Question

The Naturally Made Bath and Body store pays $550 a month for rent and utilities. The average cost for its products to be manufactured is about $3.00 an item. If the average price for a product sold in the store is $5.50, what will the break-even point be?



Let x represent the number of products sold.

The cost function that represents the situation is
.



The revenue function that represents the situation is
.

Answer 1

To find the break-even point for the Naturally Made Bath and Body store, we need to establish the cost and revenue functions.

### Cost Function

The total cost \( C(x) \) consists of both fixed costs (rent and utilities) and variable costs (cost of manufacturing the products).

1. **Fixed Costs**: $550 (rent and utilities)
2. **Variable Cost per Item**: $3.00

The cost function can be expressed as:
\[
C(x) = 550 + 3x
\]
where \( x \) is the number of products sold.

### Revenue Function

The revenue \( R(x) \) is generated from selling the products. With an average selling price of $5.50 per item, the revenue function is:
\[
R(x) = 5.50x
\]
where \( x \) is the number of products sold.

### Break-even Point

The break-even point occurs when the total cost equals total revenue, i.e., \( C(x) = R(x) \).

Setting the two equations equal to each other, we get:
\[
550 + 3x = 5.50x
\]

Now, to find \( x \), solve the equation:
1. Subtract \( 3x \) from both sides:
\[
550 = 5.50x - 3x
\]
\[
550 = 2.50x
\]

2. Divide both sides by 2.50:
\[
x = \frac{550}{2.50} = 220
\]

### Conclusion

Thus, the break-even point is at \( x = 220 \) products sold.

**Summary of Functions**:
- Cost function: \( C(x) = 550 + 3x \)
- Revenue function: \( R(x) = 5.50x \)
- Break-even point: 220 products sold.

Answer 2

Let x represent the number of products sold.

The cost function that represents the situation is
✔ C(x) = 550 + 3.00x
.



The revenue function that represents the situation is
✔ R(x) = 5.50x
.




The Naturally Made Bath and Body store pays $550 a month for rent and utilities. The average cost for its products to be manufactured is about $3.00 an item. If the average price for a product sold in the store is $5.50, what will the break-even point be? Let x represent the number of products sold.

The break-even point occurs when the cost function equals the revenue function.

550 + 3.00x = 5.50x

Naturally Made needs to sell___
products to break even.



The cost or revenue when it sells the break-even number of products will be$___
.

Answer 3

To determine how many products the Naturally Made Bath and Body store needs to sell to break even, we already found the equation:

\[
550 + 3.00x = 5.50x
\]

1. **Rearranging the Equation:**
\[
550 = 5.50x - 3.00x
\]
\[
550 = 2.50x
\]

2. **Solving for \( x \):**
\[
x = \frac{550}{2.50} = 220
\]

So, the Naturally Made Bath and Body store needs to sell **220 products** to break even.

### Cost or Revenue at Break-Even Point

Now, we can calculate the cost or revenue when the break-even number of products (220) is sold.

1. **Calculating Cost:**
\[
C(220) = 550 + 3.00(220)
\]
\[
C(220) = 550 + 660 = 1210
\]

2. **Calculating Revenue:**
\[
R(220) = 5.50(220)
\]
\[
R(220) = 1210
\]

Thus, the cost or revenue when it sells the break-even number of products will be **$1210**.

### Summary
- Naturally Made needs to sell **220 products** to break even.
- The cost or revenue when it sells the break-even number of products will be **$1210**.

Answer 4

Elliot has a total of 26 books. He has 12 more fiction books than nonfiction books. Let x represent the number of fiction books and y represent the number of nonfiction books.

The system of equations models the total costs for each.

x + y = 26
x – y = 12
Elliot added the two equations and the result was
2x = 38.

Solve the equation. How many of each type of book does Elliot have?

___fiction books

___nonfiction books

Answer 5

Let's start from the system of equations that models Elliot's books:

1. \( x + y = 26 \) (Equation 1: total number of books)
2. \( x - y = 12 \) (Equation 2: difference between fiction and nonfiction books)

### Step 1: Solve the first equation for \( y \)

From Equation 1:
\[
y = 26 - x
\]

### Step 2: Substitute for \( y \) in Equation 2

Now we substitute this expression into Equation 2:
\[
x - (26 - x) = 12
\]

### Step 3: Simplify the equation

\[
x - 26 + x = 12
\]
\[
2x - 26 = 12
\]

### Step 4: Solve for \( x \)

Add 26 to both sides:
\[
2x = 12 + 26
\]
\[
2x = 38
\]

Now, divide both sides by 2:
\[
x = 19
\]

### Step 5: Find \( y \)

Now that we know \( x \), we can find \( y \) using the first equation:
\[
y = 26 - x
\]
\[
y = 26 - 19 = 7
\]

### Conclusion

Elliot has:
- **19 fiction books**
- **7 nonfiction books**

### Final Answer:
- ___fiction books: **19**
- ___nonfiction books: **7**

Answer 6

Aro diagrams his river rafting trip, estimating the time it will take him to paddle upstream against the current, and then back downstream with the current. His campsite destination is 5.2 miles upstream. Determine how fast Aro can paddle and how fast the river water is moving.

Upstream: 5.2 miles 5 hours

Downstream: 5.2 miles 2.5 hours

Use the fact that d = rt to write the system of equations that represents the scenario. Let x be the speed of Aro’s paddling and let y be the speed of the river.



Upstream:

5.2 = (__,__)



Downstream:

5.2 = (__,__)

Answer 7

To set up the system of equations representing Aro's river rafting trip, we will use the formula \( d = rt \), where \( d \) is distance, \( r \) is the rate (or speed), and \( t \) is time.

### Upstream

1. **Distance**: 5.2 miles
2. **Time**: 5 hours
3. **Effective Speed Upstream**: When paddling upstream, Aro's effective speed is his paddling speed \( x \) minus the speed of the river \( y \).

Using the formula \( d = rt \):
\[
5.2 = (x - y) \cdot 5
\]

### Downstream

1. **Distance**: 5.2 miles
2. **Time**: 2.5 hours
3. **Effective Speed Downstream**: When paddling downstream, Aro's effective speed is his paddling speed \( x \) plus the speed of the river \( y \).

Using the formula \( d = rt \):
\[
5.2 = (x + y) \cdot 2.5
\]

### System of Equations

Now we can write the two equations:

1. **Upstream**:
\[
5.2 = (x - y) \cdot 5
\]

2. **Downstream**:
\[
5.2 = (x + y) \cdot 2.5
\]

### Expanded Equations

Expanding the equations gives us:

1. Upstream:
\[
5.2 = 5x - 5y \quad \Rightarrow \quad 5x - 5y = 5.2 \quad \Rightarrow \quad x - y = \frac{5.2}{5} \quad \Rightarrow \quad x - y = 1.04 \quad \text{(Equation 1)}
\]

2. Downstream:
\[
5.2 = 2.5x + 2.5y \quad \Rightarrow \quad 2.5x + 2.5y = 5.2 \quad \Rightarrow \quad x + y = \frac{5.2}{2.5} \quad \Rightarrow \quad x + y = 2.08 \quad \text{(Equation 2)}
\]

### Summary of System of Equations
The system of equations is as follows:

1. Upstream:
\[
x - y = 1.04
\]
2. Downstream:
\[
x + y = 2.08
\]

### Solving the System
Now we can solve this system of equations to find values for \( x \) (Aro’s paddling speed) and \( y \) (speed of the river).

1. Adding both equations:
\[
(x - y) + (x + y) = 1.04 + 2.08
\]
\[
2x = 3.12 \quad \Rightarrow x = \frac{3.12}{2} \quad \Rightarrow x = 1.56 \text{ miles per hour}
\]

2. Substitute \( x \) back into one of the equations to find \( y \):
Using Equation 1:
\[
1.56 - y = 1.04
\]
\[
y = 1.56 - 1.04 \quad \Rightarrow \quad y = 0.52 \text{ miles per hour}
\]

### Final Result
- Aro's paddling speed \( x \) is **1.56 miles per hour**.
- The speed of the river \( y \) is **0.52 miles per hour**.

Answer 8

wait a minute sorry

Upstream:

5.2 = (____) ?
(x)(y)? x/y? x+y? x-y?
for ? 2.5, 5.2 or 5.



Downstream:

5.2 = (____) ?
(x)(y)? x/y? x+y? x-y?
for ? 2.5, 5.2 or 5.

Answer 9

No problem! Let's clarify the equations in the context of Aro's river rafting trip by identifying the distance, rate (speed), and time for both upstream and downstream travel.

### Upstream

- **Distance**: \( 5.2 \) miles
- **Time**: \( 5 \) hours
- **Effective speed upstream**: \( x - y \) (Aro’s paddling speed minus the river's current speed)

Using the formula \( d = rt \):
\[
5.2 = (x - y) \cdot 5
\]

Thus, the completed statement is:

**Upstream:**
\[
5.2 = (x - y)(5)
\]

---

### Downstream

- **Distance**: \( 5.2 \) miles
- **Time**: \( 2.5 \) hours
- **Effective speed downstream**: \( x + y \) (Aro’s paddling speed plus the river's current speed)

Using the formula \( d = rt \):
\[
5.2 = (x + y) \cdot 2.5
\]

Thus, the completed statement is:

**Downstream:**
\[
5.2 = (x + y)(2.5)
\]

### Summary

In summary, the system of equations based on the scenario is:

1. Upstream:
\[
5.2 = (x - y)(5)
\]

2. Downstream:
\[
5.2 = (x + y)(2.5)
\]

Thank you for your understanding! Is there anything else you'd like to clarify or ask?

Answer 10

Aro diagrams his river rafting trip, estimating the time it will take him to paddle upstream against the current, and then back downstream with the current. His campsite destination is 5.2 miles upstream. Determine how fast Aro can paddle and how fast the river water is moving. Round to the nearest hundredth as needed.

Upstream: 5.2 miles 5 hours

Downstream: 5.2 miles 2.5 hours

Let x be the speed of Aro’s paddling and let y be the speed of the river.

Upstream: 1.04 = x – y

Downstream: 2.08 = x + y



Aro can paddle at a speed of____
miles per hour.



The river’s speed is___
miles per hour.

Answer 11

To find Aro's paddling speed (\(x\)) and the river's speed (\(y\)), we can use the equations you provided:

1. Upstream: \( 1.04 = x - y \) (Equation 1)
2. Downstream: \( 2.08 = x + y \) (Equation 2)

### Step 1: Set Up the System of Equations

We can rewrite the equations for clarity:

1. \( x - y = 1.04 \) (Equation 1)
2. \( x + y = 2.08 \) (Equation 2)

### Step 2: Solve the System

We can solve these equations by addition or substitution. Let's use the elimination method by adding both equations to eliminate \(y\).

**Adding the two equations:**
\[
(x - y) + (x + y) = 1.04 + 2.08
\]
\[
2x = 3.12
\]
\[
x = \frac{3.12}{2} = 1.56
\]

### Step 3: Solve for \(y\)

Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). We'll use Equation 1:

\[
1.56 - y = 1.04
\]
\[
y = 1.56 - 1.04 = 0.52
\]

### Final Results
1. Aro can paddle at a speed of **1.56 miles per hour**.
2. The river’s speed is **0.52 miles per hour**.

### Summary
- Aro can paddle at a speed of **1.56 miles per hour**.
- The river's speed is **0.52 miles per hour**.