The business manager of a 90 unit apartment building is tryi

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 rent is set at $400, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue? What is that maximum total revenue?Find the maximum revenue (or income)of the apartment building. What is the rent coincides with this maximum revenue? What is the outcome if the rent hike of $20 results in 2 additional vacancies instead of 1?

Solution

Let x units be vacant for the maximum revenue. Then the number of units occupied is 90 - x and the rent per unit is 400 + 20 x . The total revenue ( say y) is ( 90 - x)( 400 + 20x) = 36000 -400x +1800x - 20x² = - 20x² + 1400x + 36000. If the revenue (y) is to be maximum then dy/dx = 0 or, - 40x + 1400 = 0 or, 40 x = 1400 or, x = 35. Thus the rent charged will be 400 + 35(20) = 400 + 700 = $ 1100 per unit and the maximum revenue is (90 - 35)(1100) = 55*1100 = $ 60500. We can verify this result by increasing and decreasing the no. of vacant units by 1.

First, let us increase the no. of vacant units by 1 so that only 54 units are occupied.Then the rent per unit is $ 1120 (1100 + 20) so that the total revenue is 54*1120 = $ 60480.

Now, let us reduce the no. of vacant units by 1 so that the no. of units occupied is 56. Then the rent per unit is 1100 - 20 = $ 1080 and the total revenue is 56*1080 = $ 60480. Since 60500 > 60480, our answer that 55 units occupied @ $ 1100 per unit For a total revenue of $ 60500, is correct.

Now, if we assume that 2x units fall vacant for every $ 20 increase in rent, then the revenue (y) =( 90 - 2x)(400 + 20x) = 36000 - 800x + 1800x -40x² or, y = - 40x² + 1000x + 36000. Now, for maximum revenue, dy/dx = 0 or,

-80x + 1000 = 0 or x = 1000/80 = 12.5 Therefore 2x = 25. Thus, the rent per unit will be 400 + 12.5 ( 20) = $ 650 and the no. of units occupied is 90 -2x = 90 -25 = 65. The maximum revenue is 65* 650 = $ 42250.