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Math Help - revenue/profit problem

  1. #1
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    revenue/profit problem

    You manage a 280 unit motel. All units will be occupied, on average, when
    you charge $50/per day per unit. Experience shows that for each $1 per day per unit that you increase the price, you will accrue 2 vacancies. Each occupied room costs $4 a day to clean and supply. On this basis, what price, in dollars, should you charge per unit per day to maximize your profit?
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  2. #2
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    Hello,

    Quote Originally Posted by gumi View Post
    You manage a 280 unit motel. All units will be occupied, on average, when
    you charge $50/per day per unit. Experience shows that for each $1 per day per unit that you increase the price, you will accrue 2 vacancies. Each occupied room costs $4 a day to clean and supply. On this basis, what price, in dollars, should you charge per unit per day to maximize your profit?
    Let X the number $1 you add to the original price.
    The price per day & per unit is 50+X.

    The profit of each occupied room is (50+X)-4=46+X, where 4 is the cost for cleaning and supplying.
    So the number of occupied rooms is 280-2X (each time you increase of 1$, there are 2 vacancies).
    We assume that an empty room makes no profit.


    Therefore, the total profit is P(X)=(46+X)(280-2X)=-2X+188X+12880.
    We want it to be maximum.

    --> P'(X)=-4X+188, which annulates at X=188/4=47.

    So the price of the rooms has to be 50+47=97.


    I hope I didn't make any mistake
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