Maximum profit: a real estate office handles a 50-unit apartment complex. When the rent is $580 per month, all units are occupied. For each increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $45 per month for service and repairs.
In the office unit the number of units occupied when the rent is $580 is 50. The revenue from rent is 580*50 = 29000. The cost involved in service and repair for each unit is $45. The cost for 50 units is $ 2250. This gives the total profit as 29000 - 2250 = $26750.
For each increase in rent of $1, the occupancy decreases by 1. If the rent is raised by $x, the profit earned is P(x) = (580+x)(50-x) - (50-x)*45
= 29000 - 530x - x^2 - 2250 + 45x
= -x^2 - 485x + 26750
To determine the rent that maximizes the profit solve P'(x) = 0
=> -2x - 485 = 0
=> x = -485/2
But reducing the rent does not increase the number of units occupied as all 50 are occupied when it is $580; neither is the cost decreased.
For maximum profit the rent should be kept at $580. And the maximum profit made is $26750