Call the number of $15 rent increases, x
So....
The number of apartments rented can be expressed as (100 - x)
The revenue earned per apartment is (800 + 15x)
The total revenue is Number of apartments rented * Revenue per apartment
The total maintenance cost is hazy....I'm taking it to mean $60 * number of apartments rented (since unrented apartments shouldn't require any maintenance )
So.....the Total Profit per month is Total Revenue per month - Total Maintenance cost per month
So we have
P(x) = (100 - x)(800 + 15x) - (60)(100-x)
P(x) = (100 - x) (800 + 15x - 60 )
P(x) = (100 - x) (15x + 740 )
Which simplifies to
P(x) = -15x^2 + 760x + 74000
Take the derivative of this and set to 0
P'(x) = -30x + 760
-30x + 760 = 0
760 = 30x divide both sides by 30
x = 25.333 ≈ 25 = number of $15 increases
This means that the max profit is made when about (100 - 25) = 75 apartments are rented
And the max profit is achieved when the rent per apartment is (800 + 15(25) ) ≈
$1125 per month
And the max revenue is about
-15 (25)^2 + 760 (25) + 74000 = $83625
Note that if all 100 apartments were rented..... x = 0 ..... and he would only make ≈ $74000
