What rent should the manager charge to maximize revenue

Question

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 420 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 3 dollar increase in rent. Similarly, one additional unit will be occupied for each 3 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

(420+3x)(110-x) where x=# of $3 increases

(420-3x)(110+x) where x=# of $3 decreases

46,200+330x-420x-3x^2 (increase)

46,200-90x-3x^2

divide by 3

15,400-30x-x^2

46,200-330x+420x-3x^2 (decrease)

46,200+90x-3x^2

divide by 3

15,400+30x-x^2

set this equation equal to zero

0=15,400+30x-x^2

this is a parabola that opens downward with vertex (h,k)

h=-b/2a and k=maximum revenue

-b/2a=-30/-2=15

15 $3 decreases maximizes the revenue

(420-3*15)(110+15)=(420-45)(125)=(375)(125)=$46,875

16 decreases yields (420-48)(110+16)=(372)(126)=$46,872

14 decreases yields (420-42)(110+14)=(378)(124)=$46,872

any increases will decrease maximum revenue

1 increase: 423*109=$46,107

10 increases: 450*100=$45,000