The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 360 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 9 dollar increase in rent. Similarly, one additional unit will be occupied for each 9 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Question

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 360 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 9 dollar increase in rent. Similarly, one additional unit will be occupied for each 9 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

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Answer 2

To determine the rent that will maximize revenue, we need to find the point at which the number of occupied units multiplied by the rent amount will yield the maximum value.

Let's begin by calculating the number of units that will be occupied for different rent amounts.

At $360 rent per month, 90 units are occupied.

For each $9 increase in rent, one additional unit remains vacant. Therefore, for a $9 increase in rent from $360, the number of occupied units decreases by one. This means at $369 rent per month, there will be 89 occupied units.

Similarly, for each $9 decrease in rent, one additional unit is occupied. So, for a $9 decrease in rent from $360, the number of occupied units increases by one. This means at $351 rent per month, there will be 91 occupied units.

We can continue this pattern to find the number of occupied units for different rent amounts:

- At $342 rent per month, there will be 92 occupied units.
- At $333 rent per month, there will be 93 occupied units.
- At $324 rent per month, there will be 94 occupied units.

Now, let's calculate the revenue for each rent amount. Revenue is calculated by multiplying the number of occupied units by the rent amount.

- At $360 rent per month, the revenue is 90 units * $360 = $32,400.
- At $369 rent per month, the revenue is 89 units * $369 = $32,841.
- At $351 rent per month, the revenue is 91 units * $351 = $31,941.
- At $342 rent per month, the revenue is 92 units * $342 = $31,464.
- At $333 rent per month, the revenue is 93 units * $333 = $30,969.
- At $324 rent per month, the revenue is 94 units * $324 = $30,456.

From these calculations, we can see that the maximum revenue is achieved at $369 rent per month, with 89 occupied units, resulting in a revenue of $32,841.

Therefore, the manager should charge $369 per month to maximize revenue.