What rent should the manager charge to maximize revenue?

Question

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 448 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one additional unit will be occupied for each 7 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

Let x times rent increase by $7. Then revenue R(x) = (448 + 7x)(90 - x) = 40320 + 182x - 7x²;

R'(x) = 182 - 14x = 0; x = 182/14 = 13; max R(x) = R(13) = 41503.

Let y times rent decrease by $7 dollars. Then revenue R(y) = (448 - 7y)(90 + y) = 40320 - 182y - 7y²;

R'(y) = - 182 - 14y = 0; y = - 13. So, this means we have to increase rent 13 times and max rent = $41503