In a guest house with 50 rooms when the rent is $45 per day all 50 are occupied. For a $5 increase in rent, one room becomes vacant. If the daily maintainence of each occupied room requires $3 what should the rent be to maximize the profit?
The guest house has a total of 50 rooms. When the rent of each room is $45 all the rooms are occupied. The maintenance costs for each occupied room is $3 per day. A $5 increase in the rent leads to one room becoming vacant.
Let the number of rooms that are vacant when the profit is maximum be x. The rent charged in this case is 45 + 5*x. The rent from the rooms occupied is (50 - x)*(45 + 5*x) and the maintenance is (50 - x)*3. The profit earned is P = (50 - x)*(45 + 5*x) - (50 - x)*3
= 2250 - 45x + 250x - 5x^2 - 150 + 3x
= 2100 +208x - 5x^2
To maximize profit solve P' = 0 for x.
P' = -10x + 208
-10x + 208 = 0
=> x = 20.8
The profit earned when 30 rooms are occupied is $4260 and when 29 rooms are occupied it is $4263.
29 rooms are occupied when the rent is $155.
The rent that should be charged to maximize profits is $155