The business manager of a 90 unit apartment building is tryi

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? What is that maximum total revenue?

Solution

Number of units in apartment building = 90

Rent for which all the units will be full = $400/-

For every $20 increase , there is an additional unit vacant.

Now let us consider, the rent is increased by n*20 dollars.

That means , n units are vacant.

So total income, when the flat rent is 400+20n, and n units are vacant is (90-n)(400-20n)

The maximum total revenue is obtained when this value is maximum

R = (90-n)(400-20n)

R = 20n^2-2200n+36000

This is in quadratic form. The maxima is at -b/2a = -(-2200)/(2*20)= 220/4 = 110/2 = 55

So to have maximum revenue n = 55

So rent to be charged is 400+55*20 = 400+1100=$1500/- per flat.

The total revenue = 1500(90-55) = $52,500/-