PIECEWISE PROBLEMS



Piecewise Functions – Application Problems

1. You will be renting a car for a six-day trip and are comparing two rental options. The first plan charges $50 per day, allows 500 miles for free, and charges 25 cents for each additional mile. The second plan charges $30 per day, allows 200 miles for free, and charges 35 cents for each additional mile.

a. If you’re going to drive 300 miles over the course of the six-day trip, which plan should you choose?

b. If you’re going to drive 800 miles over the course of the six-day trip, which plan should you choose?

c. Write a piecewise function for the cost of Plan 1 for a six-day trip of x miles.

d. Write a piecewise function for the cost of Plan 2 for a six-day trip of x miles.

e. Make a graph of both functions on the same set of axes.

f. After how many miles does Plan 1 become cheaper?

2. A certain country taxes the first $20,000 of an individual’s income at a rate of 15%, and all income over $20,000 is taxed at 20%.

a. Al makes $16,000. Betty makes $36,000. How much is each taxed?

b. Write a piecewise function T that specifies the total tax on an income of x dollars.

c. Make a graph of T. Be sure to plot the points from part a!

d. Catina is taxed $5000. What is her income?

3. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold, and 15% on any additional copies.

a. When the 6,000th book is sold, how much will the author earn on that sale?

Also, what will the author’s total royalties be at that point?

b. When the 12,000th book is sold, how much will the author earn on that sale?

Also, what will the author’s total royalties be at that point?

c. Let x be the number of copies sold. Write a piecewise function for R (the royalty

payment earned on that sale) in terms of x.

d. Write a piecewise function for T (the total amount of royalties earned).

e. Make a graph of T(x).

f. How many copies have to be sold in order for the author to have earned $30,000?

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