Answer:
The best rate for the hotel for profit maximization is = 340 $/room
Explanation:
Given that
A hotel room has = 50 rooms
The rate per night = $180
More room are available when the rate is increased by = $10
A maintenance fee of =$20
Now
We find the best rate for the hotel in order to have for profit
Thus,
When no rate increase is found we have the following,
Cost = ( 180 $/room * 50) = $ 9000
Thus,
When there is a rate increase for a room, we have the following
10x $/ room
The new cost becomes = (180 + 10x) $/room * (50 - x)
which is = 9000 = 500x - 180 x - 10x²
= 9000 + 320 x - 10x²
To get the new profit, we have the following :
Thus,
Profit = (New cost) - (cost)
Profit = (9000 + 320 x - 10x²) - (9000)
= 320x - 10x²
By applying maximization
dp/dx = 0 = 320 -10 * 2x = 0
So,
x = 16
Therefore,the best rate for the hotel for profit maximization is = (180 + 10 * 16)
=340 $/room
