A real estate manages 80 apartment units. When the rent of each unit is $180 per month, all units are occupied. However, for each $6 increase in rent, one of the units becomes vacant. Each occupied unit requires an average of $18 per month for service and repairs. What rent should be charged to realize the most profit?
number of $6 increases ---- n
right now:
number rented = 80
rent = 180
after increase:
rent = 180+6n
number rented = 80-n
revenue = (180+6n)(80-n) - 18(80-n)
=14400 + 300n - 6n^2 - 1440 + 18n
= -6n^2 + 318n + 12960
= -6(n^2 - 53n - 2160)
d(revenue)/dn = -6(2n - 53) = 0 for a max of revenue
n = 53/2
.....was expecting a whole number, check if typed correctly.
Yes the question is typed correctly but thank you!
To determine the rent that should be charged to realize the most profit, we need to calculate the point at which the additional revenue from increased rent surpasses the loss of revenue from vacant units, taking into account the cost of service and repairs.
Let's break down the problem step by step:
1. Calculate the current total revenue:
Total units = 80
Rent per unit = $180
Current total revenue = Total units * Rent per unit
Total revenue = 80 * $180 = $14,400 per month
2. Determine the number of vacant units for each $6 increase in rent:
For each $6 increase in rent, one unit becomes vacant.
So, the number of vacant units = Total units / $6
Number of vacant units = 80 / 6 = 13.33 ~ 13 units (rounded down to the nearest whole number)
3. Calculate the revenue and expenses for each rent increase:
We need to calculate the total revenue and subtract the cost of service and repairs for each potential rent amount.
Let's consider different rent increases starting from the current rent of $180 and calculate the profit for each:
a) For the current rent of $180:
Number of vacant units = 0
Revenue = Total units * Rent per unit = 80 * $180 = $14,400
Expenses for service and repairs = Total units * Cost per unit = 80 * $18 = $1,440
Profit = Revenue - Expenses = $14,400 - $1,440 = $12,960
b) For a $6 rent increase ($186):
Number of vacant units = 13
Revenue = (Total units - Number of vacant units) * Rent per unit = (80 - 13) * $186 = $10,548
Expenses for service and repairs = Total units * Cost per unit = 80 * $18 = $1,440
Profit = Revenue - Expenses = $10,548 - $1,440 = $9,108
c) Repeat the above calculation for each subsequent $6 rent increase until the profit starts decreasing.
Calculate the profit for $192, $198, $204, and so on, until the profit starts to decrease.
4. Compare the profits and determine the rent that achieves the highest profit:
Determine the rent amount that yields the highest profit by comparing the calculated profits for different rent increases. The highest profit will correspond to the optimal rent.
It is important to note that this process assumes a linear relationship between rent increase and the number of vacant units. In reality, the relationship might be more complex, and market factors should also be taken into account. Consulting with a real estate professional would help in making an informed decision.