) There is a hotel which has 50 rooms. When the rate is $180/night, all rooms will be booked up. There will be one more room available when the rate is increased by $10. For example, when it is $200/night, 48 rooms will be booked. For each room booked, the hotel has to pay $20 for maintenance. Whats the best rate for the hotel in order to maximize the profit

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

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