Defining a revenue function and finding maximum revenue.

**mathcalculushelp** · Nov 18th 2009, 04:16 PM

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.

**Scott H** · Nov 18th 2009, 07:16 PM

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

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

We know that the cost of one ticket is $p$ , 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 $p=8$ , the number purchased is $990$ , but for every additional dollar after $p=8$ , this total drops by $60$ . In symbols,

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

Therefore,

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

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

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