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Ms. Platinum has a hit on her hands. A survey she commissioned shows that she can sell 100 tickets to her smashing new musical for $40 each, and for each $1 she drops the ticket price, she can sell 10 more tickets. Too bad the theater she rented only seats 200 people! How much should she charge for tickets to maximize her revenue? Homework Help ✎

Before we even start, notice that there is boundaries to the domain of this story. There are only 200 seats to rent! We will return to that at the end of the problem.

So, we have not answered the question. Sometimes, when a function has a restricted domain, the maximums is NOT where the derivative equals 0, but at an endpoint (where the derivative does not exist). Go back and determine how much Ms. Platinum will need to charge to sell 200 tickets.

Let x = number of times she decreases the price. Write a Revenue function, R(x).

Revenue = (seats sold)(price per seat)
R(x) = (100 + 10x)(40 − 1x)

Maximize the revenue.

R'(x) = 10(40 − x) − 1(100 + 10x) Product Rule
= 300 − 20x
0 = 300 − 20x
x = 15

What does this mean?

She should charge 40 −15 dollars, or $25 and she will sell 100 + 10(15) tickets = 450 tickets

But this is impossible because the theater only has 200 seats.