Question

The manager of an 110​-unit apartment complex knows from experience that at a rent of​ $300, all the units will be full. On the​ average, one additional unit will remain vacant for each​ $20 increase in rent over​ $300. Furthermore, the manager must keep at least 50 units rented due to other financial considerations.​ Currently, the revenue from the complex is ​$72 comma 000. How many apartments are​ rented?

Answer 1

The number of rented apartments is 80

Explanation:

Hi there!

The revenue is calculated as the number of units rented times the value of the rent. Let y be the revenue, then:

y = 300 · 110

For each $20 increase, one unit will remain vacant. Then, if the increase is 20x then the number of vacant apartments will be x. In this case, the revenue will be calculated as:

y = (300 + 20x)(110 - x)

Apply distributive property:

y = 33000 - 300x + 2200x - 20x²

y = -20x² + 1900x + 33000

If the revenue is $72000, then:

72000 = -20x² + 1900x + 33000

0 = -20x² + 1900x + 33000 - 72000

0 = -20x² + 1900x - 39000

Solving the quadratic equation using the quadratic formula:

a = -20

b = 1900

c = -39000

x = [-b±√(b² - 4ac)]/2a

x1 = 30 and x2 = 65

Since at least 50 units have to be rented, the numer of apartments rented is 110 - 30 = 80 (because 110 - 65 = 45<50).