The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 460 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one additional unit will be occupied for each 10 dollar decrease in rent. What rent should the manager charge to maximize revenue?
let the number of $10 increases be n
new rent = 460+10n
number rented= 80-n
Revenue = (80-n)(460+10n)
= 36800 + 340n - 10n^2
d(Rev)/dn = 340 - 20n
= 0 for a max/min
20n=340
n = 17
So there should be an increase of 17(10) or 170
and the new rent should be 630
and 63 units would be rented
check:
let n = 16
rent = 620
units rented = 64
Rev = 620(64) = 39680
let n = 17
rent = 630
units rented = 63
Rev = 39690 , a bit more , our maximum
let n = 18
rent = 640
units rented = 62
Rev = 39680 , not as much as the 39690 at n = 17
jnjkn
To maximize revenue, the manager needs to find the rent amount that will result in the highest occupancy rate while considering the market survey data.
Let's calculate the rent and occupancy for different scenarios:
Scenario 1: Rent decreases by $10
In this case, the occupancy will increase by one unit. So, if the rent decreases by $10, the number of occupied units will be 80 + 1 = 81.
Scenario 2: Rent increases by $10
In this case, the occupancy will decrease by one unit. So, if the rent increases by $10, the number of occupied units will be 80 - 1 = 79.
Based on the given information, we know that for every $10 change in rent, the occupancy changes by one unit.
Now, we need to determine the rent amount that should be set to maximize revenue. To do this, we can consider a range of rent increases and decreases from the initial rent of $460 per month and calculate the corresponding revenue for each case.
Let's consider different scenarios:
Scenario 1: Rent decreases to $450
Occupancy: 81 units
Revenue: 81 * $450 = $36,450
Scenario 2: Rent decreases to $440
Occupancy: 82 units
Revenue: 82 * $440 = $36,080
...
We can continue this process for different rent amounts and calculate the corresponding revenues. Then, we can compare the revenues to find the one that maximizes revenue.
Note: It is important to consider the demand and competition in the market while setting the rent. Market research should be conducted to determine the optimal rent that will attract tenants while maximizing revenue.
To maximize revenue, the manager needs to determine the rent at which the maximum number of units will be occupied. This can be done by analyzing the relationship between the rent amount and the number of units occupied.
Let's break down the information provided:
- At a rent of $460 per month, 80 units are occupied.
- For each $10 increase in rent, one additional unit remains vacant.
- For each $10 decrease in rent, one additional unit is occupied.
Now, we can set up a relationship between the rent (R) and the number of units occupied (N) as follows:
N = 80 + (R - 460)/10
This equation represents the occupancy rate based on the rent amount. The term (R - 460)/10 accounts for the decrease or increase in the number of units based on the change in rent.
To maximize revenue, the manager needs to find the rent at which the number of units occupied is maximized. Since revenue is a product of the rent and the number of units occupied, the maximum revenue occurs when the number of units occupied is maximized.
To find the maximum number of units occupied, we can take the derivative of the equation with respect to the rent (dN/dR) and set it to zero, as this will give us the critical points where the rate of change of occupancy is zero (maxima or minima).
dN/dR = 1/10
Setting dN/dR to zero and solving for R, we get:
1/10 = 0
However, this equation has no solution, indicating that the occupancy rate keeps increasing as the rent increases indefinitely.
Therefore, to maximize revenue, the manager should charge the highest possible rent. In this case, the highest possible rent would be the amount at which all units are occupied.
Since 80 units are occupied at a rent of $460 per month, the manager should charge $460 per month to maximize revenue.