A property management company manages an apartment block containing 150 units. All 150 units are rented at a monthly rate of $460 per unit and each occupied unit costs the

Question

A property management company manages an apartment block containing 150 units.

All 150 units are rented at a monthly rate of $460 per unit and each occupied unit costs the

property management company $72.50/month per unit for utilities and repairs. For every $25

rent increase, four fewer apartments are occupied. What rent should be charged in order to

realize the most profit?

Answer 1

r represents the number of $25 increases

P(r) = (150 - 4r)(387.50 + 25r)

Expand, convert to vertex from (Algebra 2),

Answer 2

Profit/month (P) = The number of rented apartments (N) * (rent (r) - 72.5)

1) P = N(r-72.5) (profits/month)

We are also told that

2) dN/dr = -4/25 (units/rent $/month)

Integrating (2)

3) N-N(r=460) = -4/25(r-460) N(460) = 150

So the relationship between N and r is

N = 150 -4/25 (r-460) = -.16r + 223.6

Now substitute into the profit equation (1)

P = (.16r+223.6)(r-72.5)

I'll leave the rest to you: Find r^* for dP/dr = 0 and check that P(r*) is greater than P(460)