Question

A real estate company owns 120 apartments which are fully occupied when the rent is $650 per month. Analytics indicate that for each $25 increase in rent, 4 apartments will become unoccupied. What rent should be charged in order to obtain the largest gross income

Answer 1

Check the explanation

Step-by-step explanation:

This is a question under Calculus (which is another segment of mathematics and it is utilized for understanding the changes between values that are related by a function. Calculus is used in a lot of diverse fields of study such as astronomy, physics, biology, economics, medicine, engineering and sociology).

The step by step solution to the above question can be seen in the attached image below:

Answer 2

The rent that should be charged to maximize gross icome is $700.

Step-by-step explanation:

The rent can be increased by $25 steps.

The rent equation can be written as,

Rent = 650 + 25x

The Apartments occupied will fall by 4 apartments everytime the rent is increased by $25.

The apartment occupancy equation can be written as,

Apparment occuoancy = 120 - 4x

Where, x is the number of times the rent is increased by $25.

The revenue will be,

Revenue = Rent * No of apartments occupied

Revenue = (650 + 25x) * (120 - 4x)

Solving the equation we get quadratic equation,

Revenue = 78000 - 2600x + 3000x - 100x²

0 = 78000 + 400x - 100x²

As we need the change, we need to take the derivative of this equation.

d/dx = 0 + 1 * 400x° - 2 * 100x

400 - 200x = 0

400 = 200x

400 / 200 = x

x = 2

Thus, the rent that should be charged to amximize gross income is,

Rent = 650 + 25 * (2) = $700 per apartment

The revenue at this rent will be,

Revenue = 700 * [120 - 4 * (2)] = $78400