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2. An apartment complex has a demand given by p = -5x + 860,
where x is the number of apartments rented and p is the price
for each apartment, in dollars. The cost of maintaining the
complex is $120 per rented apartment. There are fixed costs of
$6000.
a) Find the revenue equation.
b) Find the cost equation.
c) Find the profit equation.
d) Find the number of apartments that must be rented to break
even.

Answer :

FICHOHGO

The required equations computed from the data given are :

  • R(x) = 860x - 5x²
  • C(x) = 120x + 6000
  • P(x) = 740x - 5x² - 6000
  • Break even point = 139 gallons and 9 apartment

Given the Parameters :

  • Price per apartment, p = - 5x + 860
  • x = number of apartments rented
  • Fixed cost = 6000
  • Maintainance cost per apartment = $120

The Revenue Function, R(x) :

  • Price per apartment × Number of apartments rented

  • (-5x + 860)x ;
  • R(x) = 860x - 5x²

The Cost equation, C(x) :

  • Fixed cost + Maintenance cost per apartment
  • C(x) = 6000 + 120x

The Profit function, P(x) ::

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

P(x) = 860x - 5x² - (6000 + 120x)

P(x) = 860x - 5x² - 6000 - 120x

P(x) = 740x - 5x² - 6000

Number of apartments required to break even :

At break even ;

P(x) = 0

740x - 5x² - 6000 = 0

Divide through by 5

-x² + 148 - 1200 = 0

Using a quadratic calculator :

The roots are (139.39, 8.60)

Therefore. The company will break even at 139 apartments rented or 9 apartments.

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