  ### Home > APCALC > Chapter 5 > Lesson 5.3.1 > Problem5-115

5-115.

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?

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, $R\left(x\right)$.

$\text{Revenue} = \left(\text{seats sold}\right)\left(\text{price per seat}\right)$
$R\left(x\right) = \left(100 + 10x\right)\left(40 − 1x\right)$

Maximize the revenue.

$R'\left(x\right) = 10\left(40 − x\right) − 1\left(100 + 10x\right)$ 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 $100 + 10\left(15\right) \text{ tickets} = 450 \text{ tickets}$

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