The owner of an apartment building can rent all 60 apartments if she charges $1,800 per month, but she rents one fewer apartment for each $60 increase in monthly rent.
(a) Construct a table that gives the revenue generated if she charges $1,800, $1,860, and $1,920.
No. of Apts	 	 	Rent	 	 	Total Revenue
60	 	 	$1,800	 	 	$ 
59	 	 	$1,860	 	 	$ 
58	 	 	$1,920	 	 	$ 

(b) Does her revenue from apartment rentals increase or decrease as she increases the rent from $1,800 to $1,920?
revenue increases
revenue decreases    

(c) Write an equation that gives the revenue R, from apartment rentals if she makes x increases of $60 in the rent.
R(x) = 
 

(d) Find the revenue if she increases rent to $3720.
$ 
 per month

Let's go step by step to address each part of the problem.

### (a) Construct a table that gives the revenue generated.

The revenue from apartment rentals is calculated using the formula:

\[
\text{Revenue} = (\text{No. of Apts}) \times (\text{Rent})
\]

- **At $1,800 rent**: 60 apartments are rented.
- **At $1,860 rent**: 59 apartments are rented.
- **At $1,920 rent**: 58 apartments are rented.

| No. of Apts | Rent  | Total Revenue |
|-------------|-------|---------------|
| 60          | $1,800| $ 60 \times 1800 = 108,000 $ |
| 59          | $1,860| $ 59 \times 1860 = 109,740 $ |
| 58          | $1,920| $ 58 \times 1920 = 111,360 $ |

Thus, the table is:

| No. of Apts | Rent  | Total Revenue |
|-------------|-------|---------------|
| 60          | $1,800| $108,000      |
| 59          | $1,860| $109,740      |
| 58          | $1,920| $111,360      |

### (b) Does the revenue increase or decrease from $1,800 to $1,920?

As we can see from the table:

- At $1,800, the revenue is $108,000.
- At $1,860, the revenue is $109,740.
- At $1,920, the revenue is $111,360.

The revenue **increases** as the rent increases from $1,800 to $1,920.

### (c) Write an equation that gives the revenue $ R $ from apartment rentals if she makes $ x $ increases of $60 in the rent.

Let:
- $ x $ be the number of $60 increases in the rent.
- The initial rent is $1,800, so the rent after $ x $ increases will be $ 1800 + 60x $.
- The number of apartments rented decreases by 1 for each $60 increase, so the number of apartments rented is $ 60 - x $.

The revenue $ R $ is the product of the number of apartments rented and the rent per apartment:

\[
R(x) = (60 - x)(1800 + 60x)
\]

This is the equation for the revenue.

### (d) Find the revenue if she increases the rent to $3,720.

First, let's find how many $60 increases there are between $1,800 and $3,720.

\[
3,720 = 1,800 + 60x
\]

Solving for $ x $:

\[
60x = 3,720 - 1,800 = 1,920
\]
\[
x = \frac{1,920}{60} = 32
\]

So, $ x = 32 $.

Now substitute $ x = 32 $ into the revenue equation $ R(x) = (60 - x)(1800 + 60x) $:

\[
R(32) = (60 - 32)(1800 + 60 \times 32)
\]
\[
R(32) = 28 \times (1800 + 1920) = 28 \times 3720
\]
\[
R(32) = 104,160
\]

Thus, the revenue if she increases the rent to $3,720 is **$104,160** per month.

---

Do you want any further details or explanations? 

Here are some related questions to expand your understanding:
1. What happens to the revenue if she increases the rent beyond $3,720?
2. How would the equation change if the decrease in number of rented apartments was not linear?
3. What is the maximum revenue she can generate based on this model?
4. How would different apartment vacancy rates affect the total revenue equation?
5. How would the solution change if she could rent more apartments at a lower rent?

**Tip:** In revenue problems, always check for the point at which increasing the price starts reducing the total revenue. This can often be found by analyzing or graphing the revenue equation.

The owner of an apartment building can rent all 60 apartments if she charges $1,800 per month, but she rents one fewer apartment for each $60 increase in monthly rent. (a) Construct a table that gives the revenue generated if she charges $1,800, $1,860, and $1,920. (b) Does her revenue from apartment rentals increase or decrease as she increases the rent from $1,800 to $1,920? (c) Write an equation that gives the revenue R, from apartment rentals if she makes x increases of $60 in the rent. (d) Find the revenue if she increases rent to $3,720.

Learn how to calculate the optimal revenue from apartment rentals when increasing rent. Explore the effect of rent increases on the number of rented apartments, and understand how to form the equation R(x) = (60 - x)(1800 + 60x) to maximize revenue. Suitable for Grades 9-12.

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