The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 424 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 2 dollar increase in rent. Similarly, one additional unit will be occupied for each 2 dollar decrease in rent. What rent should the manager charge to maximize revenue?
let the number of $1 increases be n
rent per unit = 424 + 2n
number of units rented = 100 - n
Rev = (424+2n)(100-n) = 42400 - 224n - 2n^2
d(Rev)/dn = -224 - 4n = 0 for a max/min of Rev
4n = -224
n = -56
He should actually reduce the rent to 424-2(56) = $312
and rent out 156 units
check:
n, number , rent
0 100 424 100*424 = 42,400
...
55 155 314 155*314 = 48,670
56 156 312 156*312 = 48,672 <------ max revenue
57 157 310 157*310 = 48.670