a complex has 205 apartment units. When the rent is $635 per month all 205 units are occupied.Assume that for each $75 increase in rent 15 units becomes vacent. Assume that the number of occupied units is a linear function of the rent. Each occupied unit also requires an average of $60 per month for sevice charges. The rent charge should maximize the profits. Obtain a formula to show the profit.
To obtain a formula to show the profit, we need to consider the following factors:
1. Total Rent Revenue: This is calculated by multiplying the number of occupied units by the monthly rent per unit. Since all 205 units are occupied when the rent is $635 per month, the revenue from rent can be given as:
Total Rent Revenue = Number of Occupied Units * Rent per Unit
2. Total Service Charge: This is calculated by multiplying the number of occupied units by the monthly service charge per unit. The service charge for each occupied unit is $60 per month. Therefore, the total service charge can be given as:
Total Service Charge = Number of Occupied Units * Service Charge per Unit
3. Vacancy Loss: As the rent increases, some units become vacant. For each $75 increase in rent, 15 units become vacant. We can calculate the number of vacant units based on the rent increase by the formula:
Number of Vacant Units = (Rent Increase / $75) * 15
4. Total Expenses: This includes the cost of service charges for occupied units and any potential vacancy loss. We can calculate the total expenses as:
Total Expenses = Total Service Charge + (Number of Vacant Units * Service Charge per Unit)
5. Profit: Finally, the profit can be calculated by subtracting the total expenses from the total rent revenue:
Profit = Total Rent Revenue - Total Expenses
By combining these formulas, we can obtain a formula to show the profit as a function of the rent.
Let's define some variables:
x = Rent Increase (in dollars)
P = Profit (in dollars)
The formulas can now be written as:
Number of Occupied Units = 205 - ((x / $75) * 15)
Total Rent Revenue = (205 - ((x / $75) * 15)) * (635 + x)
Total Service Charge = (205 - ((x / $75) * 15)) * $60
Number of Vacant Units = (x / $75) * 15
Total Expenses = (205 - ((x / $75) * 15)) * $60 + ((x / $75) * 15) * $60
Profit = (205 - ((x / $75) * 15)) * (635 + x) - ((205 - ((x / $75) * 15)) * $60 + ((x / $75) * 15) * $60)
This equation represents the profit as a function of the rent increase (x). By analyzing this equation, we can find the rent increase that maximizes the profit.