Question

The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 472 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 2 dollar increase in rent. Similarly, one additional unit will be occupied for each 2 dollar decrease in rent. What rent should the manager charge to maximize revenue?

Answer 1

The manager should charge a rent of $588 to maximize revenue.

To maximize revenue, the manager should find the optimal rent that will lead to the highest number of occupied units. We can start by defining a revenue function based on the given information and then find the rent that maximizes this function.

Let's denote:

- R as the total revenue.

- x as the number of 2-dollar increases in rent (compared to $472).

From the information given, we know that for each $2 increase in rent, one additional unit remains vacant, and for each $2 decrease in rent, one additional unit is occupied. Therefore, we can express the number of occupied units as 120 - x since x represents the number of 2-dollar increases in rent.

Now, let's calculate the revenue for this situation. Revenue is calculated as the product of the rent and the number of occupied units:

`\[R = (\text{Rent}) * (\text{Number of Occupied Units})\]`

Substituting in the expressions we found for the number of occupied units:

`\[R = (\text{Rent}) * (120 - x)\]`

Now, we need to express R in terms of x, so we can maximize it. Given that the rent is $472 plus $2 for each x, the rent can be expressed as 472 + 2x. Therefore, our revenue function becomes:

`\[R(x) = (472 + 2x) * (120 - x)\]`

Now, we want to find the value of x that maximizes R(x), which is our revenue function.

To find the maximum, we can take the derivative of R(x) with respect to x, set it equal to zero, and solve for x:

`\[R'(x) = 0\]`

First, let's find R'(x) by applying the product rule:

`\[R'(x) = (472 + 2x) \cdot (-1) + (120 - x) \cdot 2\]`

Now, set R'(x) equal to zero and solve for x:

`\[(472 + 2x) \cdot (-1) + (120 - x) \cdot 2 = 0\]`

Now, simplify and solve for x:

-472 - 2x + 240 - 2x = 0

Combine like terms:

-2x - 2x - 472 + 240 = 0

-4x - 232 = 0

Add 232 to both sides:

-4x = 232

Now, divide by -4:

`\[x = -58\]`

So, x = -58.

Since x represents the number of 2-dollar increases in rent compared to $472, a negative value doesn't make sense in this context. Therefore, we should ignore the negative value for x

Now, let's find the corresponding rent when x = -58. Remember that each unit corresponds to a $2 decrease in rent, so we need to add
`\(58 * $2\) to $472` to find the optimal rent:

`\[\text{Optimal Rent} = 472 + (58 * $2) = 472 + $116 = $588\]`