Question

A community theater sells about 150 tickets to a play each week when it charges $20 per ticket.

For each $2 decrease on price about 25 more tickets per week are sold. Write a quadratic function to
find how the theater can maximize weekly revenue. What should the theater charge to maximize
revenue? What is the maximum weekly revenue? (6 pts) (Show all work.)

What is the equation:
What should the theater charge?
What is the maximum revenue?

Answer 1

• r = 12.5p(32 -p)
• $16 per ticket
• $3200 maximum revenue

Explanation:

The number of tickets sold (q) at some price p is apparently ...

q = 150 + 25(20 -p)/2 = 150 +250 -12.5p

q = 12.5(32 -p)

The revenue is the product of the price and the number of tickets sold:

r = pq

r = 12.5p(32 -p) . . . . revenue equation

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The maximum of revenue will be on the line of symmetry of this quadratic function, which is halfway between the zeros at p=0 and p=32. Revenue will be maximized when ...

p = (0 +32)/2 = 16

The theater should charge $16 per ticket.

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Maximum revenue will be found by using the above revenue function with p=16.

r = 12.5(16)(32 -16) = $3200 . . . . maximum revenue

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Additional comment

The number of tickets sold at $16 will be ...

q = 12.5(32 -16) = 200

It might also be noted that if there are variable costs involved, maximum revenue may not correspond to maximum profit.