# 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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