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# Algebra

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### Quadratic Function Model

Annual profit in thousands of dollars is given by the function, P(x) = -.1x2 + 50x - 300, where x is the number of items sold, x ? 0. Describe the meaning of the number -.1 in the formula, in terms of its meaning in relation to the profit. Describe the meaning of the number -300 in the formula, in terms of its meaning in rel

### You want to fence in a rectangular area, but first you must determine the dimensions of the area.

Congratulations, you have just purchased your first house. Unfortunately, the builder forgot to put a fence around the property and you must now put a fence up so that you can let your pet out in the backyard, without fear of it running away. (My pet would be a large dog such as a Mastiff). You want to fence in a rectangul

### Recipe Word Problems

1. You are doubling a receipe that calls for 3/4 of a cup of milk. How much milk will you need? 2. Write a number using the following instructions: put a 5 in the hundreds place divide 24/8 and put the result in the tenths place put the number in the hundreds place also in the ones place subtract 3 from the numbe in the t

### Finding the Value of a Variable Based on Equations

You are choosing between two health clubs. Club A offers membership for a fee of \$50 plus a monthly fee of \$25. Club B offers a membership fee of \$25 plus a monthly fee of \$30. a. Using variables of your choice, set-up an equation for each club's membership cost. Make sure your variables are defined clearly. b. After how ma

### Quadratic Function: Word Problem

1. Write a word problem involving a quadratic function. How would you explain the steps in finding the solution to someone else? Provide a detailed example. You may use the Internet for help/ideas. Cite your source if it is not an original idea. 2. Quadratic equations, which are expressed in the form of ax2 + bx + c = 0, where

### arithmetic sequence for depreciation

Depreciation: For tax purposes, businesses frequently depreciate equipment. Two methods of depreciation are straight-line depreciation and sum-of-the-years'-digits. Suppose that a college student buys a \$3000 computer to start a business that provides Internet services. This student estimates the life of the computer at 4 years,

### Telescoping Series Proofs

Prove telescoping series: Let a_n from n=0 to infinity be a sequence of real numbers which converge to 0, i.e. lim n-->infinity a_n=0. Then the series of the sum from n=0 to infinity of (a_n - a_n+1) converges to a_0. Hint: first work out what the partial sums of the sum from n=0 to N of (a_n - a_n+1) should be, and prove yo

### Series and Infinite Series

Is the series sum from n=1 to infinity of (-1)^n convergent or divergent? Justify your answer. Can you now resolve the difficulty of the following: Divergent series: S=1+1/2+1/4+1/8+1/16+... You have probably seen the following trick to sum this series: if we call the above sum S, then if we multiply by 2, we obtain: 2S=2

### Key Components of Functions and Graph Verbally

The height h(t) in feet above the ground of a golf ball depends on the time, t (in seconds) it has been in the air. Ed hits a shot off the tee that has a height modeled by the velocity function h(t) = 0.5at^2 + vt + s where a is -32 ft/sec^2, v is the initial velocity, and s is the initial height. 1. Write a function that mod

### Network Models

Minfly Golf Company manufactures golf balls at Philadelphia, Chicago, and Denver. Cases of golf balls are shipped to warehouses in Orlando, Dallas, and Los Angeles. The following table provides the weekly capacities of the factories and the minimum weekly requirements of the warehouses along with the cost per case to ship betwee

### Fence material and needs to enclose an area

Jim has 400 ft of fence material and needs to enclose an area of 10,000 square feet for his garden as the diagram depicts. (The length of the house is 62 ft, and the fence extends the same distance on either side of the house.) What must the dimensions of the rectangular enclosed region be? That is, what are the values of x and

### Scalar projection

Please see the attached file for the full problem. How can I use a scalar projection to show that the distance from a point? P(x, y) to the line 3*x-4*y+5 = 0 is > abs(a*x+b*y+c)/sqrt(a^2+b^2);

### Surface area of a cone

1. Solve by factoring: 6t^2 - t - 1 = 0 2. Solve the attached equation. 3. Solve each formula for the indicated letter. Assume that all variables represent nonnegative numbers. A = pr^2 + prs, for r

### In solving the equation (x + 1)(x - 2) = 4

1.) In solving the equation (x + 1)(x - 2) = 4, Eric stated that the solution would be x + 1 = 4 => x = 3 or (x - 2) = 4 => x = 6. However, at least one of these solutions fails to work when substituted back into the original equation. Why is that? Please help Eric to understand better; solve the problem yourself, and explain

### Fields - Cardinality and Splitting Fields.

We know that F_2[x]/(x^2 + x + 1) is a field of cardinality 4; call it F_4. Find an irreducible quadratic f(y) [element of] F_4[y]. What is the cardinality of the finite field F_4[y]/(f(y))? This field is isomorphic to the splitting field for x^2^n - x over F_2[x]. What is the appropriate n in this case? Show that x^4 + x

### Proof of two Legendre Symbol identities

1) Use that (Z/pZ)* is cyclic to directly prove that (-3/p) = 1 when p = 1 (mod 3). 2) If p = 1 (mod 5), directly prove that (5/p) = 1. I know that with exercise 1, we show that there is an element c in (Z/pZ)* of order 3, and that (2c+1) ^2 = -3. Similarly for exercise 2, there is a c in (Z/pZ)* of order 5, and [(c + c

### Aerobics Instructor: Water Consumption Problem

An aerobics instructor says she drinks one cup (8 ounces) of water before she starts her classes and then she drinks 10 ounces every 10 minutes. a. How much water does she drink when she teaches 2 one-hour classes in a row? b. Let W represent ounces of water and N the number of 10-minute intervals the aerobics instructor s

### Solving Quadratic Equations With Factoring

Solving Quadratic Equations by Factoring is Chapter 13.7 page 955 1) Quadratic Equation is in STANDARD FORM (ax^2 + bx + c = 0 where a is positive) Factor trinomial: x^2 + bx +c =(X + m)( X +n)=0 ,which :(mn=C),and (m+n=b) (X + m)=0 ---->X = - m , and (X + n)=0----> X= - n 2) Pythagorean Theorem in Right triangle (

### Maximal and Principal Ideals

Let delta=sqrt(-3) and R=Z[delta]. This is not the ring of integers in the imaginary quadratic number field Q[delta]. Let A be the ideal (2,1+delta). a) Show that A is a maximal ideal and identify the quotient ring R/A. b) Show that A contains the principal ideal (2) but that A does not divide (2).

### Groups and Representations

Please help answer the following algebra questions. Provide step-by-step solutions. (i) Now let G be a group with the presentation G=<a,b|a^7=e,b^3=e,b^(-1) ab=a^2> You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation p:G --> GL(3,C) with p(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(

### Forming a linear function from a story

The relation between the cost of a certain gem and its weight is linear. In looking at two gems, we find that one of the gems weighs 0.2 carats and costs \$3,548, while the other gem weights 0.5 carats and costs \$4,374. Find C(x), the linear function that relates the price of gem to its weight x, treating weight as the independen

### Cyclic Modules and Generators

For cyclic Z-modules Zm and Zn with generators a and b, respectively, show that Zm ⊗Z Zn is isomorphic to Z(m,n) with generator a ⊗ b, where(m,n) is the greatest common divisor of m and n.

### Prove the following Proposition

Prove the following Proposition: On Z_n, both * and + are commutative and associative. The identity for Z_n with * is  and the identity for + is . Please submit response as either a PDF or MS Word file. Infinite thanks. See attachment.

### Apply the distributive property for a negative monomial times a trinomial with different signs on the terms.

Give an example of using the distributive property for a negative monomial times a trinomial with different signs on the terms [for example: -3x(2xy + 3y - 2x)] and show each step of the distribution. Why do you think many students make sign errors on this type of problem? What would be your advice to a student who has trouble w

### vector question

Under what condition is there a solution to the simultaneous vector equations alpha* x+ beta*y = a and x ^ y = b, for the vectors x and y in terms of given non-zero scalars alpha and beta and given non-zero vectors a and b? Find the general solution to these equations when this condition is satisfied. n.b[ "^" represents t

### Working with functions

Let y = f(x) describe the speed y of an automobile after x minutes if the cruise control is set at 60 miles per hour. (a) Represent f symbolically and graphically over a 15-minute period for 0 < x > 15. (b) Construct a table of f for x = 0, 1, 2, . . . ., 6. (c ) What type of function is f?

### Proving that a Function is Irreducible

Let p be a prime congruent to -1 mod 4. Show that X^2 + 1 is irreducible in Z_p[X], and hence K = Z_p[X] / (X^2 + 1) is the field of order p^2. Note that K has a multiplication similar to that of the complex numbers.

### Nonlinear System

Give all solutions of each nonlinear system of equations, including those with nonreal complex components: 1. X^2+y=2 X-y=0 2. X^2+y^2=10 2X^2-y^2=17 3. -5xy+2=0 X-15y=5

### Algebra: Substitution and Elimination

Solve each system by substitution: 1. 4x+5y=7 9y=31+2x Solve each system by elimination: 1. 12x-5y=9 3x-8y= -18

### Investment Time and Interest Rate

1. Find t to the nearest hundredth if \$1786 becomes \$2063 at 2.6%, with interest compounded monthly. 2. At what interest rate, to the nearest hundredth of a percent, will \$16,000 grow to \$20,000 if invested for 5.25 yr and interest is compounded quarterly?