defined by the following truth table:
-------
Let $p_1,p_2,p_3,p_4,p_5,p_6$ be propositions such that $\left[ {(p_1 \veebar p_2 ) \imp... | {
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A=1, B=1, (A => B) = 1
A=0, (A => B) = 1
A=1, B=0, (A => B) = 0
For the first one, find the values of p1 to p4 such as the proposition is false. Then you can determine the table of truth of i. ii. and iii. according to the values of p1 to p4...
5. $p\downarrow q$ is $(\sim p)\wedge (\sim q)$; hence $q\wedge p$ is $(\... | {
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8. Hello, Krizalid!
Express the propositions $p\to q$ and $q\wedge p$ using only $\downarrow$ and $\sim$
where $p\downarrow q,$ is defined by the following truth table:
. . . $\begin{array}{cc|c} p & q & p \downarrow q \\ \hline
T & T & F \\ T & F & F \\ F & T & F \\ F & F &T \end{array}$
I don't know of any direct... | {
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${\neg P} \equiv {P \downarrow P}$
${P \wedge Q} \equiv {\left( {P \downarrow P} \right) \downarrow \left( {Q \downarrow Q} \right)}$
${P \vee Q} \equiv {\left( {P \downarrow Q} \right) \downarrow \left( {P \downarrow Q} \right)}$
${P \to Q} \equiv {\left[ {\left( {P \downarrow P} \right) \downarrow Q} \right] \downarr... | {
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. . Since the first part is false, the implication is true. | {
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# Nice Puzzle
#### Amer
##### Active member
Arrange the numbers: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
such that the summation of each two successive numbers is a complete square easy interesting question
#### MarkFL
Staff member
8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9
#### Bacterius
##### Well-known memb... | {
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#### Bacterius
##### Well-known member
MHB Math Helper
I solved it too using
a matrix (I named my matrix A) with $$\displaystyle A_ab=1 if a+b=perfect square$$
It's pretty much done once you have the matrix, you see that 8 as the be the first (or last) number.
I ended with the same solution as MarkFL, and I was won... | {
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n = 18 to n = 22 have no solutions.
n = 23 has the following solutions:
(22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2, 14, 11, 5, 20, 16, 9, 7, 18)
(18, 7, 2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 5, 20, 16, 9)
(2, 23, 13, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3, 22, 14, 11, 5, 20, 16, 9, 7... | {
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# What is the maximum y-value of the following function?
What is the maximum y-value of the following function?
$$y=8t - \frac{t^2}{2} -24$$
It can be done by using the parabolic equation , setting the equation is equal zero. But is there any other straight forward or shortest method to do this?
You can complete th... | {
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# How do I get Mathematica to write the coefficients of polynomial as fractions rather than decimal expansions? [duplicate]
I am trying to have Mathematica find the polynomial of degree $n$ in a single variable which fits the given points $(x_1,y_1),\dots, (x_{n+1},y_{n+1})$. In the examples I am concerned with, the p... | {
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(* 10. +16.5 x+7.5 x^2+1. x^3 *)
(* 10+(33 x)/2+(15 x^2)/2+x^3 *)
or use InterpolatingPolynomial:
InterpolatingPolynomial[{{0,10},{1,35},{2,81},{3,154}},x]
Expand @ %
(* 10+x (25+(-1+x) (17/2+x)) *)
(* 10+(33 x)/2+(15 x^2)/2+x^3 *) | {
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That is, if $$P$$ =$$[p_{ij}]_{m×n}$$ and $$Q$$ =$$[q_{ij}]_{r×s}$$ are two matrices such that$$P$$ = $$Q$$, then: Let us now go back to our original matrices A and B. That’s because their order is not the same. Hence, for a matrix A. To calculate the transpose of a matrix, simply interchange the rows and columns of th... | {
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reserved. C++ Program to Find Transpose of a Matrix. Join our newsletter for the latest updates. In this program, the user is asked to enter the number of rows r and columns c. Their values should be less than 10 in this program. Transpose of a matrix: Transpose of a matrix can be found by interchanging rows with the c... | {
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is A = B? Commands Used LinearAlgebra[Transpose] See Also LinearAlgebra , Matrix … We can treat each element as a row of the matrix. For the transposed matrix, we change the order of transposed to 3x2, i.e. Transpose of a matrix can be calculated by switching the rows with columns. the screen. The algorithm of matrix t... | {
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Himalayan Quail Habitat, Juniper Name Popularity 2019, Wcvh4800k2ww Stacking Kit, Vendakka Theeyal Kerala Style, Prentice Surname Scotland, Johns Hopkins Coronavirus Map, Which Diagram In Uml Emphasizes The Time-ordering Of Messages, Ath-ad900x Vs Ath-ad700x, Egyptian Symbols Ankh, Rolling Stones Sesame Street, | {
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# Find time, when angle between the minute and hour hands are given
The question which I was trying to solve is:
Find the time between four and five o' clock when the angle between the hour hand and the minute hand is $78^\circ$.
My approach:
At four o' clock, the hour hand and the minute hand of the clock will be ... | {
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• Are you saying that there can be two instances between 4 o' clock and 5 o' clock when the separation between both the hands will be $78^\circ$? Is my solution wrong or just an alternative? – NINCOMPOOP Mar 1 '17 at 19:03
• Per Hour: $$\text{H}=\frac{360^{\circ}}{12\space\text{hours}}=30^{\circ}\text{/}\space\text{hou... | {
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# Bag A Contains 2 white and 3 black balls, Bag B Contains 5 white and 7 black balls. Ball is selected at random and is black.
Bag A Contains 2 white and 3 black balls, Bag B Contains 5 white and 7 black balls. Ball is selected at random and is black. What is probability that the ball came from bag A?
Here's my appro... | {
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Let $S_i$ be the event that you selected bag $i$; $W,B$ that you selected a white or black ball. I assume that be 'randomly selected' they selecting either bag is equally likely. Then you are being asked \begin{align*} P(S_A|B)&= \frac{P(S_A,B)}{P(B)}\tag 1\\ &=\frac{P(B|S_A)P(S_A)}{P(S_A,B)+P(S_B,B)}\tag 2\\ &=\frac{P... | {
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How to do exponential growth and decay word problems
## How to do exponential growth and decay word problems | {
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" The little "o" is a zero for time = 0 when you start. 33(1. 4: Exponential Growth and Decay Practice HW from Stewart Textbook (not to hand in) p. Determine the value of . Exponential Growth And Decay Graphs. The meaning of doubling time and half-life. Find a bank account balance if the account starts with $100, has a... | {
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exponential, Graphing exponential functions, 4 1 exponential functions and their graphs, Exponential growth, 1 exploration identifying graphs of As a rule, the formatting styles are standardized and accepted by the majority of educational and research solving exponential growth and decay problems institutions. In this ... | {
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it is growth or decay change, use the form The variable b represents the growth or decay factor. * time is usually in hours or years Let's just do one -- they're really easy! In 1950, the world's population was 2,555,982,611. Dylan invested$600 in a savings account at a 1. 314 Start with the basics and move up the expo... | {
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is that they are so useful in real applications: population growth, exponential decay, and compound interest. Whenever an exponential function is decreasing, this is often referred to as exponential decay. Find a bank account balance if the account starts with $100, has an annual rate of 4%, and the money is left in th... | {
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and Decay Word Problems Find a bank account balance if the account starts with$100, has an annual rate of 4%, and the money left in the account for 12 years. x 0 is the initial value at time t=0. Two word problem examples: one about a radioactive decay, and the other the exponential growth of a fast-food chain. In fact... | {
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we also have been noticed that 23 Exponential Growth And Decay Worksheet Answers is being just about the most popular field on the subject of document template sample right now. Jan 2007 26 0. 4) The half-life of a medication is the amount of time for half of the drug to be eliminated from the body. Note that we studie... | {
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Solving Problems Using the Exponential Decay Model — Solving Word Problems Using the Exponential Decay Model. Example: Atmospheric pressure (the pressure of air around you) decreases as you go higher. Model solving convex optimizations solving them, you can fix that problem to solve various equations logarithmic and ex... | {
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Academy is a 501(c)(3) nonprofit organization. May 16, 2009 · I need help answering this exponential growth/decay question. The value of the van depreciates at a rate of 9% each year. of exponential growth and decay and we used different bases Do you remember the formula to calculate compound interest? To solve problem... | {
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math notes on word problems involving exponential growth and decay. I'm just going to change the letters a little: The is pronounced "P not. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. On... | {
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and’re trying to get a remedy to a issue and data available. Exponential functions are used to model relationships with exponential growth or decay. Interpret and rewrite exponential growth and decay functions. 180 = 90*e^(3k) 2 = e^(3k) take the natural log (ln) of both sides. Exponential Growth & Decay Word Problems ... | {
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rate when r<0, in percent. And please add step by step instructions on how you answered it please. Walk around and check their work, ask questions. Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time. 5 • o. Keep the answer exact or give decimal approx... | {
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Lesson 13. The equation for "continual" growth (or decay) is A = Pe rt, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). Solving Word Problem... | {
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Growth and Decay Algebra 2 Consider this situation: You decided to purchase a car for$28,500. The half-life of Advil or ibuprofen is represented by the equation, where R is the amount of Advil remaining in the body, M is the initial dosage, and t is time in hours. Properties of Exponential Functions (EdPuzzle Video) Pr... | {
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in the town of Huntersville is presently 38,300. 2) Get the logarithms of both sides of the equation. Apr 28, 2012 · Best Answer: In exponential growth, the quantity at time t from the initial measurement is given by: Q(t) = Q(0)e^(kt) where k is a constant. y = 80 Jun 18, 2018 · Truly, we have been noticed that 11 Exp... | {
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can't get through the water. • If you’re given a number that’s used for repeated multiplication, multiplier = that number. In exponential growth, the quantity at time t from the initial measurement is given by: Q(t) = Q(0)e^(kt) where k is a constant. Aug 27, 2016 · Growth and exponential growth and problems; solving t... | {
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equations, Exponential growth and decay work, Exponential word problems, Exponential Generally, in exponential growth and decay problems, you will use an equation that looks like: € f(x) = (starting amount) ⋅ (multiplier)x There are several ways to find the multiplier, depending on the way that the change is described ... | {
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4 The population of a certain city was 292000 in 1998, and the observed relative growth rate is 2 Section 8. This little section is a tiny introduction to a very important subject and bunch of ideas: solving differential equations. Section 8. Price is one of the main advantages that distinguish our company solving expo... | {
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problem solving solve word problems, amount of money invested, and a larger investment grows faster. steepness. An example of an exponential growth word problem is the following: '$1000 is invested at 9% interest compounded annually. The exponential decay of an element is given by the function: In this function, is the... | {
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- E3 - MrArt" If you look for word problems about exponential growth or decay, in text books or on the internet, certain settings arise again and again: bank accounts, populations of animals, people or bacteria, and. !Writetheexponential!decayfunctionfor!a200 Vmg! Oct 25, 2016 · After tackling some word problems, we di... | {
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to almost worthless? Well, it has something to do with what's to solve exponential growth and decay word problems. You can use any bases for logs. Solve word problems requiring exponential models. Four variables (percent change, time, the amount at the beginning of the time period, and the amount at the end of the time... | {
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# Summation of arctangents
#### Pranav
##### Well-known member
Problem:
Evaluate
$$\lim_{n\rightarrow \infty} \left(\sum_{r=1}^n (\arctan(2r^2))-\frac{n\pi}{2}\right)$$
Attempt:
I tried evaluating the summation but couldn't. Had the problem involved $\arctan(1/(2r^2))$, I could rewrite it as
$$\arctan\left(\frac{2r+... | {
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http://mathhelpboards.com/math-notes-49/series-inverse-functions-8530.html
... we have $\displaystyle G(u,v) = \frac{u - v}{1 + u\ v}$ and choosing the sequence $c_{k}= \frac{k}{k+1}$ which is stricktly increasing and tends to 1, we obtain $\displaystyle G(c_{k},c_{k-1}) = \frac{1}{2\ k^{2}}$ and finally...
$\display... | {
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scale a number between a range [duplicate]
I have been trying to achieve a system which can scale a number down and in between two ranges. I have been stuck with the mathematical part of it.
What im thinking is lets say number 200 to be normalized so it falls between a range lets say 0 to 0.66 or 0.66 to 1 or 1 to 1.... | {
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3. Multiplying this by $(t_{\text{max}}-t_{\text{min}})$ maps $m$ to $[0,t_{\text{max}}-t_{\text{min}}]$.
4. Finally, adding $t_{\text{min}}$ shifts everything and maps $m$ to $[t_{\text{min}},t_{\text{max}}]$ as desired.
• Great explanation! I'm in a scenario, where I want to minimize $m$ and want $m$ to take the va... | {
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Note that each component of the moment of inertia tensor can be written as either a sum over separate mass elements, or as an integral over infinitesimal mass elements. Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational m... | {
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beam deflection and shaft torsion problems. Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects) with an invariant circular cross section and no significant warping or out-of-plane deformation. The Moment of Inertia for a thick-walled Cylindric... | {
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uniform cylinder. In this lesson, we will derive the moment of inertia of a semicircle about an axis perpendicular. Polar moment of inertia (denoted here as I p ) can also be found by summing the x and y planar moments of inertia (I x and I y ). 9 Thin walled hollow sphere rotating about the central axis. The moment of... | {
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about a particular axis. Here are some of the most common moments of inertia: Solid cylinder or disk of radius r rotating about its axis of symmetry: Hollow cylinder of radius r rotating about its axis of […]. It is required to compute the twist of a beam subjected to a torque. However, the moment of inertia I looked u... | {
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these masses are closer to the axis than the full radius of the cylinder, makes it so that the total moment of inertia is 1/2 the total mass of the cylinder times the total radius of the cylinder squared. Numbers can still be entered in these cells but are not necessary. A quantity expressing a body's tendency to resis... | {
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inertia synonyms, moment of inertia pronunciation, moment of inertia translation, English dictionary definition of moment of inertia. If an object has non-uniform. It is the rotational analogue to mass. When should the Effective Moment of Inertia be used? When calculating deflection? for concrete beams, if the Cracking... | {
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of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. In the principal axes frame, the moments are also sometimes denoted , , and. 030 m pivots on a frictionless bearing. Define moment of inertia. Even moments of inertia for more complicated solids can... | {
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object has. In the preceding subsection, we defined the moment of inertia but did not show how to calculate it. of an object to be the sum of. The matrix of the values is known as the moment of inertia tensor. Rotational Inertia or Moment of Inertia If we have a rigid body rotating at an angular speed Ω about a fixed a... | {
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the distance from the rotation axis. The moment of inertia, I, is the rotational equivalent of mass. Facts regarding rolling motion and the moment of inertia are covered by this quiz. Derivation of moment of inertia of an uniform solid sphere. Moment of inertia is an important concept for JEE exam. It is the rotational... | {
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Education; Show more Show less. As expected the moment of inertia of the disk is smaller than the one of the ring. However, if we found the moment of inertia of each section about some. half the value of the moment of inertia about the central axis to the value of the moment of inertia about the base plane. Lecture Not... | {
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denoted with the letter J, instead of I, but its units are the same as those for planar moment of inertia: m 4 or in 4. The moment of inertia of an object is a numerical value that can be calculated for any rigid body that is undergoing a physical rotation around a fixed axis. Rotational inertia. An object that is rota... | {
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Now, for the above figure we have the axis 0 given and hence we can calculate the moment of area by summing together l 2 dA for all the given elements of area dA in the yellow region. Rolling Rod, Cylinder or Disc Measure the radius of the object from the center to the edge in centimeters; enter this figure into the ca... | {
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is displayed and can be inserted anywhere in the drawing. As a preliminary, the volume of the cone is. Observe from figure 2, that this moment of inertia has been calculated about #z# axis. This article will discuss the concept of the area moment of inertia and polar moment of inertia and their application in practical... | {
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(Moment of Inertia) - I - is a measure of an object's resistance to change in rotation direction. Note that for an object where the mass is all concentrated at the same distance from the axis of rotation, such as a small ball being swung in a circle on a string, the moment of inertia is simply MR 2. Moment of inertia s... | {
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not $1/4$). Moment of Inertia (I) Calculator Common Shapes Rotational Inertia Calculator getcalc. As a preliminary, the volume of the cone is. The particular skills you will need are in establishing. Moment of inertia (), also called "angular mass" (kg·m 2), is the inertia of a rotating body with respect to its rotatio... | {
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of inertia about transverse (perpendicular) axis passing through its center. Moment of inertia, also known as rotational inertia, is analogous to the inertia of linear motion. Problem 817 Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube... | {
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is a physical quantity which describes how easily a body can be rotated about a given axis. What is the formula for calculating the moment of inertia (about the central axis) of an annular cylinder? Again, this will be in terms of the dimensions of the cylinder. So what I'd like you to do is, for the cylinder, I'd like... | {
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shape and its relative point of rotation - Radius of Gyration Sponsored Links Mass Moment of Inertia (Moment of Inertia) - I - is a measure of an object's resistance to change in rotation direction. Calculate the angular acceleration of the cylinder if a mass of 0. Moment of Inertia and Radius of Gyration Moment of Ine... | {
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(about the central axis) of an annular cylinder? Again, this will be in terms of the dimensions of the cylinder. 790 kg is attached to the string wrapped around the pulley. Find moment of inertia of a uniform hollow cylinder Home Problems and Answers Classical Mechanics Find moment of inertia of a uniform hollow cylind... | {
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known as the moment of inertia tensor. Moment of Inertia - Conversion Moment of inertia, also called rotational inertia, mass moment of inertia, or polar moment of inertia of mass, is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation. Second... | {
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5. For basic shapes there are tables that contain area moment of inertia equations which can be viewed below. In the preceding section, we defined the moment of inertia but did not show how to calculate it. When should the Effective Moment of Inertia be used? When calculating deflection? for concrete beams, if the Crac... | {
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mass. Area Moment of Inertia: The second moment of the area around a given axis is called the area moment of inertia. Moments of Inertia by Integration. The units of the area moment of inertia are meters to the fourth power (m^4). In its inertial properties, the body behaves like a circular cylinder. | {
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# Thread: Combination problem involving book selling.
1. ## Combination problem involving book selling.
16. A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How many choices are possible if
(b) the books are to be on different subjects?
So for the first pass one can choos... | {
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In any of the three cases the number outcomes is dependent on the pool sizes. So, taking the first case as an example, the math pile has 6 books and the science pile has 7 books; meaning that there are 6 different possibilities for the first pass and 7 possibilities for the second pass. Or, 6*7, outcomes.
Finally sinc... | {
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# Integrating over a hypercube, not a hypersphere
Denote $$\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$$ be an $$m$$-dimensional cube.
It is all too familiar that $$\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$$.
QUESTION. If $$\Vert\cdot\Vert$$stands for the Euclidean norm, th... | {
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More generally, we can consider
$$C_{p-1}(-p)=\int_{[0,1]^{p-1}}(1+|\vec{r}|^2)^{-p/2}\,d\vec{r}=\frac{1}{p}\,{\rm Res}_p=\frac{1}{p} \frac{1}{2^{p-1}}\frac{\pi^{p/2}}{\Gamma(p/2)}.$$ This formula holds for even and odd $$p\geq 2$$, as surmised by Gro-Tsen in a comment.
There are similarly remarkable hypercube integr... | {
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This derivation obviously holds whenever $$2n-1$$ is a positive integer.
More generally, in view of Bernstein's theorem on completely monotone functions, one can quite similarly express the box integral $$$$\int_{I^n} g(\|x\|^2)\,dx$$$$ as an ordinary integral over $$[0,\infty)$$ -- for any completely monotone functio... | {
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# Solving “coin exchange” for coins of power values by greedy algorithm
When solving the problem of coin exchange by greedy algorithm, why will we will always have the correct result if the coin values are $$1, a, a^2, \cdots, a^n$$, where $$a\ge 2$$ and $$n\gt 0$$?
For example, if $$a = 3$$, $$n =3$$, we get the fol... | {
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Here is the general theorem.
(Greedy algorithm works for power coins) Let $$a\ge2$$ and $$n\ge0$$. If $$e_na^n+(e_{n-1}e_{n-2}\cdots e_0)_a=\sum_{i=0}^nc_ia^i$$, then $$\sum_{i=0}^ne_i\le\sum_{i=0}^nc_i$$.
Proof by mathematical induction on $$n\ge0$$.
The base case, when $$n=0$$ is trivial.
Suppose it is true for s... | {
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## Multivariable calculus
### Unit 1: Lesson 2
Vectors and matrices
# Visualizing matrices
Lea... | {
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What is the action of a matrix? What does a matrix look like? These questions may seem nonsensical, but we will answer them both by visualizing how 2, times, 2 matrices move the 2D plane.
Here is a drawing of the plane, along with the unit vectors start color #11accd, \imath, with, hat, on top, end color #11accd and st... | {
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The unit vector start color #ca337c, \jmath, with, hat, on top, end color #ca337c didn't move because it started at start color #ca337c, left parenthesis, 0, comma, 1, right parenthesis, end color #ca337c. The unit vector start color #11accd, \imath, with, hat, on top, end color #11accd moved upward one unit, and this ... | {
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To sum up, the action of a matrix is to move the entire grid. We can understand it by thinking about how it moves the unit vectors. We can visualize what this looks like by drawing a modified 2D grid.
These ideas extend into three dimensions as well. The third row of the matrix contains z-coordinates for all the unit v... | {
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We already know how a given matrix moves the unit vectors start color #11accd, \imath, with, hat, on top, end color #11accd and start color #ca337c, \jmath, with, hat, on top, end color #ca337c (just look at the columns), but how can we find where a matrix moves any arbitrary vector? Let's consider a specific example u... | {
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Here's what the whole process looks like in symbols.
\begin{aligned} A \left[ \begin{array}{c} \goldD{1} \\ \goldD{2} \end{array} \right] &= A\left( \goldD{1} \left[ \begin{array}{c} \blueD{1} \\ \blueD{0} \end{array} \right] + \goldD{2} \left[ \begin{array}{c} \maroonD{0} \\ \maroonD{1} \end{array} \right] \right) \\ ... | {
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Problem 3
Let $B = \left[ \begin{array}{cc} \blueD{0} & \maroonD{2} \\ \blueD{1} & \maroonD{-1} \end{array} \right]$.
Where does B move the vector $\left[ \begin{array}{c} 3 \\ 1 \end{array} \right]$?
$B \left[ \begin{array}{c} 3 \\ 1 \end{array} \right] = ($
comma
right parenthesis
Like in the section above, the idea... | {
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With the perspective of matrices as movement, we have the tools to understand what it means to multiply two matrices. The core idea is composition.
\begin{aligned} A &= \left[ \begin{array}{cc} \blueD{1} & \maroonD{1} \\ \blueD{0} & \maroonD{1} \end{array} \right] \\ \\ B &= \left[ \begin{array}{cc} \blueD{0} & \maroon... | {
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$\left[ \begin{array}{cc} \blueD{1} & \maroonD{1} \\ \blueD{0} & \maroonD{1} \end{array} \right] \left[ \begin{array}{cc} \blueD{0} & \maroonD{-1} \\ \blueD{1} & \maroonD{0} \end{array} \right] = \left[ \begin{array}{cc} \blueD{1} & \maroonD{-1} \\ \blueD{1} & \maroonD{0} \end{array} \right]$
To conclude, we can think ... | {
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## What's next
Now that we have a solid grasp of how matrices move space, we are in a prime position to understand the final concept that we'll cover in this review series: the determinant.
## Want to join the conversation?
• does the explanation for a general formula of 2x2 matrix multiplication skip steps of exzpl... | {
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# 6 fair coin flips: probability of exactly 3 heads
When a certain coin is flipped, the probability of heads is $0.5$. If the coin is flipped $6$ times, what is the probability that there are exactly $3$ heads?
The answer is $\frac5{16}$. I wonder why it isn't $\frac12$. Since a fair coin flip results in equally like... | {
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So the answer is $20/64 = 5/16$.
The error you made is thinking that "number of outcomes with exactly 3 heads" is equal to "half of the total number of outcomes of 6 tosses." If this were the case then logically, "exactly 3 tails" must also be exactly half of the total outcomes. This means that "exactly 3 heads or exa... | {
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What's the probability of this happening? $\frac12$, either it will happen or it won't.
It is true that each sequence of heads and tails is equally likely to occur – with probability $\frac1{64}$, in this case. However, the number of those sequences having exactly three heads is not 32, but $\binom63=20$, which leads ... | {
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# Thread: Probability Distribution of Random Variables
1. ## Probability Distribution of Random Variables
Hi there,
Let me begin by introducing myself. My name is Zoheb and I'm a currently studying for a BSc Computer science degree in London (my hometown). I use to love maths (back in GCSE years) but then started to... | {
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So far, everything is straight forward. But my question/problem is: what steps are to be taken in order to obtain the distribution of $X$? A solution is given (below) as I stated, but I haven't quite worked out why it has ben done this way... Can someone please explain the methodology that is taken here?
There are $\b... | {
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# log2
Base 2 logarithm and floating-point number dissection
## Description
example
Y = log2(X) computes the base 2 logarithm of the elements of X such that ${2}^{Y}=X$.
example
[F,E] = log2(X) returns arrays F and E such that $X=F\cdot {2}^{E}$. The values in F are typically in the range 0.5 <= abs(F) < 1.
## E... | {
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Data Types: single | double
Complex Number Support: Yes
## Output Arguments
collapse all
Base 2 logarithm values, returned as a scalar, vector, matrix, or multidimensional array of the same size as X.
Mantissa values, returned as a scalar, vector, matrix, or multidimensional array of the same size as X. The values ... | {
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# Surface area of a sphere dilemma
I recently found that surface area of a sphere can't be found with the following method. What's the flaw in it? First, I have taken a very thin ring of thickness $$dx$$ at a distance of $$x$$ from the centre. Then, I integrated it, using substitution of $$x=R\sin\theta$$
But this is... | {
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• I am asking, why the other method is incorrect? – Yash Mittal Apr 11 at 17:51
• @YashMittal that is because $dx$ is not the right thickness of the ring. If you post your entire derivation, I can take a look at the incorrect areas. There could be more than one but the thickness is one of them. – user1952500 Apr 11 at ... | {
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# Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?
Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the
nth x = √(1st x √ 2nd x ... √nth x);
Then
$$\text{the "infinith" } x = x$$
Example:
$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$
Try it yourself, type calc in Google... | {
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Therefore, $\sqrt{xy} = y$, and solving we have $xy = y^2 \implies x = y$.
• Mh, be careful with that reasoning. $X = 1-1+1-1+1-1+...$, so $X=1-(1+1-1+1-...) = 1-X$, therefore $X=1/2$ May 30, 2014 at 22:06
• You are begging the question—which is fine, but it should be a little more explicit. The point is to suppose th... | {
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Another Approach
Not as rigorous, but perhaps more intuitive. Take the logarithm of both sides and we get \begin{align} \log\left(\sqrt{n\sqrt{n\sqrt{n\dots}}}\right) &=\frac12\left(\log(n)+\frac12\left(\log(n)+\frac12\left(\log(n)+\vphantom{\frac12}\dots\right)\right)\right)\\ &=\frac12\log(n)+\frac14\log(n)+\frac18\... | {
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Suppose $y = \sqrt{x\sqrt{x\sqrt{x\cdots}}}$. Then obviously $y^2=xy$, whence $y=x$ or $y=0$. But $y \gt 0$, so $y=x$. | {
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# $M$ is a point in an equalateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$.
$$M$$ is a point in an equilateral triangle $$ABC$$ with the area $$S$$. Prove that $$MA, MB, MC$$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713887451681,
"lm_q1q2_score": 0.8470020955475573,
"lm_q2_score": 0.8596637487122111,
"openwebmath_perplexity": 183.64810865693417,
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"tag... |
Reflect $$M$$ with respect to the sides of $$ABC$$. You get an hexagon whose area is $$2S$$: The hexagon can be decomposed as the union between $$A'B'C'$$ (whose side lengths are $$\sqrt{3}MA,\sqrt{3}MB,\sqrt{3}MC$$) and the isosceles triangles $$CA'B',BC'A',AB'C'$$. It follows that $$2S = \frac{\sqrt{3}}{4}(AM^2+BM^2+... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713887451681,
"lm_q1q2_score": 0.8470020955475573,
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Doing the same thing to $$a^4 + b^4 + c^4$$, we get \begin{align}a^4 + b^4 + c^4 &= \sum_{k=0}^2 (\rho^2 + 1 + (z\omega^k + \bar{z}\omega^{-k}))^2\\ &= \sum_{k=0}^2\left[ (\rho^2 + 1)^2 + (z\omega^k + \bar{z}\omega^{-k})^2\right]\\ &= 3(\rho^2 + 1)^2 + 6\rho^2\end{align} Combine these, we obtain
$$16S'^2 = 3(\rho^2+1)... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713887451681,
"lm_q1q2_score": 0.8470020955475573,
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"openwebmath_perplexity": 183.64810865693417,
"openwebmath_score": 0.955951452255249,
"tag... |
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