Math

A real estate company owns 218 efficiency apartments, which are fully occupied when the rent is $940 /month. The company estimated that for each $25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income?

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  1. let the number of $25 increases be n

    rent = 940 + 25n
    number rented = 218-5n

    income
    = (940+25n)(218-5n)
    = 204920 +750n - 125n^2

    If you know Calculus:
    d(income) = 750 - 250n = 0 for a max of income
    250n=750
    n = 3

    if you don't know Calc
    complete the square:
    -125(n^2 - 6n + 9 - 9) + 204920
    = -125( (n-3)^2 - 9) + 204920
    = -125(n-3)^2 + 1125 + 204920
    = -125(n-3)^2 + 206045
    vertex is (3, 206045)

    there should be 3 increases for a rent of 940+75 = 1015

    or
    the x of the vertex is -b/(2a) = -750/-250 = 3
    for 3 increases.

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