Calculus - Finding demand and revenue functions

Question

--Manager of a 100-unit apartment complex knows that all units will be occupied if the rent is $1,000 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $50 increase in rent. What rent should the manager charge to maximize revenue?

a) find demand function and revenue functions

b) to the nearest dollar, what rent should the manager charge to maximize the revenue? How many units will be vacant?

Answer 1

If we lose 1 unit/$50, the slope is -1/50

D = mP + b and, when P = 1000, D = 100

100 = -1/50 (1000) + b

100 = -20 + b

b = 120

D = -1/50 P + 120

R = D P = (-1/50 P + 120) P = -1/50P² + 120 P

b) Since this a calculus problem, we maximize by setting dR/dP = 0

dR/dP = -1/25 P + 120

-1/25 P + 120 = 0

1/25 P = 120

P = 25*120 = 3000

At this price, D = 1/50(3000) + 120 = 60

100 - 60 = 40 units are vacant

R = 3000 * 60 = 180000

Note that d²R/dP² = -1/25 < 0, so this is a maximum.