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optimization calculus

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a real estate office manages 50 apartments in 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 be real estate office charge to maximize profits?

  • optimization calculus - ,

    let the number of $25 increases be n
    new rent = 900 + 25n
    number rented = 50-n

    profit = (50-n)(900+25n) - 75n
    = 45000+275n - 25n^2

    d(profit)/dn = 275 - 50n = 0 for max/min of profit
    50n = 275
    n = 275/50 = 5.5

    The question is not clear whether the increases have to be in exact jumps of $25.
    if so, then n=5 or n=6 will yield the same profit
    e.g.
    if n=6, profit = 45000+1650-900 = 45750
    if n=5 , profit = 45000+1375-625 = 45750

    if a partial increase is allowed, then
    n = 5.5
    new rent = 1037.50
    number rented = 50-5.5 = 44.5 ???? (makes no real sense)
    but mathematically profit
    = 45000+5.5(275)-25(5.5)^2 = 45756.25

    I would go with either n=5 or n=6

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