Question

the manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $700 per month. a market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. what rent (in dollars per month) should the manager charge to maximize revenue?

Answer 1

To maximize revenue for a 100-unit apartment complex with a base rent of $700 per month and a loss of one unit occupied for every $10 increase in rent, the manager should charge $850 per month.

Step-by-step explanation:

The student is dealing with a question related to price optimization in microeconomics—an application of mathematical modeling used in business. Since the manager knows that at $700 per month all units are occupied, but each $10 increase results in one vacant unit, we can use a quadratic equation to represent the revenue as a function of the number of $10 increases over $700.

Let x be the number of $10 increases. The rent will be R(x) = (700 + 10x), and the number of occupied units will be O(x) = 100 - x. Therefore, the revenue function is given by Revenue = R(x) × O(x) = (700 + 10x)(100 - x).

To maximize revenue, we need to find the vertex of the parabola represented by this quadratic equation, which occurs at -b/2a when the equation is in standard form ax^2 + bx + c. By expanding and rearranging, we find the revenue function in standard form and calculate this vertex. After calculating, we find that the maximum revenue occurs at x = 15. Thus, the manager should charge $850 per month to maximize revenue (since 15 units of $10 increases over $700 adds up to $850).

Answer 2

To maximize revenue, the manager should calculate the total revenue for each potential rent increase and find the rent value that results in the highest revenue, which can be done by setting up a revenue function and finding its maximum value.

Step-by-step explanation:

The student has asked how to determine the optimal rent that an apartment complex manager should charge to maximize revenue considering that every $10 increase in rent above $700 leads to one unit remaining vacant. To find the rent that maximizes revenue, we need to use a mathematical model for revenue that takes into account the price increase per unit and the associated decrease in the number of units rented.

Let's denote the number of $10 increases over the base rent of $700 as x. For each additional $10 increase in rent, one unit becomes vacant, so the number of units rented out is (100 - x). The new rent charged will be ($700 + 10x). Therefore, the total revenue R can be calculated as:

R = (100 - x) × (700 + 10x)

To maximize revenue, we need to find the value of x that results in the highest R. By taking the derivative of R with respect to x and setting it to zero, we can find the maximum point for R. However, this requires knowledge of calculus, which is not provided in the excerpt.

If this calculus approach is not possible, the question could be approached by creating a table or graph to manually find the rent value that produces the highest revenue, which may be ideal for a high school context.