The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 370 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 5 dollar increase in rent. Similarly, one additional unit will be occupied for each 5 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 370 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 5 dollar increase in rent. Similarly, one additional unit will be occupied for each 5 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

let the number of $5 increases by n

(e.g. if n=1 rent will be 375, if n = 4 rent will be 390 etc)

so the rent will be $(370+5n)
number of units occupied will be 90 - n

Revenue = (370+5n)(90-n)

expand and simplify, take the first derivative, set that equal to zero and solve for n

I did not finish it, but if n is positive there will be an increase in rent and a decrease in the number of units rented.
If n should turn out negative, there would be a decrease in the rent and an increase in the number of units rented.

Answer 2

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Answer 3

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 300 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 2 dollar increase in rent. Similarly, one additional unit will be occupied for each 2 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 4

To maximize revenue, the manager needs to find the rent that will result in the maximum number of units occupied.

Let's break down the problem step by step:

1. Start with the given information:
- The current rent is $370 per month, and there are 90 units occupied.
- For each $5 decrease in rent, one additional unit will be occupied.
- For each $5 increase in rent, one additional unit will remain vacant.

2. Determine the relationship between the rent and the number of units occupied.
- For each $5 decrease in rent, the number of occupied units will increase by 1.
- For each $5 increase in rent, the number of occupied units will decrease by 1.

3. Set up a function to represent the revenue based on the number of units occupied.
- Revenue = Rent x Occupied Units

4. Calculate the revenue for the given rent of $370.
- Revenue = $370 x 90 units = $33,300

5. Determine the changes in revenue for each $5 increase and decrease in rent:
- For each $5 decrease in rent, the number of occupied units will increase by 1, resulting in an increase in revenue.
- For each $5 increase in rent, the number of occupied units will decrease by 1, resulting in a decrease in revenue.

6. Start adjusting the rent from $370 and calculate the corresponding revenue:
- Decrease the rent by $5 and calculate the new revenue.
- Increase the rent by $5 and calculate the new revenue.
- Repeat this process until you find the maximum revenue.

7. Compare the revenues obtained from each adjusted rent:
- Identify the rent that corresponds to the highest revenue.

Using this method, the manager can find the rent that maximizes revenue by considering the relationship between rent and the number of units occupied.

let the number of $5 increases by n

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344

To maximize revenue, the manager needs to find the rent that will result in the maximum number of units occupied.