I am using derivatives to find this. It is business calculus.

Question

A hotel has 300 rooms.  All rooms are occupied when the hotel charges $80 per day for a room. Research show that for every one dollar increase in the daily room rate, the number of vacant rooms also increases by one. Each occupied room costs $22 per day to service and maintain. Assuming the fixed costs to operate is $1000 what should the hotel charge per day in order to maximize profit? Find that maximum profit and give units. Hint: find the demand equation

Answer 1

So for every $1 increase in price, the number of rooms goes down by 1.

 

So the income can be represented by R(x) = (300-x)(80+x)

 

From that, you need to subtract 22(300-x) since it costs $22 per occupied room.

AND there is a fixed cost of $1000 per day.

 

So your profit function is P(x) = (300-x)(80+x) - 22(300-x) - 1000

Simplified, this is P(x) = -x² +242x + 16400.

 

P'(x) = -2x + 242    Set it = to 0

x = 121 which represents the number of vacant rooms.

 

P(121) = -14641 + 29282 + 16400 = $31041

 

You can check by calculating P(120) and P(122) which both equal $31040.

 

Hope this helps.