As the manager of a 60-unit motel you know that all units are occupied when you
ID: 2441377 • Letter: A
Question
As the manager of a 60-unit motel you know that all units are occupied when you charge $100 a day per unit. Each occupied room costs $76 for service and maintenance a day. You have also observed that for every x dollars increase in the daily rate above $100, there are 2x units vacant. Determine the daily price that you should charge in order to maximize profit. Calculate the number of occupied units. Assuming that fixed cost is $562 calculate optimal profit.
(Hint: You may like to determine the demand function first.
Explanation / Answer
Let x = number of dollar increases in the rent per night
60 - 2x = total number of rooms rented (quantity)
100 + x = rent per night (Price)
Revenue = P*Q = (100+x)*(60-2x) = 6000 -140x - 2x2
Cost = Fixed cost + Variable cost
Cost = 562 + 76Q
Cost = 562 + 76 (60-2x) = 562 + 4560 - 152x
Cost = 5122 - 152x
Profit = Revenue - Cost
Profit = 6000 - 140x - 2x2 - 5122 + 152x
Profit = 878 +12x - 2x2
To maximize profit, put d(profit)/dx = 0,
12 - 4x = 0
X = 3
Also we do double derivative, which is equal to -4 means less than 0, it means that x= 3 maximize the profit.
The daily price that manager should charge in order to maximize profit, 100+x
Put x = 3,
So the manager should charge $103 as the daily price.
The number of occupied units = 60 - 2x
Put x = 3
The number of occupied units = 54
The optimal profit :
Profit = 878 + 12x - 2x2
Put x = 3
Profit = 878 + 12*3 - 2*9 =$896.
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