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.
(a) Let represent the number of increases. Find an expression for the rent for each apartment.
Th) Find an expression for the number of apartments rented.
The number of apartments rented is given by .
© Find an expression for the total revenue received from all rented apartments.

Ask by Chen Evans. 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.
(a) Let represent the number of increases. Find an expression for the rent for each apartment.
Th) Find an expression for the number of apartments rented.
The number of apartments rented is given by .
© Find an expression for the total revenue received from all rented apartments.

Ask by Chen Evans. in the United States

Dec 15,2024

UpStudy AI · Accepted Answer

To solve the problem, we will define the variables and derive the required expressions step by step.

### (a) Expression for the rent for each apartment

Let $$ x $$ represent the number of $$ \$ 25 $$ increases. The initial rent is $$ \$ 850 $$. Each increase of $$ x $$ results in an increase of $$ 25x $$ in rent. Therefore, the expression for the rent $$ R(x) $$ for each apartment is:

$$
R(x) = 850 + 25x
$$

### (b) Expression for the number of apartments rented

At a rent of $$ \$ 850 $$, all 70 units are rented. For each $$ \$ 25 $$ increase in rent, one additional unit becomes vacant. Therefore, the number of apartments rented decreases by $$ x $$ for each increase. The expression for the number of apartments rented $$ N(x) $$ is:

$$
N(x) = 70 - x
$$

### (c) Expression for the total revenue received from all rented apartments

Total revenue $$ R $$ is calculated by multiplying the rent per apartment by the number of apartments rented. Using the expressions we derived in parts (a) and (b):

$$
R(x) = \text{(Rent per apartment)} \times \text{(Number of apartments rented)}
$$

Substituting the expressions we found:

$$
R(x) = (850 + 25x)(70 - x)
$$

Now, we can expand this expression:

$$
R(x) = (850 \cdot 70) + (850 \cdot -x) + (25x \cdot 70) + (25x \cdot -x)
$$
$$
R(x) = 59500 - 850x + 1750x - 25x^2
$$
$$
R(x) = 59500 + 900x - 25x^2
$$

Thus, the expression for the total revenue received from all rented apartments is:

$$
R(x) = -25x^2 + 900x + 59500
$$

### Summary of Answers

(a) Rent for each apartment: $$ R(x) = 850 + 25x $$

(b) Number of apartments rented: $$ N(x) = 70 - x $$

(c) Total revenue received from all rented apartments: $$ R(x) = -25x^2 + 900x + 59500 $$
(a) Rent for each apartment: $$ R(x) = 850 + 25x $$ (b) Number of apartments rented: $$ N(x) = 70 - x $$ (c) Total revenue received from all rented apartments: $$ R(x) = -25x^2 + 900x + 59500 $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions