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09be2dd0-33ed-49f6-810e-2fa57362386f | # Infinite Expected Value of a Random Variable
## How can a positive random variable \(X\) which never takes on the value \(+\infty\), have expected value \(\mathbb{E}[X] = +\infty\)?
This question explores the concept of infinite expected values in the context of random variables. A random variable \(X\) has an infi... | {
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8089f149-fe80-475d-b98b-0bea01edd64d | # GMAT Club Forum Index
## GMAT Quantitative Section: GMAT Data Sufficiency (DS)
### Is \( k \) greater than 3?
**Question:**
Determine whether \( k \) is greater than 3 given:
1. \((k - 3)(k - 2)(k - 1) > 0\)
2. \(k > 1\)
**Discussion:**
**tonebeeze** (Current Student)
Joined: 19 Nov 2009
Posts: 127
GMAT 1... | {
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fe108331-4c9c-496a-82cf-15adeed85359 | # Sandwich Theorem
## Thread: Sandwich Theorem
### Post 1: Morgan82
- **Date**: Sep 9th, 2009, 05:58 PM
- **Status**: Offline, Member
- **Joined**: Jan 2008
- **Posts**: 118
My teacher just touched on this today. I am trying to figure out some problems. I will show you one, and show how I did it. Could you tell me i... | {
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be985844-09b9-4969-9998-1b5b336ddc3b | # How Much Sand Do I Need?
## Problem Statement
I have a 20' x 20' square. I want to fill it with 2.5" of sand. How many tons of sand am I going to need? Thank you.
### Calculation
You have 400 ft² of area and the depth is 2.5/12 or 0.208 ft, so you need 83.33 ft³.
Sand is sometimes sold by the ft³, by the yd³, o... | {
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3f008878-9eb6-447c-8e94-f98da091be62 | # A State Representative's Estimate on Women Representatives
A state representative wishes to estimate the mean number of women representatives per state legislature. A random sample of 17 states is selected, and the number of women representatives is shown. Based on the sample, what is the point estimate of the mean?... | {
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1da415d8-c7ca-4bcf-bd47-9d41e68362a4 | ## Similarity of Rectangles
The question at hand explores the condition under which rectangles \(ABCD\) and \(WXYZ\) are similar. To address this, we delve into the mathematical definition of similarity as it pertains to geometric figures, particularly rectangles.
### Definition of Similarity
Two geometric figures a... | {
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137fe6e9-1375-46cf-9d84-80d2f76cbe9f | # Abstract Algebra: Hypercomplex Numbers
## Hypercomplex Numbers
Hypercomplex numbers are numbers that use the square root of -1 to create more than one extra dimension. The most basic hypercomplex number is the quaternion, which consists of four dimensions and is commonly used in vector mathematics. Higher dimension... | {
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3507f600-8e68-42e4-939a-2eb81451c8f9 | # 4th Order Runge-Kutta Method for 2nd Order ODE
## Introduction
The 4th order Runge-Kutta method is a numerical technique used to solve ordinary differential equations (ODEs). This method provides a powerful tool for approximating solutions to differential equations that cannot be solved analytically.
## Mathematic... | {
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01098e42-34f7-4f42-99ef-37ab219da6de | # Why do I have to flip the equation in this problem?
### Problem Discussion
**JR Sauerland, Jul 27, 2015**
This problem involves similar triangles and trigonometric functions. The issue arises when the problem involves flipping the equation involving cosine, transforming it from adjacent over hypotenuse to hypotenu... | {
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23a417ff-1da4-488f-866b-6cff63e38609 | # Calculation Formula for Spiral Flight Form for Screw Conveyor
## Flat Form of a Flight of Screw Conveyor Formula
The flat form of a flight of screw conveyor formula provides the steps to calculate the flat form of a flight of screw. The formula calculates the outside diameter by subtracting the inside flight diamet... | {
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e2bef54a-b090-4f04-bde5-8580341a93fb | # Compare Fractions
## Compare \( \frac{4}{2} \) and \( \frac{7}{3} \)
1st number: \( 2 \frac{0}{2} \), 2nd number: \( 2 \frac{1}{3} \)
\( \frac{4}{2} \) is smaller than \( \frac{7}{3} \).
## Steps for Comparing Fractions
1. **Find the least common denominator (LCD) or least common multiple (LCM) of the two denomi... | {
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870bd011-0ca7-4c39-ba09-0467c76e750b | # Help (story problems)
## Problem 1
The length of a rectangular playing field is 5 ft less than twice its width. If the perimeter of the playing field is 230 ft, find the length and width of the field.
**Solution:**
Let \( l \) be the length of the rectangle and \( w \) be the width of the rectangle. Then the perim... | {
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3ad88629-6411-4e31-9e4d-991d3f73e337 | # How to Simplify Algebraic Expression
Posted on July 14th, 2012, 01:52 PM by daigo
To simplify the algebraic expression:
$$
\frac{1}{\left(\frac{x - 3}{x - 2}\right)^{\frac{1}{2}}} \cdot \frac{1}{2}\left(\frac{x - 3}{x - 2}\right)^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}}
$$
The steps are as follows:
1. Simplify... | {
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a2c224f6-a71e-461d-bb1d-611973e0261f | **Homework Help: Physics**
**Posted by Katie on Wednesday, March 28, 2012, at 12:44pm.**
A 390.0 g block is dropped onto a vertical spring with a spring constant \( k = 252.0 \, \text{N/m} \). The block becomes attached to the spring, and the spring compresses 0.15 m before momentarily stopping.
- While the spring ... | {
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1ace3284-3be9-45a6-83f4-717012ed5cac | # How To Calculate Cutting Length of Stirrups for Beam and Column
by Civil Lead
## Table of Contents
- How To Calculate Cutting Length of Stirrups
- Method for Calculating Cutting Length of Stirrups
- Important Formulas
- Example
- Standard Length of Hook and Bend
- Cutting Length of Rectangular Stirrup
- Cutting Len... | {
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d364b4dd-cc55-4023-9022-fe45955cd91d | # Middle School Archive: Probability
## Topics
- Probability
- See also Dr. Math FAQ: Probability
- Internet Library: Probability
### Selected Answers to Common Questions
#### Drawing Marbles [02/01/1997]
A jar contains 2 red, 3 blue, and 4 green marbles. Niki draws one marble from the jar, and then Tom draws a ma... | {
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42d19602-e318-4618-b3a8-c2ecdc240b00 | # Precalculus: Composition of Functions
**Posted by Julie on Sunday, February 10, 2013 at 10:35pm**
The application of one function followed by the application of a second function to the result of the first is called composition of functions. The two functions need not be inverses of each other. In a diagram, the ou... | {
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fc78db75-a33c-47b0-8182-240c486503c5 | **Homework Help: Math**
**Posted by shanquessa on Tuesday, November 4, 2008, at 1:18 pm.**
Hi, I am taking Algebra 2 and I need help with some substitution method problems:
1. \( x + 12y = 68 \)
2. \( x = 8y - 12 \)
3. \( x + 3y = 7 \)
4. \( 2x - 4y = 24 \)
5. \( 3a + b = 3 \)
6. \( 2a - 5b = -15 \)
7. \( -6 = 3x - ... | {
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fa291124-3b04-435a-86de-3d12807e8046 | # Combination / Permutation Question
**Question:**
A part has a code consisting of a digit, a letter, and a digit, such as 5C7 or 9G2. The factory made 5000 parts. Find the minimum number of parts that must have the same serial number.
**Discussion:**
Assuming that a part could be numbered 0U0, digits can be repeat... | {
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2df03547-c9e1-4770-8278-b353e13c2673 | # What is the Woman's Age?
A woman says, “If you reverse my own age, the figures represent my husband’s age. He is, of course, senior to me and the difference between our ages is one-eleventh of their sum.” The woman’s age is:
1. 23 years
2. 34 years
3. 43 years
4. 45 years
**Answer:** (4) 45 years
**Explanation:**... | {
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8dee0160-f2b1-4f5d-b50a-1e22becc9cc5 | # Equilateral Triangle Formula
**Post by Steve Jacobs, Mon May 04, 2009 1:33 pm**
Hi there,
How do you calculate the area of an equilateral triangle? What is the formula?
---
**Re: Equilateral Triangle Formula, Post by GMATguru, Mon May 04, 2009 1:50 pm**
There are two things that you need to know in order to ded... | {
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3b596828-943f-442b-916b-cbcf630660fc | # Problem 2441: Bernoulli's Equation
Bernoulli's equation is a principle of fluid dynamics that describes the behavior of a fluid under varying conditions of flow and height. It is expressed mathematically as:
\[
P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}
\]
Where:
- \( P \) is the static pressure within t... | {
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dfaebcdd-f274-4f4e-bcbb-8ea920ed3f9a | # Which Fraction Is Equivalent To
**September 10, 2020**
First, divide the numerator and denominator by 2. Multiplication and division work for obtaining equivalent fractions because multiplying and dividing by fractional forms of the number 1 (like 2/2, 3/3, etc.) give answers that are equivalent to the starting fra... | {
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97b93aa6-fdc3-4d13-8b84-7bbbc6d977f2 | # Which Day of the Week Was It?
Article by NRICH team
Published February 2011.
There is a mathematical method for working out what day of the week each particular date fell on back as far as 1700. For example, what day was it on Christmas Day in 1801? Neil Armstrong stepped onto the moon on July 21st, 1969. What da... | {
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148629ed-d3c4-40de-b29c-3e9c8e22b074 | # Showing a Sequence is Decreasing
### Introduction
This discussion focuses on demonstrating that the sequence defined by \(x_{n+1} = 2 - \frac{1}{x_n}\) is decreasing, given \(x_1 > 1\). The goal is to show the sequence is monotone and convergent, and then find its limit.
### Showing Monotonicity and Convergence
##... | {
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493f0024-e987-4d5e-aa5c-e4d592329499 | # Dancing Triangular Squares: The Process of Creating a Mathematical Dance
**Summary:**
The author brings her experience of ballet to the mathematics classroom. She describes how her learners embrace the creativity of dance, using mathematics as the basis for choreography, where the students become the numbers.
## ... | {
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ac070a96-1e05-4caf-9557-b65eb29cdfa5 | # Sum of Angles Formula
## What Is the Sum of Angles Formula?
The formula for interior angles can be determined by multiplying the number of triangles by \(180^\circ\). The total number of triangles is always two less than the number of sides of a polygon. The sum of angles formula for a given polygon can be expresse... | {
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f5824770-fe5b-4ada-9cce-2373db18e28f | ## Greatest Common Factor (GCF) of 65 and 85
### What is the Greatest Common Factor of 65 and 85?
The greatest common factor (GCF) of 65 and 85 is 5.
\[ \text{GCF}(65,85) = 5 \]
We will calculate the prime factors of 65 and 85 and find the greatest common factor by matching the biggest common factor of 65 and 85.
... | {
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8e71b0a1-9723-4f81-bb8a-d6cebddbe13c | ## Homework Help: Physics
**Posted by Nuwaya on Monday, January 16, 2012 at 2:32 am**
A piston in a car engine has a mass of 0.75 kg and moves with motion which was approximately simple harmonic. If the amplitude of this oscillation is 10 cm and the max. safe operating speed of the engine is 6000 rev. per min, calcul... | {
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762acdb6-7563-408e-add9-0ff6cbb7157e | ### Homework Help: Physics
**Posted by Johnson on Sunday, April 20, 2008 at 11:51am.**
The pressure and volume of a gas are changed along a path ABCA in the figure. The vertical divisions on the graph represent \(3.50 \times 10^5\) Pa, and the horizontal divisions represent \(2.00 \times 10^{-3} \, \text{m}^3\). Dete... | {
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93004bea-9af6-4d91-8c30-62071d77d938 | **Statement of a Problem № 3836**
(a) A current \(I\) flows in a straight wire of length \(L\). Show that the magnitude of the magnetic field at a perpendicular distance \(x\) from the midpoint of the wire is given by
\[
|B| = \frac{\mu_0 I}{4\pi} \cdot \frac{1}{L} \cdot \frac{1}{\sqrt{\left(\frac{L}{2}\right)^2 + x... | {
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31b91499-428a-4cbc-be07-5b3888301ca8 | # Find the Factors
## A Multiplication Based Logic Puzzle
### 319 and Level 2
* 319 is a composite number.
* Prime factorization: \(319 = 11 \times 29\)
* The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get \((1 + 1)(1 + 1) = 2 \times 2 = 4\). Therefore, 319 has exactly 4 ... | {
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7e54e7ed-8cf9-4def-9089-35457105e83c | # Monitor Screen Problem
The monitor screen is initially blank. Four new wheels appear on the screen every seven seconds when the beep sounds. Simultaneously, three wheels disappear from the screen every eleven seconds. If both actions occur simultaneously, the number of circles on the screen will not change.
A) Spec... | {
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caef4692-2cb1-4477-86a2-692e1584b899 | # X And Divide By 10 And 100 Worksheet
**By Kristina Fe**
*Published on May 27, 2020*
*264 views*
## Overview
The worksheet includes a variety of tasks designed to enhance your child's understanding of multiplication and division by 10 and 100. These tasks are developmentally appropriate for children of differen... | {
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1069d332-83bf-40d2-91d8-787ff3102afc | # Intermediate Algebra, 5th Edition
## Overview
Intermediate Algebra is a subject tailored for high school and college students who wish to deepen their understanding of algebraic concepts. This book, authored by Elayn Martin Gay, provides comprehensive coverage of fundamental and advanced algebraic topics.
## Key F... | {
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496c8fe0-3605-456f-a029-d9f6b8e652bb | # How many cubic feet is a Kenmore refrigerator model?
Answer this question.
---
## Related Answers
### Kitchen Aid Side By Side Refrigerator - What to do if Sears stops making
GE has gotten some very good ratings.
+ Posted by Ol' Yeddar 4 months ago
### How do you convert something that is 0.625 in diameter an... | {
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df94ee54-e671-40be-9733-4e00ef9af02e | ## 8.6: Fractional Exponents
### What is a Fractional Exponent?
Fractional exponents are a way to represent both roots and powers of a number in a single expression. For example, consider the expression \(\frac{x^{\frac{5}{8}}}{x^{\frac{1}{3}}}\). How can we simplify it? Understanding fractional exponents will allow ... | {
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42069c22-eb1d-4279-8327-8e57001f0006 | # Math Help Forum: Approximating the Value of a Logarithm by Trial and Error
## Exclamation: Approximating the value of a logarithm by trial and error
**Use systematic trial and error to evaluate $\log_{10}(50)$.**
### Discussion
#### math123456
*Posted on April 21st, 2009, 11:20 PM*
Use systematic trial and error... | {
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01061aa9-e1fe-48fd-b747-43fd538b08c4 | # Homework Help: College Algebra
**Posted by Anonymous on Monday, October 24, 2011 at 9:14pm.**
Show all work.
A projectile is fired from a cliff 100 ft above the water at an inclination of 45 degrees to the horizontal, with a muzzle velocity of 260 ft/second. The height of the projectile above the water is given by... | {
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f767c679-c932-4a1e-a753-2b2bcae86af7 | **Motion with Constant Acceleration**
Starting from rest, a boat increases its speed to \(4.12 \, \text{m/s}\) with constant acceleration.
(a) What is the boat's average speed?
(b) If it takes the boat \(4.77 \, \text{s}\) to reach this speed, how far has it traveled?
**Answer**
Starting from rest, a boat increa... | {
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b3755220-d3b0-4388-9962-8a5f714a9e02 | # Algebraic Identities and Their Applications
## Introduction
Algebraic identities are equations that are true for all values of the variables involved. They play a fundamental role in simplifying algebraic expressions, solving equations, and proving various mathematical theorems. This document explores some key alge... | {
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1b87c523-1ab8-4af3-bfdd-a99d941805c1 | **A can contains a mixture of two liquids A and B in the ratio 7:5. When 9 litres of the mixture are drawn off and the can is filled with liquid B, the ratio of A to B becomes 7:9. How many litres of liquid A was contained by the can initially?**
**Options:**
- [A]. 10
- [B]. 20
- [C]. 21
- [D]. 25
**Answer: Option ... | {
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f21359f2-21d4-4a96-ac7c-10d66c45ec16 | Since the provided text primarily consists of navigational buttons, lists of page links, and search terms used by Google users, it contains no meaningful content that adheres to the guidelines provided. As a result, there is no main content or mathematical material to extract and process.
**Output:**
```
NO USEFUL CO... | {
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09ec5df8-ea3d-494c-bc04-0af7a6095005 | ## Introduction to Advanced Calculus
Calculus is a branch of mathematics that studies continuous change. It is divided into differential calculus and integral calculus, with the former focusing on the concept of the derivative and the latter on the concept of the integral.
### The Fundamental Theorem of Calculus
The... | {
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9b8842a8-4cdd-48bb-9ab9-4a764013c3f8 | # Linear Search
## General Info
### Introduction and Summary
Linear search is an algorithm used to find a specific element in a list. It works by checking each element in the list sequentially until the desired element is found or the end of the list is reached. Linear search is straightforward but can be inefficien... | {
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aeb23b7c-27e5-45d9-afdb-f4fdbe67b5c4 | # Homework Help: Math
### General Solution
Consider the equation:
\[ \cos^2 x + 3\sin x = -3 \]
We can rewrite \(\cos^2 x\) using the Pythagorean identity:
\[ 1 - \sin^2 x + 3\sin x + 3 = 0 \]
Simplifying, we get:
\[ \sin^2 x - 3\sin x - 4 = 0 \]
Factoring the quadratic equation:
\[ (\sin x - 4)(\sin x + 1) = ... | {
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3d57c9fc-7f9e-4950-a6c8-1b725c64127d | # Time Intervals Between Customers
The time interval between two successive customers entering a store in a mall is exponentially distributed with a mean of 6.55 seconds.
### Questions
a. **What is the probability that the time interval is more than 10 seconds?**
b. **What is the probability that the time interval ... | {
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db93727c-543a-4751-948f-e44e8bd75528 | # Common Core Standards: Math
## High School: Statistics and Probability
These standards describe the skills that students need to analyze and interpret data effectively.
### Interpreting Categorical and Quantitative Data
One sentence can be interpreted in a million different ways. Imagine how many ways we can inte... | {
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e6dca10a-e107-4434-8398-3878a995a94a | # 279 Percent of What Number is 80?
## Problem Statement
279 percent of what number is 80?
## Solution
80 is 279% of 28.67.
## Steps to Solve
1. We have the equation:
\[
279\% \times x = 80
\]
2. This can be rewritten as:
\[
\frac{279}{100} \times x = 80
\]
3. Multiplying both sides by 100 and... | {
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eb800d93-4376-4196-b242-a7826e5a41b1 | # 2D Shape with 1 Side?
A Möbius strip, like the one you make out of paper, is a 2D shape with one side. How is it possible for a 2D shape to have one side, or really any n-dimensional shape to have one side? The closest that I see to one side is a single straight line, and even then it is zero sides. A circle or its ... | {
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8d1186f5-1082-458b-9f76-8fe3941d9365 | ## Arcs and Chords
In \(\Delta ABC\), the perpendiculars from vertices \(A\) and \(B\) to their opposite sides meet (when produced) the circumcircle of the triangle at points \(D\) and \(E\) respectively. Prove that arc \(CD\) equals arc \(CE\).
### Discussion
**Agnishom** (Real Member, Riemann Sphere, Registered: 2... | {
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