CPM Homework Help : CALC Problem 5-115

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

Hint:

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.

Answer:

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.

Step 1:

Let $x =$ number of times she decreases the price. Write a Revenue function, $R(x)$ .
Revenue $=$ (seats sold)(price per seat)
$R(x) =$ ( $100 + 10x$ )( $40$ minus: $1x$ )

Step 2:

Maximize the revenue.
$R^\prime(x) = 10(40 − x) − 1(100 + 10x)$ Product Rule
$= 300 − 20x$
$0 = 300 − 20x$
$x = 15$

Step 3:

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.