Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Race Analysis of Runners A and B

#### Graph Description:
On the coordinate plane, the horizontal axis is labeled \(t\) (seconds) and the vertical axis is labeled \(s\) (meters). A line and a curve are graphed to represent the performance of two runners in a race.

##### Line A (Red Line):
- Begins at the origin \((0,0)\)
- Proceeds up and to the right in a straight line.
- Ends at the point \((14,100)\)

##### Curve B (Blue Curve):
- Starts at the origin \((0,0)\)
- Curves upward and becomes progressively steeper as it goes right.
- Passes through approximately \((9.5, 55)\) where it is furthest from Line A.
- Ends at the point \((14,100)\).

#### Analysis:
Based on the graphical representation and explanations provided, here are the conclusions about the runners' performances:

1. **Runner A** starts and maintains a constant velocity throughout the race, which is depicted by the straight, linear representation of Line A.
2. **Runner B** starts at a lower velocity, as indicated by the gentler initial slope of the curve, and gradually increases speed, finishing at the same endpoint as Runner A.

#### Multiple Choice Question:
Describe and compare how the runners run the race. Select the correct answer.

- [ ] Runner A starts the race at a slower velocity than runner B but finishes the race faster.
- [ ] Runner A starts the race at a faster velocity than runner B but finishes the race slower.
- [ ] Runner A runs the entire race at the same velocity. Runner B starts the race slow and speeds up gradually.
- [ ] Runner A runs the entire race at the same velocity. Runner B does not change speed.
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Transcribed Image Text:### Race Analysis of Runners A and B #### Graph Description: On the coordinate plane, the horizontal axis is labeled \(t\) (seconds) and the vertical axis is labeled \(s\) (meters). A line and a curve are graphed to represent the performance of two runners in a race. ##### Line A (Red Line): - Begins at the origin \((0,0)\) - Proceeds up and to the right in a straight line. - Ends at the point \((14,100)\) ##### Curve B (Blue Curve): - Starts at the origin \((0,0)\) - Curves upward and becomes progressively steeper as it goes right. - Passes through approximately \((9.5, 55)\) where it is furthest from Line A. - Ends at the point \((14,100)\). #### Analysis: Based on the graphical representation and explanations provided, here are the conclusions about the runners' performances: 1. **Runner A** starts and maintains a constant velocity throughout the race, which is depicted by the straight, linear representation of Line A. 2. **Runner B** starts at a lower velocity, as indicated by the gentler initial slope of the curve, and gradually increases speed, finishing at the same endpoint as Runner A. #### Multiple Choice Question: Describe and compare how the runners run the race. Select the correct answer. - [ ] Runner A starts the race at a slower velocity than runner B but finishes the race faster. - [ ] Runner A starts the race at a faster velocity than runner B but finishes the race slower. - [ ] Runner A runs the entire race at the same velocity. Runner B starts the race slow and speeds up gradually. - [ ] Runner A runs the entire race at the same velocity. Runner B does not change speed.
**Analyzing Race Performance Using Graphical Data**

The provided graph illustrates the position of two runners, A and B, over a 12-second interval during a 100-meter race. The x-axis represents time in seconds (t), while the y-axis indicates position in meters (s).

**Graph Analysis**:
- **Runner A** is denoted by a red line.
- **Runner B** is represented by a blue line.
- Both runners start at 0 meters and 0 seconds.
- The graph shows their position in intervals of 2 seconds up to 12 seconds.

**Questionnaire**:

**(a) Describe and compare how the two runners run the race.**

- **Option 1**: Runner A starts the race at a slower velocity than runner B but finishes the race at a faster velocity. Runner B runs the entire race at the same velocity.
- **Option 2**: Runner A starts the race at a faster velocity than runner B but finishes the race at a slower velocity. Runner B runs the entire race at the same velocity.
- **Option 3**: Runner A runs the entire race at the same velocity. Runner B starts the race at a slower velocity than runner A but finishes the race at a faster velocity.
- **Option 4**: Runner A runs the entire race at the same velocity. Runner B starts the race at a faster velocity than runner A but finishes the race at a slower velocity.
- **Option 5**: Runner A and runner B both start and end the race at the same velocities.

**(b) How long after the race began (in seconds) do the runners have the same velocity?**

_Box for Answer:__ ____________ s_

**(c) How long after the race began (in seconds) is the distance between the runners the greatest?**

_Box for Answer:__ ____________ s_

---

By understanding the graph, students can analyze the performance and speed variations of two competitors in a race. This exercise provides practical insight into interpreting motion and velocity from graphical data.
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Transcribed Image Text:**Analyzing Race Performance Using Graphical Data** The provided graph illustrates the position of two runners, A and B, over a 12-second interval during a 100-meter race. The x-axis represents time in seconds (t), while the y-axis indicates position in meters (s). **Graph Analysis**: - **Runner A** is denoted by a red line. - **Runner B** is represented by a blue line. - Both runners start at 0 meters and 0 seconds. - The graph shows their position in intervals of 2 seconds up to 12 seconds. **Questionnaire**: **(a) Describe and compare how the two runners run the race.** - **Option 1**: Runner A starts the race at a slower velocity than runner B but finishes the race at a faster velocity. Runner B runs the entire race at the same velocity. - **Option 2**: Runner A starts the race at a faster velocity than runner B but finishes the race at a slower velocity. Runner B runs the entire race at the same velocity. - **Option 3**: Runner A runs the entire race at the same velocity. Runner B starts the race at a slower velocity than runner A but finishes the race at a faster velocity. - **Option 4**: Runner A runs the entire race at the same velocity. Runner B starts the race at a faster velocity than runner A but finishes the race at a slower velocity. - **Option 5**: Runner A and runner B both start and end the race at the same velocities. **(b) How long after the race began (in seconds) do the runners have the same velocity?** _Box for Answer:__ ____________ s_ **(c) How long after the race began (in seconds) is the distance between the runners the greatest?** _Box for Answer:__ ____________ s_ --- By understanding the graph, students can analyze the performance and speed variations of two competitors in a race. This exercise provides practical insight into interpreting motion and velocity from graphical data.
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