Revenue Word Problem

Question

You are managing a 1600 unit apartment complex. Currently, 800 units are rented at $300/month. A market survey has indicated that for each $5 you decrease the rent you will get by 20 new leases.

a) Determine the function R(x) that models the total rental income, where x is the number of $5 decreases in monthly rent. 

Answer 1

R(x) = (number of units rented)(price per unit)

 

R(x) = (800 + 20x)(300 - 5x)

 

where x = the number of $5 decreases in rent.  x must be less than or equal to 40 because you only have 1600 units total.  Multiplying out the two binomials in R(x) gives:

 

R(x) = -100x² + 2000x + 240,000

 

R(x) is an inverted parabola, so the vertex will be the maximum revenue.  Complete the square on R(x) to put it into the vertex form y = a(x-h)²+k where (h, k) is the location of the vertex.:

 

R(x) = -100(x² - 20x + (-10)²) + 240,000 + (100)(10²)

R(x) = -100(x-10)² + 250,000

 

The vertex is located at (10, 250,000).  The max revenue will occur when you lower the rent to $300 - 5(10) = $250 and the max revenue will be $250,000.