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Basic Algebra

Determining Correlation and Distribution

6) Suppose we have a building with a floor shaped like an isosceles right triangle. The two sides adjacent to the right triangle have length 100 feet. Think of the right angle being at the origin, and other two corners at (100, 0) and (0, 100). The overhead crane is located at the origin and needs to travel to a point (X, Y), w

Scientific notation

A number written in scientific notation is doubled. Explain why the exponent of 10 may or may not change.

Scientific notation

A human being has about 2.5 X 10(to the 13th power) red blood cells in her bloodstream. There are about 2 white blood cells for every 1,000 red blood cells. How many white blood cells are in a human's bloodstream? Please show your answer in scientific and standard notation.

Serial Problem

At Very Long Hotel in Florida, there are n rooms located along a very long corridor and numbered consecutively from 1 to n. One night after a party, n people, who have been likewise numbered from 1 to n, arrived at this hotel and proceeded as follows: Guest 1 opened all the doors. Then Guest 2 closed every second door beginnin

Show a pair has a midpoint with integer coordinates

Let p = {(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)} be a set of five distinct points in the plane , each of which has integer coordinates. Show that some pair has a midpoint that has integer coordinates.

Functions: Proof by Induction

Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) If n is even, then f is strictly increasing hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) If n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. "This needs to be a proof by induction, provin

Nontrivial Central Extension

Find a nontrivial central Z_2 extension of the group A_4, meaning an extension of the form: 1 --> Z_2 --> G --> A_4 --> 1 Also, is it unique? The trivial extension is just the direct product of Z_2 and A_4.

Algebra: Logarithms - Solve for X, Solve for Base

Please see the attached file for the fully formatted problems. 1. The value of e ln 1+ ln 2+ ln 3 is ? 2. If logN(25) - logN(81) = 2, then N is? 3. Solve for x: log5( x) + log5(2x +13) = log5(24) 4. Find the coefficient of the fourth term in the expansion of (2x +3y)^11

Algebra Problem of the week

There is a hallway that is infinitely long, with a series of lightbulbs that are all turned off. Someone enters and pulls the string on every light bulb, turning them all on. Another person enters, pulling the string on every other lightbulb. A third person enters, pulling the string on every third bulb. This continues indef

Central Extension Problem

Describe all nonisomorphic central extensions of Z_n by Z_2 x Z_2 meaning central group extensions of the following form 1 --> Z_n --> G --> Z_2 x Z_2 --> 1 Meaning, determine those nonisomorphic groups G that can be described by such an extension. Please also explain how you came up with the answer.

Working with the sum of cubes.

Please see the attached file for the fully formatted problems. Problem 3: This problem itself is directly creating perceptual curiosity and at the cognitive level it is against what is known in the mathematics. The problem is: 13+23+33+.... +n3 = (1+2+3+...+n) 2 Visual representation of problem: Sum of the cubes

Getting three thousand bananas across a one thousand mile desert

You have three thousand bananas that you have to get to a destination 1000 miles away. You can only carry 1000 bananas at a time. You also must eat 1 banana per mile for energy. Assuming you design your trip as efficiently as possible, how many bananas will you have left when you arrive at the destination? (apparently someon

Algebraic Equations : Reciprocal Equations

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order ax^4 + bx^3 + cx^2 + dx + f=0 (*) is the maximum order type of algebraic equations the solution to which one can write down in radical expressions. Among all the equations of fourth

Algebra: Word Problem - Relative Velocity

A mo-ped can travel 60 miles in 2 hours less time than a bicycle can travel 50 miles. The mo-ped is traveling at a rate of 10 miles per hour faster than the bicycle. 1. How fast in mph is each traveling? 2. How long will it take each to travel their respective distance?

Algebra : Puzzle Problems

Please see the attached file for the fully formatted problems. 1) Have you ever seen the written form of the Sanskrit language? If so, you probably are amazed at how different this ancient language from India looks from ours. Some English words, however, are based on Sanskrit. For example, cup comes from the Sanskrit work kup

Miner and Bickford's Fuse : How to get a Fuse of Specific Length

You are a miner and you have three pieces of Bickford's fuse of equal length. You need only 3/4 of one of them. You have no ruler or other measurement device with you. You cannot also bend the fuses as they are old and can be broken at any point while being bent. You only can ignite them from any end and extinguish at any moment

Algebra : Remainders

What is the remainder when the product of one hundred 5's is divided by 7? Please be detailed in your response.

Exponential Equations : Word Problem

The population of the green deer in 1999 was 17000. In 2003 there were only 15000. Write an exponential equation to express the poulation decrease p(t) in terms of t years.

Perturbation theory

Find the real roots of the equation x^5 + ex-1=0 approximately to O(e^2) usind perturbation theory. Compare the accuracy of the perturbative solution for e=0.001, 0.1, and 1

Perturbation Theory

Find the real root of the equation x^5 + ex - 1= 0 approximately to O(^2) using perturbation theory.

Algebra : Word Problem - Sums and Products

During the census, a man told the census-taker that he had three children. When asked their ages he replied, " The product of their ages is 72. The sum of their ages is my house number." The census taker turned around and ran outside to look at the house number displayed over the door. He then re-entered the house and said, "

Algebra: Word Problem - System of Equations

Three people play a game in which one person loses and two people win each game. The one who loses must double the amount of money that each of the other players has at that time. The three players agree to play three games. At the end of the three games, each player has lost one game and each has $8. What was the original st

Algebra : Word problem - Time and Distance

Two swimmers start at opposite ends of a pool 89 feet long. One person swims at the rate of 19 feet per minute and the other swims at a rate of 53 feet per minute. How many times will they meet in 33 minutes? Plese try to give a detailed response as my answer is not as important as the thought processes that I must understa

Find a solution in the form of a power series for an ODE

Please see the attached file for the fully formatted problems. Find a solution in the form of a power series for the equation y" - 2*x*y' = 0 (ie find 2 linearly independent solutions y1(x) and y2(x)). After doing that, note that the equation can also be solved directly by integration: y"/y' = 2x ln(y') = x^2 +