A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?

I don't even know the answer to this one. I don't understand at all, please help me step by step

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  1. right now:
    price of rent = 900
    number of units rented = 50

    Let the number of $25 increases be n
    (e.g. If n= 2 , new rent is 900+2(25) = 950
    if n = 5 , the new rent 900 + 5(25) = 1025

    so the new rent = 900 + 25n
    number rented = 50-n
    maintenance cost = 75(50-n)

    Profit = P = (900+25n)(50-n) - 75(50-n)
    = 45000 + 350n - 25n^2 - 3750 + 75n
    = 41250 + 425n - 25n^2

    d(profit)/dn = 425 - 50n
    = 0 for a max of P
    50n = 425
    n = 8.5

    The question did not say if increases are in whole multiples of 25 , but I will assume that. We could not rent 50-8.5 or 41.5 units.

    when n = 8 or n = 9

    if n = 8
    number rented = 42
    rent = 900+8(25) = 1100
    maintenace cost = 75(42) = 3150
    Profit = 42x1100 - 3150 = 43050

    if n = 9
    number rented = 41
    rent = 1125
    maintenance cost = 3075
    Profit = 41x1125 - 3075 = 43050 , as expected.

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  2. Wow thankyou sooooo much! you saved me xD

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