# 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...

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.

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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**