The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $900 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

If 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 will be $1,000

x = $10 increases

Price=$900+10x

Quantity=100-x

First step is to find revenue by coming up with an equation

Revenue=Price × Revenue

R(x)=(900+10x) (100-x)

Second step is to maximize the revenue

R(x)=90,000 +200x-10x²

Third step is to find the derivative and set it to zero

R(x)=200-20x

0=200-20x

20x=200

Divide both side by 20x

x=200/20

x=10

Fourth step is to  determine rent that the manager should  charge to maximize revenue

Price=$900+10x

Price=$900+10(10)

Price=$900+$100

Price=$1,000

Inconclusion if 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 will be $1,000

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$950 in order to maximize the revenue.

Explanation:

The computation of monthly rent in order to maximize revenue is shown below:-

R (x) = Rent price per unit × Number of units rented

= ($900 + $10 x) × (100 - x)

= $90,000 - 900 x + 1000 x - 10 x^2

R (x) = -10 x^2 + 100 x + $90,000

Here to maximize R (x), we will find derivative and equal it to zero

R1 (x) = -20 x + 100 = 0

20 x = 100

x = 5

Therefore the monthly rent is p(5) = $900 + 10(5)

= $900 + 50

= $950 in order to maximize the revenue.