The manager of a 70 -unit apartment complex is trying to decide what rent to charge. Experience has shown that at a rent of all of the units
will be full. On average, one additional unit will be vacant for each increase in rent.
Answer parts (a) through (e) below.
(d) What value of leads to maximum revenue?
The maximum revenue will happen when .
(Round to the nearest whole number as needed.)
(e) What is the maximum revenue?
The maximum revenue is

Ask by Mckenzie Ross. in the United States

Dec 15,2024

Question

The manager of a 70 -unit apartment complex is trying to decide what rent to charge. Experience has shown that at a rent of all of the units
will be full. On average, one additional unit will be vacant for each increase in rent.
Answer parts (a) through (e) below.
(d) What value of leads to maximum revenue?
The maximum revenue will happen when .
(Round to the nearest whole number as needed.)
(e) What is the maximum revenue?
The maximum revenue is

Ask by Mckenzie Ross. in the United States

Dec 15,2024

UpStudy AI · Accepted Answer

To solve this problem, we need to express the revenue as a function of the rent charged and then find the value of $$ x $$ that maximizes this revenue.

Let:
- $$ R $$ be the revenue,
- $$ x $$ be the number of $$ \$25 $$ increases in rent.

The rent charged can be expressed as:
$$
\text{Rent} = 850 + 25x
$$

The number of units rented will decrease by 1 for each $$ \$25 $$ increase in rent. Therefore, the number of units rented can be expressed as:
$$
\text{Units rented} = 70 - x
$$

The revenue $$ R $$ can be expressed as:
$$
R = \text{Rent} \times \text{Units rented} = (850 + 25x)(70 - x)
$$

Now, we can expand this equation:
$$
R = (850 \times 70) + (25x \times 70) - (850x) - (25x^2)
$$
$$
R = 59500 + 1750x - 850x - 25x^2
$$
$$
R = 59500 + 900x - 25x^2
$$

This is a quadratic equation in the form of:
$$
R = -25x^2 + 900x + 59500
$$

To find the value of $$ x $$ that maximizes revenue, we can use the vertex formula for a quadratic equation $$ ax^2 + bx + c $$, where the vertex $$ x $$ is given by:
$$
x = -\frac{b}{2a}
$$

In our case:
- $$ a = -25 $$
- $$ b = 900 $$

Substituting these values into the vertex formula:
$$
x = -\frac{900}{2 \times -25} = \frac{900}{50} = 18
$$

So, the value of $$ x $$ that leads to maximum revenue is:
$$
x = 18
$$

Now, to find the maximum revenue, we substitute $$ x = 18 $$ back into the revenue equation:
$$
R = -25(18^2) + 900(18) + 59500
$$
Calculating $$ 18^2 $$:
$$
18^2 = 324
$$
Now substituting:
$$
R = -25(324) + 900(18) + 59500
$$
Calculating each term:
$$
-25(324) = -8100
$$
$$
900(18) = 16200
$$
Now substituting these values:
$$
R = -8100 + 16200 + 59500
$$
$$
R = 8100 + 59500 = 67600
$$

Thus, the maximum revenue is:
$$
R = 67600
$$

To summarize:
(d) The maximum revenue will happen when $$ x = 18 $$.
(e) The maximum revenue is $$ \$ 67600 $$.
(d) The value of $$ x $$ for maximum revenue is 18. (e) The maximum revenue is $67,600.

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