Problema Solution
The manager of a 50-unit apartment complex is trying to decide what rent to charge. He knows that all units will be rented if he charges $600 per month, but for every increase of $25 over the $600, there will be a unit vacant. Find the number of occupied units if his total monthly rental income is $32,625 and the rent per unit is less than $1000.
Answer provided by our tutors
For $600 per month all 50 apartments are rented;
For $600 + $25 per month 50 - 1 apartments are rented;
For $600 + $25*x per month 50 - x apartments are rented, where x is non-negative integer;
Lets find x such that
(600 + 25*x)(50 - x) = 32625
rent per unit is less than $1000 that means $600 + $25*x < 1000 that is x < 16.
Lets solve the quadratic equation
(600 + 25*x)(50 - x) = 32625
there are 2 roots of the equation
x1 = 21 this is not a solution since 21 > 16
x2 = 5 this is the solution we are looking for
Indeed
45 units are rented by the price of 600 + 5*25 = $725 so the total monthly rental income is 45*725 = $32,625
the rent $725 per unit is less than $1,000
The number of occupied units is 45.