math

miss Thao owns a small hotel with 30 identical rooms. she finds that if she charges a price of 12 dollars ( or less) per room per day, all the rooms are rented. for every 1 dollar in price, 2 rooms remain vacant. each day, maintenance cost 4 dollars per rented room and 1 for unrented room. what price should miss thao charge n order to maximize her daily profit? hint: profit = revenue - costs
Help me, thanks a lots

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  1. current number of rooms = 30
    current price per room = 12

    let the number of additional increases of $1 be n

    numbers of rooms rented = 30 - 2n
    number of rooms not rented = 30 - (30-2n) = 2n
    price per room = 12 + n

    Profit = (12+n)(30-2n) - 4(30-n) - 1(2n)
    = 360 + 6n -2n^2 - 120 + 4n - 2n
    = -2n^2 + 8n + 240

    by Calculus:
    d(profit)/dn = -4n + 8
    = 0 for a max of profit
    4n = 8
    n = 2

    the price per room should be 12+n = 12+2
    = $ 14.00

    if you don't know calculus:

    profit = -2n^2 + 8n + 240
    the n of the vertex = -8/-4 = 2
    so the price should be $ 14.0

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  2. tks Reiny so much :x

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