How to figure out maximum profits using profit parabola

Question

The business manager of a 90 unit apartment building is trying to determine the rent to be charged. From past experience with similar buildings, when the rent is set at $400, all units are full. For every $20 increase in rent,one additional unit remain vacant. What rent should be charged for a maximum total revenue? What is the maximum total revenue?

 

Must solve using the profit parabola.

Answer 1

x = number of increases

R = revenue

 

R = (90 - x)(400 + 20x)

R = 3600 - 380x - 20x²

R = -20x² + 380x + 3600              Use completion of the square

R = -20(x² - 19x + (19/2)²) + 3600 + 1805

R = -20(x - 19/2)² + 5405

 

Maximum revenue is at 9.5 increases.

 

You must interpret this in the context of the real world.