# revenue/profit problem

• May 4th 2008, 04:49 PM
gumi
revenue/profit problem
You manage a 280 unit motel. All units will be occupied, on average, when
you charge \$50/per day per unit. Experience shows that for each \$1 per day per unit that you increase the price, you will accrue 2 vacancies. Each occupied room costs \$4 a day to clean and supply. On this basis, what price, in dollars, should you charge per unit per day to maximize your profit?
• May 5th 2008, 12:18 AM
Moo
Hello,

Quote:

Originally Posted by gumi
You manage a 280 unit motel. All units will be occupied, on average, when
you charge \$50/per day per unit. Experience shows that for each \$1 per day per unit that you increase the price, you will accrue 2 vacancies. Each occupied room costs \$4 a day to clean and supply. On this basis, what price, in dollars, should you charge per unit per day to maximize your profit?

Let X the number \$1 you add to the original price.
The price per day & per unit is 50+X.

The profit of each occupied room is (50+X)-4=46+X, where 4 is the cost for cleaning and supplying.
So the number of occupied rooms is 280-2X (each time you increase of 1\$, there are 2 vacancies).
We assume that an empty room makes no profit.

Therefore, the total profit is P(X)=(46+X)(280-2X)=-2X²+188X+12880.
We want it to be maximum.

--> \$\displaystyle P'(X)=-4X+188\$, which annulates at X=188/4=47.

So the price of the rooms has to be 50+47=97.

I hope I didn't make any mistake (Wondering)