Posted by **Deepa** on Thursday, April 24, 2014 at 8:56am.

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent should the real estate office charge to maximize profits?

I don't even know the answer to this one. I don't understand at all, please help me step by step

- Calculus -
**Reiny**, Thursday, April 24, 2014 at 9:41am
right now:

price of rent = 900

number of units rented = 50

Let the number of $25 increases be n

(e.g. If n= 2 , new rent is 900+2(25) = 950

if n = 5 , the new rent 900 + 5(25) = 1025

so the new rent = 900 + 25n

number rented = 50-n

maintenance cost = 75(50-n)

Profit = P = (900+25n)(50-n) - 75(50-n)

= 45000 + 350n - 25n^2 - 3750 + 75n

= 41250 + 425n - 25n^2

d(profit)/dn = 425 - 50n

= 0 for a max of P

50n = 425

n = 8.5

The question did not say if increases are in whole multiples of 25 , but I will assume that. We could not rent 50-8.5 or 41.5 units.

when n = 8 or n = 9

if n = 8

number rented = 42

rent = 900+8(25) = 1100

maintenace cost = 75(42) = 3150

Profit = 42x1100 - 3150 = 43050

if n = 9

number rented = 41

rent = 1125

maintenance cost = 3075

Profit = 41x1125 - 3075 = 43050 , as expected.

- Calculus -
**Deepa**, Thursday, April 24, 2014 at 11:22am
Wow thankyou sooooo much! you saved me xD

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