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Math Help - optimization in price

  1. #1
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    optimization in price

    The current ticket price at a local theatre is $4, and the theatre attracts an average of 250 customers per show. every $.20 increase in ticket price reduces average attendance by 10 customers, while every $.20 decrease results in 10 extra customers.

    a) let x respresent the change in ticket price. show that revenue R from ticket sales depends of x according to R(x) = (4+x)(250-50x).

    I differentiated this and solved for x and found x=1/2 and because the rate of change is not zero the sales depend on the value of x. Correct?

    b) if the seating capacity is 400, show that -3 \le x \le 5

    I'm not sure what to do, I rewrote the function as (4+.2x)(400-10x but I just ended up with a x value of 10 when I differentiated.

    c) find the ticket price that will maximize revenue.

    so I wrote

    (4+.2x)(250-10x) then I differentiated and came to a value of x = 2.5

    then solve R(x) for x = -3, x = 2.5, x=5 and found the optimum price at x=2.5 for a price of $4.50

    solving (4-.2x)(250+10x) gave me the same result but x = -2.5

    Are A and C correct, and what am I doing incorrectly for B?
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  2. #2
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    Re: optimization in price

    1. a) You are not supposed to differentiate this. You are asked to GENERATE the equation you are given.
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  3. #3
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    Re: optimization in price

    okay is this correct then,

    Total revenue is equal to the amount of customers times the price, if R is the revenue, p is equal the price, a is equal to the increase in price, q is equal to the amount of customers and b is to the amount of customers lost and x is equal to the amount of the number of times the price is increased. For every increase in x value there is a increase in price but a decrease in customers

    R = (p+ax)(q-bx)

    and when the price is 4+the increase times x, and the average customers is 250, and loss of customers is 50 times the value of x

    R(x) = (4+x)(250-50x)

    and so R(x) is dependent on x.

    What about B and C?
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