The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 416 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue?
revenue = # units * rent/unit
If each unit is rented for x,
r = x (100-(x-416)/8)
dr/dx = 152 - x/4
dr/dx=0 when x = 608
so, max rent is 608*76 = 46208
If you had read the whole explanation, you would have seen that the answer of rent=608 was given. Sorry for giving you more than you needed.
You could have saved reposting it and looking clueless.
To maximize revenue, the manager needs to find the rent that will result in the highest number of occupied units.
Let's break it down step-by-step:
Step 1: Determine the relationship between rent and the number of occupied units.
- For every $8 increase in rent, one additional unit will remain vacant.
- For every $8 decrease in rent, one additional unit will be occupied.
Step 2: Calculate the number of units that will be occupied at different rent levels.
Start with the given information:
- Rent at $416 → 100 units occupied (from experience)
Let's calculate the number of units occupied for different rent levels.
For $408 rent:
- $8 decrease from $416 rent
- 1 additional unit occupied
- Total occupied units: 100 + 1 = 101
For $400 rent:
- $16 decrease from $416 rent
- 2 additional units occupied
- Total occupied units: 100 + 2 = 102
Similarly, we can continue this process to find the number of occupied units for each $8 decrease in rent.
Step 3: Calculate the number of units that will be occupied at different rent levels for increases in rent.
For $424 rent:
- $8 increase from $416 rent
- 1 additional unit remains vacant
- Total occupied units: 100
For $432 rent:
- $16 increase from $416 rent
- 2 additional units remain vacant
- Total occupied units: 100
Similarly, we can continue this process to find the number of occupied units for each $8 increase in rent.
Step 4: Analyze the results and find the optimal rent.
Based on the calculations, we can create a table to summarize the number of occupied units for different rent levels:
Rent | Number of Occupied Units
------------------------------------
$400 | 102
$408 | 101
$416 | 100
$424 | 100
$432 | 100
From the table, we can see that the highest number of occupied units (102 units) occurs at a rent of $400.
Therefore, the manager should charge a rent of $400 to maximize revenue by having the highest number of occupied units.
To determine the rent that the manager should charge to maximize revenue, we need to find the point at which the number of occupied units multiplied by the rent is maximized.
Let's assume that the number of additional units that will remain vacant or be occupied can be represented by the variable "x". Let's also assume that the current rent is "R" dollars.
From the information given, we know that if the rent is $416, then 100 units will be occupied. This means that if the rent increases by $8, one additional unit will remain vacant. Conversely, if the rent decreases by $8, one additional unit will be occupied.
Therefore, the number of occupied units can be expressed as:
Number of occupied units = 100 + (x / 8)
Similarly, the number of vacant units can be expressed as:
Number of vacant units = 100 - (x / 8)
Now, the revenue can be calculated by multiplying the number of occupied units by the rent:
Revenue = (100 + (x / 8)) * R
To maximize the revenue, we need to find the value of "x" that yields the maximum value for the revenue equation. To do this, we can take the derivative of the revenue equation with respect to "x" and set it equal to zero.
d(Revenue) / dx = (1/8) * R = 0
Solving this equation, we find that R = 0. Therefore, the rent has no impact on the revenue.
Since the rent has no impact on the revenue, the manager should charge the highest possible rent, which is $416 per month.