Final answer:
The problem is a profit maximization issue where we must calculate the optimal rent price considering incremental increases and maintenance costs against potential vacancies. The process involves setting up a profit function, taking its derivative, and finding the rent increase level that yields the highest profit.
Step-by-step explanation:
The student's question revolves around optimizing the monthly profit from an apartment building with a given rent increase and maintenance cost scenario. First, we must understand how the rent rate affects the number of units rented and then consider the costs involved to find the maximum profit.
To start, we presume that $400 per month is the current rent for each of the 100 units. We are told that for every $10 increase in rent, one unit becomes vacant. Since each occupied unit has a maintenance cost of $50, we need to calculate the increase in revenue against the decrease in occupied units and additional costs to find the optimal rent price for maximum profit.
Let's use the following equation to represent the monthly profit (P) in relation to the number of $10 rent increases (x): P = (400 + 10x)(100 - x) - 50(100 - x). This equation calculates the total revenue from rented units minus the month-to-month maintenance cost for each rented unit.
To maximize profit, we differentiate the profit function with respect to x, set the derivative equal to zero, and solve for x. This value of x shows the number of $10 rent increases that will yield the highest profit, considering the balance between additional revenue per unit and the loss of tenants due to higher rent.
Once x is determined, we calculate the new rent per unit as $400 + 10x and determine how many units would be rented out at this price (100 - x). With these numbers, we can determine the monthly profit. The rent increase that allows for the highest monthly profit, after accounting for maintenance and potential vacancy, is our solution.