At a price of $8 per ticket, a theater group can fill every seat in the theater, which has a seating capacityof 990. For every additional dollar charged, the number of people buying tickets decreases by 60.

a. Define a function, R(p), that gives revenue for the performance if a single ticket costs $p.

b. Using R'(p), determine the ticket price that maximizes revenue. Also, compute the maximum possible revenue for the performance.

We first find the function $\displaystyle R(p)$ of total profit if tickets cost $\displaystyle p$ dollars each. To find the profit, we use the formula

$\displaystyle \mbox{revenue}=\mbox{cost of item}\cdot\mbox{total items sold}.$

We know that the cost of one ticket is $\displaystyle p$, so all we need to do is find a formula for the total number of tickets sold based on the price of one ticket. This is given to us in the problem: at $\displaystyle p=8$, the number purchased is $\displaystyle 990$, but for every additional dollar after $\displaystyle p=8$, this total drops by $\displaystyle 60$. In symbols,

$\displaystyle \mbox{total tickets purchased}=990-60(p-8).$

Therefore,

$\displaystyle R(p)=p\cdot(990-60(p-8)).$

Now, all that remains is to find the critical points of $\displaystyle R$ in order to determine the maximum of $\displaystyle R$.