Question

Community Theater Owner A community theater currently sells 200 season tickets at $50 each. In order to increase its season-ticket revenue, the theater surveys its season ticket holders to see if they would be willing to pay more. The survey finds that for every $5 increase in the price of a season ticket, the theater would lose 10 season-ticket holders. What action, if any, should the theater owner take to increase revenue? a. Let n be the number of $5 price increases in the cost of a season ticket. Write an expression for the cost of a season ticket after n price increases, and an expression for the number of season-ticket holders after n price increases. b. Use the expressions from part a to create a revenue function, R (n) , from the survey information. c. Graph the revenue function. Be sure to label the axes with the quantities they represent and indicate the axis scales by showing numbers for some grid lines.

Answer 1

• the owner should increase prices to increase revenue (up to n=5)
• a) c(n) = 50+5n; q(n) = 200-10n
• b) R(n) =50(10+n)(20-n)
• c) see below for a graph

Explanation:

a. The cost of a ticket after n price increases of $5 each will be ...

c(n) = 50 +5n

The number of season-ticket holders (q) after n price increases will be ...

q(n) = 200 -10n

__

b. The revenue after n price increases will be ...

R(n) = c(n)·q(n) = (50 +5n)(200 -10n)

R(n) = 50(10 +n)(20 -n)

__

c. See the attachment for a graph

__

d. The theater owner should raise the price of a season ticket to increase revenue. The number of $5 price increases should not exceed 5 if revenue is to increase. Above that number, revenue will decrease.