Chapter 6

Distance traveled. Bonita drove 100 miles in \(x\) hours. Assuming she continued to drive at the same speed, write a rational expression for the distance that she traveled in the next \(\frac{3}{4}\) of an hour.

Solve each problem. Driving speed. Jeremy drives 500 miles in \(x\) hours. Find a rational function \(S(x)\) that gives his average speed in miles per hour.

Perform the indicated operations. $$ \frac{8 a}{2 a^{2}+4 a+2}-\frac{3 a-3}{a^{2}-1} $$

Solve each problem. Travel time. Marsha traveled 400 miles with an average speed of \(x\) miles per hour. Find a rational function \(T(x)\) that gives her travel time in hours.

Perform the indicated operations. $$ \frac{4 x^{2}+9}{4 x^{2}-9} \cdot \frac{4 x^{2}+12 x+9}{2 x^{2}+3 x} $$

Solve each problem. Average cost. Bobby spent \(\$ 150\) on \(x\) pieces of clothing for her child. a) Find a rational function \(C(x)\) that gives the average cost in dollars for an item of clothing. b) Find \(C(5), C(10),\) and \(C(30)\)

Solve each problem. Estimating weapons. When intelligence agents obtain enemy weapons marked with serial numbers, they use the formula \(N=(1+1 / C) B-1\) to estimate the total number of such weapons \(N\) that the enemy has produced. \(B\) is the biggest serial number obtained and \(C\) is the number of weapons obtained. It is assumed the weapons are numbered 1 through \(N\) a) Find \(N\) if agents obtain five nerve gas containers numbered \(45,143,258,301,\) and 465 b) Find \(C\) if agents estimate that the enemy has 255 tanks from a group of captured tanks on which the biggest serial number is 224.

Which of the following equations is not an identity? Explain. a) \(\frac{x^{2}-1}{2} \cdot \frac{2}{x-1}=x+1\) b) \(\frac{x-1}{x^{2}-1}=x+1\) c) \(x^{2}-1=(x-1)(x+1)\) d) \(\frac{1}{x^{2}-1} \div \frac{1}{x+1}=\frac{1}{x-1}\)

Perform the indicated operations. $$ \frac{3 a^{2}-2 a-16}{2 a^{2}+3 a-2} \cdot \frac{6 a+16}{9 a^{2}-64} $$

Solve each problem. Average profit. The weekly profit in dollars for manufacturing \(x\) bicycles is given by the polynomial \(P(x)=\) \(100 x+2 x^{2} .\) The average profit per bicycle is given by \(A P(x)=\frac{P(x)}{x} .\) Find \(A P(x) .\) Find the average profit per bicycle when 12 bicycles are manufactured.

Solve each problem. Flying high. A flying club has \(x\) members who plan to share equally the cost of a \(\$ 300,000\) airplane. a) Find a rational function \(C(x)\) that gives the cost in dollars per member. b) Find \(C(10), C(30),\) and \(C(500)\)

Writing. In this chapter the LCD is used to add rational expressions and to solve equations. Explain the difference between using the LCD to solve the equation $$\frac{3}{x-2}+\frac{7}{x+2}=2$$ and using the LCD to find the sum $$\frac{3}{x-2}+\frac{7}{x+2}$$

Perform the indicated operations. $$ \frac{w^{2}-3}{3 w^{3}+81}-\frac{2}{6 w+18}-\frac{w-4}{w^{2}-3 w+9} $$

Solve each problem. Area of a poster. The area of a rectangular poster advertising a Pearl Jam concert is \(x^{2}-1\) square feet. If the length is \(x+1\) feet, then what is the width?

Discussion. For each equation, find the values for \(x\) that cannot be solutions to the equation. Do not solve the equations. a) \(\frac{1}{x}+\frac{1}{x-1}=\frac{1}{2}\) b) \(\frac{x}{x-1}=\frac{1}{2}\) c) \(\frac{1}{x^{2}+1}=\frac{1}{x+1}\)

Perform the indicated operations. $$ \frac{a-3}{a^{3}+8}-\frac{2}{a+2}-\frac{a-3}{a^{2}-2 a+4} $$

Solve each problem. Rose Bowl bound. A travel agent offers a Rose Bowl package including hotel, tickets, and transportation. It costs the travel agent \(\$ 50,000\) plus \(\$ 300\) per person to charter the airplane. Find a rational function that gives the average cost in dollars per person for the charter flight. How much lower is the average cost per person when 200 people go compared to 100 people?

Perform the indicated operations. $$ \frac{a^{2}-6 a+9}{a^{3}-8} \div \frac{a^{2}-a-6}{a^{2}-4} $$

Solve each problem. Solid waste recovery. The amount of municipal solid waste generated in the United States in the year \(1960+n\) is given by the polynomial $$ 3.43 n+87.24 $$ whereas the amount recycled is given by the polynomial $$ 0.053 n^{2}-0.64 n+6.71 $$ where the amounts are in millions of tons (U.S. Environmental Protection Agency, www.epa.gov). a) Write a rational function \(p(n)\) that gives the fraction of solid waste that is recycled in the year \(1960+n\) b) Find \(p(0), p(30),\) and \(p(50)\)

Perform the indicated operations. $$ \frac{1}{z^{2}+4} \div \frac{z^{3}-8}{z^{4}-16} $$

Solve each problem. Higher education. The total number of degrees awarded in U.S. higher education in the year \(1990+n\) is given in thousands by the polynomial \(41.7 n+1429\) whereas the number of bachelor's degrees awarded is given in thousands by the polynomial \(25.2 n+1069\) (National Center for Education Statistics, www.nces.ed.gov). a) Write a rational function \(p(n)\) that gives the percentage of bachelor's degrees among the total number of degrees conferred for the year \(1990+n\) b) What percentage of the degrees awarded in 2010 will be bachelor's degrees?

Perform the indicated operations. $$ \frac{w^{2}+3}{w^{3}-8}-\frac{2 w}{w^{2}-4} $$

Discussion. On a test a student divided \(3 x^{3}-5 x^{2}-3 x+7\) by \(x-3\) and got a quotient of \(3 x^{2}+4 x\) and remainder \(9 x+7 .\) Verify that the divisor times the quotient plus the remainder is equal to the dividend. Why was the student's answer incorrect?

Use a calculator to find \(R(2), R(30), R(500)\) \(R(9000),\) and \(R(80,000)\) for the rational function $$ R(x)=\frac{x-3}{2 x+1} $$ Round answers to four decimal places. What can you conclude about the value of \(R(x)\) as \(x\) gets larger and larger without bound?

Discussion. Use synthetic division to find the quotient when \(x^{5}-1\) is divided by \(x-1\) and the quotient when \(x^{6}-1\) is divided by \(x-1 .\) Observe the pattern in the first two quotients and then write the quotient for \(x^{9}-1\) divided by \(x-1\) without dividing.

Use a calculator to find \(H(1000), H(100,000)\) \(H(1,000,000),\) and \(H(10,000,000)\) for the rational function $$ H(x)=\frac{7 x-50}{3 x+91} $$ Round answers to four decimal places. What can you conclude about the value of \(H(x)\) as \(x\) gets larger and larger without bound?

Attaching shingles. Bill attaches one bundle of shingles in an average of \(x\) minutes using a hammer, whereas Julio attaches one bundle in 6 minutes less time using a pneumatic stapler. a) Write a rational function \(B(x)\) that gives the number of bundles that they attach while working together for 10 hours. b) Find \(B(30)\) (Image cannot copy)

Explain why fractions must have common denominators for addition but not for multiplication.