Question

The business manager of a 90 unit apartment building is trying to determine the rent to be charged. From past experience with similar buildings, when rent is set at $400, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue

Answer 1

Answer:

$1100 per apartment

Explanation:

For each x times increase by $20, x units goes vacant.

Thus,

We can say:

Rent is 400 + 20x

The number of apartments rented then is 90 - x

The revenue would be the rent multiplied by the number of apartments rented. So revenue would be:

$R(x)=(400+20x)(90-x)\\R(x)=36000-400x+1800x-20x^2\\R(x)=-20x^2+1400x+36000\\R(x)=20x^2-1400x-36000\\R(x)=x^2-70x-1800$

This is a parabola with a = 1 , b = -70, and c = -1800

THe max value occurs at $x=-\frac{b}{2a}$

SO, that would be at:

$x=-\frac{-70}{2(1)}=35$

Hence, if there is "35" increases of $20, that would give us the max revenue.

35 number of increases of $20 means:

20*35 = 700 increase

So, rent should be set at 400 + 700 = $1100 per apartment