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 of total profit if tickets cost dollars each. To find the profit, we use the formula



We know that the cost of one ticket is , 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 , the number purchased is , but for every additional dollar after , this total drops by . In symbols,



Therefore,



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