Find the Maximized Rent in $

Question

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $500 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue?

Answer 1

Let x = number of $10 increases

R(x) = revenue = (price)(quantity) = (500 + 10x)(100 - x) = -10x² + 500x + 50000

Max when x = -500/ [2(-10)] = 25

To maximize revenue, the rent should be 500 + 10(25) = 750

Answer 2

The monthly revenue generated by renting out the apartments is given by the formula:

Revenue = Rent * Units occupied

In this case, the rent is given by:

Rent = 500 + 10x

And the number of units occupied is:

Units occupied = 100 - x

Substituting these values, we get:

Revenue = (500 + 10x) * (100 - x)

Expanding this expression, we get:

Revenue = 50000 - 500x + 1000x - 10x^2

Simplifying, we get:

Revenue = -10x^2 + 500x + 50000

To find the value of x that maximizes revenue, we can take the derivative of the revenue function with respect to x and set it equal to zero:

dRevenue/dx = -20x + 500 = 0

Solving for x, we get:

x = 25

Therefore, the number of $10 increases in rent that will maximize revenue is 25, and the corresponding rent is:

Rent = 500 + 10x = 500 + 10(25) = $750

Therefore, the manager should charge a rent of $750 to maximize revenue.