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Math Help - Defining a revenue function and finding maximum revenue.

  1. #1
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    Question Defining a revenue function and finding maximum revenue.

    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.
    Last edited by mr fantastic; November 18th 2009 at 04:34 PM. Reason: Changed post title.
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  2. #2
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    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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