The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 500 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?
To maximize revenue, the manager needs to find the rent that will result in the maximum number of units occupied.
Let's assume the rent increase or decrease is by "x" dollars.
From the information given, we know that the number of units occupied is inversely proportional to the rent.
If we let "n" be the number of units occupied and "r" be the rent, we can write the following equation:
n = 90 + (500 - r)/10
To calculate the revenue, we need to multiply the number of units occupied by the rent:
Revenue = n * r
Replacing the value of "n" from the equation above, we get:
Revenue = (90 + (500 - r)/10) * r
To maximize the revenue, we need to find the value of "r" that maximizes this equation.
We can find the maximum value using calculus by taking the derivative of the revenue equation with respect to "r" and setting it to zero.
To make this process easier, let's simplify the equation by expanding it:
Revenue = 90r + (500 - r)r/10
To maximize this equation, we differentiate it with respect to "r":
dRevenue/dr = 90 - r/10 + 50 - r/10
= 140 - 2r/10
Setting this derivative equal to zero:
140 - 2r/10 = 0
Simplifying, we get:
140 = 2r/10
140 = r/5
r = 140 * 5
r = 700
Therefore, the manager should charge a rent of $700 to maximize revenue.
To determine the rent that maximizes revenue, we need to find the number of occupied units at different rent levels and calculate the corresponding revenue.
Let's start by using the given information: when the rent is $500 per month, 90 units are occupied.
According to the market survey, one additional unit will remain vacant for each $10 increase in rent. This means that for every $10 decrease in rent, one additional unit will be occupied.
To find the optimal rent, we'll need to determine how the number of occupied units changes as rent increases or decreases.
First, let's consider the case of increasing the rent by $10. We know that one unit will remain vacant for each $10 increase. Therefore, if the rent is increased to $510, only 89 units will be occupied.
Next, consider decreasing the rent by $10. We know that one additional unit will be occupied for each $10 decrease. Therefore, if the rent is decreased to $490, 91 units will be occupied.
Based on this pattern, we can make the following observations:
- For every $10 increase in rent, one unit remains vacant.
- For every $10 decrease in rent, one additional unit is occupied.
Now, let's calculate the revenue at different rent levels. Revenue is equal to the rent multiplied by the number of occupied units.
Revenue at $500 rent:
Revenue = $500 * 90 units = $45,000
Revenue at $510 rent:
Revenue = $510 * 89 units = $45,390
Revenue at $490 rent:
Revenue = $490 * 91 units = $44,590
Based on these calculations, we can see that the revenue is highest when the rent is $510, resulting in a revenue of $45,390.
Therefore, to maximize revenue, the manager should charge a rent of $510 per month.