# Calculus- Economics?

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 unit costs the property management company $72.50/month 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?

### 3 Answers

- llafferLv 75 months agoFavorite Answer
Profit is revenue minus cost.

So when at full capacity, the revenue and costs are:

R() = 150 * 460

R() = 69000

C() = 72.5 * 150

C() = 10875

For every $25 rent increase, 4 fewer apartments are rented. Since 150 is the max, we can't have a $25 decrease from this base.

So if we let "x" be the number of $25 rent increases, then the rent becomes: (460 + 25x) and the number of units rented will be (150 - 4x)

With this we can come up with updated revenue and cost functions and come up with a profit function in terms of "x":

R(x) = (460 + 25x)(150 - 4x)

C(x) = 72.5(150 - 4x)

Now let's come up with the profit function and simplify:

P(x) = R(x) - C(x)

P(x) = (460 + 25x)(150 - 4x) - 72.5(150 - 4x)

P(x) = 69000 - 1840x + 3750x - 100x² - 10875 + 290x

P(x) = -100x² + 2200x + 58125

Now that we have a profit function where "x" is the input and it's a quadratic we can solve for the maximum by solving for the zero of the first derivative:

P'(x) = -200x + 2200

0 = -200x + 2200

200x = 2200

x = 11

So the maximum profit will be had when there are 11 increases of $25. This amount becomes:

460 + 25x

460 + 25 * 11

460 + 275

$735 per month

And they will rent out:

150 - 4x

150 - 4 * 11

150 - 44

106 units

- King LeoLv 75 months ago
.

Monthly rental p for each unit if n units are occupied

( 150, 460 )

( 146, 485 )

( p, n )

( p - 485 )( 146 - 150 ) = ( 485 - 460 ) ( n - 146 )

p = ( 1397.50 - 6.25n )

Cost of occupying n units; C = 72.50n

Revenue from n units = np = ( 1372.50n - 6.25n² )

Profit, P:

P = ( 1397.50n - 6.25n² ) - 72.50n

P = 1325n - 6.25n²

P’(n) = 1325 - 12.5n

1325 - 12.5n = 0

∴ n = 106

p = ( 1397.50 - 6.25 * 106 )

p = $735

————

- Mike GLv 75 months ago
Points given are (460,150) and (485,146)

x = Rent, y = Units occupied

Slope = -4/25 = -0.16

Linear equation

y-150 = -0.16(x-460)

y = -0.16x + 223.6

R = Revenue = xy = -0.16x^2+223.6x

C = Costs = 72.5y = -11.6x+16211

P = Profit = R-C

P = -0.16x^2+223.6x+11.6x-16211

P = -0.16x^2+235.2x-16211

dP/dx = -0.32x+235.2 = 0 for maximum

x = 235.2/0.32 = 735

Answer $735 per month