Answer:
Explanation:
To construct a mathematical model that relates the profit made by the owners of the apartment complex to the number of unoccupied units, we need to consider the revenue and cost components.
Let's denote the number of unoccupied units as "x". As the rent increases by $25, the number of rented units decreases by one. Therefore, the number of rented units can be represented as "50 - x".
The revenue generated from the rented units can be calculated by multiplying the number of rented units by the rent per unit. So, the revenue is given by 800 * (50 - x).
The total cost of the complex consists of two parts: maintenance costs and fixed costs. The maintenance cost is $50 per occupied unit, and since there are (50 - x) rented units, the maintenance cost is 50 * (50 - x). The fixed costs are given as $20,000 per month.
To calculate the profit, we subtract the total cost from the revenue:
Profit = Revenue - Cost
Profit = 800 * (50 - x) - (50 * (50 - x) + 20,000)
To determine the number of unoccupied units that maximizes the profit, we need to find the value of "x" that maximizes the profit function. This can be done by taking the derivative of the profit function with respect to "x" and setting it equal to zero. Solving this equation will give us the optimal number of unoccupied units.
To find the maximum profit for this apartment complex, substitute the value of "x" that maximizes the profit into the profit function and calculate the resulting profit.
Please note that without specific numerical values for the costs, it is not possible to determine the exact optimal number of unoccupied units or the maximum profit. The above steps outline the general approach to constructing the mathematical model and finding the optimal solution.