Question
Read each of the following descriptions, and identify the two quantities that
are involved in the linear relationship.
• the distance traveled by a car traveling at 40 miles per hour is
proportional to the time spent traveling
• the total cost of buying bottles of water at $2.50 per bottle and a
water jug at $20
• the area of a rectangle is proportional to the length and width of the
rectangle
• the volume of a rectangular prism is proportional to the length, width,
and height of a rectangular prism
© 2024 Pearson Education, Inc. All rights reserved. 3
B. For each description in part A, write a linear equation to model the
relationship between the two quantities.
C. For each equation in part B, explain the meaning of the slope and 𝑦𝑦-intercept
in the context of the situation described.
D. For each equation in part B, create a table of values and use the table of
values to graph the relationship between the two quantities.
are involved in the linear relationship.
• the distance traveled by a car traveling at 40 miles per hour is
proportional to the time spent traveling
• the total cost of buying bottles of water at $2.50 per bottle and a
water jug at $20
• the area of a rectangle is proportional to the length and width of the
rectangle
• the volume of a rectangular prism is proportional to the length, width,
and height of a rectangular prism
© 2024 Pearson Education, Inc. All rights reserved. 3
B. For each description in part A, write a linear equation to model the
relationship between the two quantities.
C. For each equation in part B, explain the meaning of the slope and 𝑦𝑦-intercept
in the context of the situation described.
D. For each equation in part B, create a table of values and use the table of
values to graph the relationship between the two quantities.
Answers
GPT-4o mini
Let's break down each of the descriptions provided, identify the quantities involved, formulate linear equations, and analyze their slope and y-intercept.
### A. Identification of Quantities
1. **Distance Traveled by a Car**:
- **Quantities**: Distance (d), Time (t)
2. **Total Cost of Buying Bottles of Water**:
- **Quantities**: Total Cost (C), Number of Bottles (n)
3. **Area of a Rectangle**:
- **Quantities**: Area (A), Length (l), Width (w)
4. **Volume of a Rectangular Prism**:
- **Quantities**: Volume (V), Length (l), Width (w), Height (h)
### B. Linear Equations
1. **Distance Traveled by a Car**:
- **Equation**: \( d = 40t \)
2. **Total Cost of Buying Bottles of Water**:
- **Equation**: \( C = 2.5n + 20 \)
3. **Area of a Rectangle** (assuming a fixed width):
- **Equation**: \( A = l \times w \). If \( w \) is constant: \( A = kw \), where \( k \) is the width.
4. **Volume of a Rectangular Prism** (assuming a specific height):
- **Equation**: \( V = l \times w \times h \). If \( h \) is constant: \( V = k \times (l \times w) \).
### C. Explanation of Slope and y-intercept
1. **Distance Traveled by a Car** \( d = 40t \):
- **Slope**: 40 (miles per hour) – it represents the speed of the car.
- **y-intercept**: 0 (the car starts at 0 distance when time is 0).
2. **Total Cost of Buying Bottles of Water** \( C = 2.5n + 20 \):
- **Slope**: 2.5 (dollars per bottle) – it represents the cost of each bottle of water.
- **y-intercept**: 20 (the cost of the water jug) – this is the fixed cost before buying any bottles.
3. **Area of a Rectangle** \( A = lw \) with constant width \( w \):
- **Slope**: \( w \) (constant) – the area increases linearly as the length increases.
- **y-intercept**: 0 (when length is 0, the area is 0).
4. **Volume of a Rectangular Prism** \( V = lwh \) with constant height \( h \):
- **Slope**: \( wh \) (product of width and height) – it represents how the volume increases with more length.
- **y-intercept**: 0 (when there is no length, the volume is 0).
### D. Creating Tables and Graphs
**1. Distance Traveled by a Car: `d = 40t`**
| Time \( t \) (hours) | Distance \( d \) (miles) |
|-----------------------|--------------------------|
| 0 | 0 |
| 1 | 40 |
| 2 | 80 |
| 3 | 120 |
| 4 | 160 |
**Graph**: A straight line through (0,0) with a slope of 40.
---
**2. Total Cost of Buying Bottles of Water: `C = 2.5n + 20`**
| Number of Bottles \( n \) | Total Cost \( C \) (dollars) |
|----------------------------|-------------------------------|
| 0 | 20 |
| 1 | 22.50 |
| 2 | 25 |
| 3 | 27.50 |
| 4 | 30 |
**Graph**: A straight line starting from (0,20) with a slope of 2.5.
---
**3. Area of a Rectangle (Assuming width \( w = 2 \)): `A = 2l`**
| Length \( l \) (units) | Area \( A \) (square units) |
|-------------------------|------------------------------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
**Graph**: A straight line through (0,0) with a slope of 2.
---
**4. Volume of a Rectangular Prism (Assuming width \( w = 2 \) and height \( h = 3 \)): `V = 6l`**
| Length \( l \) (units) | Volume \( V \) (cubic units) |
|-------------------------|-------------------------------|
| 0 | 0 |
| 1 | 6 |
| 2 | 12 |
| 3 | 18 |
| 4 | 24 |
**Graph**: A straight line through (0,0) with a slope of 6.
### Summary
- Each description relates two quantities in a linear way.
- The slope typically represents the rate of change, while the y-intercept indicates the starting value when the independent variable (time, number of bottles, length, etc.) is zero.
- Values were tabulated to facilitate graphing, showing the linear relationship visually.
### A. Identification of Quantities
1. **Distance Traveled by a Car**:
- **Quantities**: Distance (d), Time (t)
2. **Total Cost of Buying Bottles of Water**:
- **Quantities**: Total Cost (C), Number of Bottles (n)
3. **Area of a Rectangle**:
- **Quantities**: Area (A), Length (l), Width (w)
4. **Volume of a Rectangular Prism**:
- **Quantities**: Volume (V), Length (l), Width (w), Height (h)
### B. Linear Equations
1. **Distance Traveled by a Car**:
- **Equation**: \( d = 40t \)
2. **Total Cost of Buying Bottles of Water**:
- **Equation**: \( C = 2.5n + 20 \)
3. **Area of a Rectangle** (assuming a fixed width):
- **Equation**: \( A = l \times w \). If \( w \) is constant: \( A = kw \), where \( k \) is the width.
4. **Volume of a Rectangular Prism** (assuming a specific height):
- **Equation**: \( V = l \times w \times h \). If \( h \) is constant: \( V = k \times (l \times w) \).
### C. Explanation of Slope and y-intercept
1. **Distance Traveled by a Car** \( d = 40t \):
- **Slope**: 40 (miles per hour) – it represents the speed of the car.
- **y-intercept**: 0 (the car starts at 0 distance when time is 0).
2. **Total Cost of Buying Bottles of Water** \( C = 2.5n + 20 \):
- **Slope**: 2.5 (dollars per bottle) – it represents the cost of each bottle of water.
- **y-intercept**: 20 (the cost of the water jug) – this is the fixed cost before buying any bottles.
3. **Area of a Rectangle** \( A = lw \) with constant width \( w \):
- **Slope**: \( w \) (constant) – the area increases linearly as the length increases.
- **y-intercept**: 0 (when length is 0, the area is 0).
4. **Volume of a Rectangular Prism** \( V = lwh \) with constant height \( h \):
- **Slope**: \( wh \) (product of width and height) – it represents how the volume increases with more length.
- **y-intercept**: 0 (when there is no length, the volume is 0).
### D. Creating Tables and Graphs
**1. Distance Traveled by a Car: `d = 40t`**
| Time \( t \) (hours) | Distance \( d \) (miles) |
|-----------------------|--------------------------|
| 0 | 0 |
| 1 | 40 |
| 2 | 80 |
| 3 | 120 |
| 4 | 160 |
**Graph**: A straight line through (0,0) with a slope of 40.
---
**2. Total Cost of Buying Bottles of Water: `C = 2.5n + 20`**
| Number of Bottles \( n \) | Total Cost \( C \) (dollars) |
|----------------------------|-------------------------------|
| 0 | 20 |
| 1 | 22.50 |
| 2 | 25 |
| 3 | 27.50 |
| 4 | 30 |
**Graph**: A straight line starting from (0,20) with a slope of 2.5.
---
**3. Area of a Rectangle (Assuming width \( w = 2 \)): `A = 2l`**
| Length \( l \) (units) | Area \( A \) (square units) |
|-------------------------|------------------------------|
| 0 | 0 |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
**Graph**: A straight line through (0,0) with a slope of 2.
---
**4. Volume of a Rectangular Prism (Assuming width \( w = 2 \) and height \( h = 3 \)): `V = 6l`**
| Length \( l \) (units) | Volume \( V \) (cubic units) |
|-------------------------|-------------------------------|
| 0 | 0 |
| 1 | 6 |
| 2 | 12 |
| 3 | 18 |
| 4 | 24 |
**Graph**: A straight line through (0,0) with a slope of 6.
### Summary
- Each description relates two quantities in a linear way.
- The slope typically represents the rate of change, while the y-intercept indicates the starting value when the independent variable (time, number of bottles, length, etc.) is zero.
- Values were tabulated to facilitate graphing, showing the linear relationship visually.