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Sagot :
Let's analyze the given problem in a step-by-step manner, focusing on constructing the distance-time graph and answering the questions about average speed, stops, and intervals of same speed.
### Step 1: Constructing the Distance-Time Graph
We are given the following data for time (in seconds), Sally's distance (in meters), and Alonzo's distance:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Time (s)} & \text{Sally's Distance (m)} & \text{Alonzo's Distance (m)} \\ \hline 1 & 2 & 1 \\ 2 & 4 & 2 \\ 3 & 6 & 2 \\ 4 & 8 & 4 \\ \hline \end{array} \][/tex]
Plotting the graph:
1. On the x-axis, plot the time (s).
2. On the y-axis, plot the distance (m).
3. For Sally, plot points (1,2), (2,4), (3,6), and (4,8).
4. For Alonzo, plot points (1,1), (2,2), (3,2), and (4,4).
### Step 2: Calculating the Average Speed of Each Runner
The average speed of each runner is calculated as the total distance traveled divided by the total time taken.
#### Sally's Average Speed:
[tex]\[ \text{Total Distance} = 8 \text{ meters} - 2 \text{ meters} = 6 \text{ meters} \][/tex]
[tex]\[ \text{Total Time} = 4 \text{ seconds} - 1 \text{ second} = 3 \text{ seconds} \][/tex]
[tex]\[ \text{Sally's Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6 \text{ meters}}{3 \text{ seconds}} = 2.0 \text{ meters per second} \][/tex]
#### Alonzo's Average Speed:
[tex]\[ \text{Total Distance} = 4 \text{ meters} - 1 \text{ meter} = 3 \text{ meters} \][/tex]
[tex]\[ \text{Total Time} = 4 \text{ seconds} - 1 \text{ second} = 3 \text{ seconds} \][/tex]
[tex]\[ \text{Alonzo's Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{3 \text{ meters}}{3 \text{ seconds}} = 1.0 \text{ meters per second} \][/tex]
### Step 3: Identifying the Runner Who Stops Briefly
To determine if either runner stops, we look for intervals where their distance does not change.
By examining the data:
- For Sally, the distances steadily increase (2, 4, 6, 8 meters), indicating no stops.
- For Alonzo, the distance stays the same between 2 seconds and 3 seconds (both are 2 meters). Thus, Alonzo stops briefly between 2 and 3 seconds.
### Step 4: Identifying the Time Intervals Where Both Run at the Same Speed
We calculate the speed in each interval and compare them to find intervals where the speeds are equal.
#### Speeds in Each Interval:
For Sally:
- From 1s to 2s: Speed = [tex]\( \frac{4-2}{2-1} = 2 \text{ m/s} \)[/tex]
- From 2s to 3s: Speed = [tex]\( \frac{6-4}{3-2} = 2 \text{ m/s} \)[/tex]
- From 3s to 4s: Speed = [tex]\( \frac{8-6}{4-3} = 2 \text{ m/s} \)[/tex]
For Alonzo:
- From 1s to 2s: Speed = [tex]\( \frac{2-1}{2-1} = 1 \text{ m/s} \)[/tex]
- From 2s to 3s: Speed = [tex]\( \frac{2-2}{3-2} = 0 \text{ m/s} \)[/tex] (Alonzo stops)
- From 3s to 4s: Speed = [tex]\( \frac{4-2}{4-3} = 2 \text{ m/s} \)[/tex]
Both Sally and Alonzo run at the same speed (2 m/s) during the interval from 3 seconds to 4 seconds (3s to 4s).
### Final Summary:
- Sally's average speed: 2.0 meters per second
- Alonzo's average speed: 1.0 meters per second
- Runner who stops briefly: Alonzo, between 2 seconds and 3 seconds
- Time interval where both run at the same speed: From 3 seconds to 4 seconds
Thus, this completes the detailed step-by-step analysis of the problem.
### Step 1: Constructing the Distance-Time Graph
We are given the following data for time (in seconds), Sally's distance (in meters), and Alonzo's distance:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Time (s)} & \text{Sally's Distance (m)} & \text{Alonzo's Distance (m)} \\ \hline 1 & 2 & 1 \\ 2 & 4 & 2 \\ 3 & 6 & 2 \\ 4 & 8 & 4 \\ \hline \end{array} \][/tex]
Plotting the graph:
1. On the x-axis, plot the time (s).
2. On the y-axis, plot the distance (m).
3. For Sally, plot points (1,2), (2,4), (3,6), and (4,8).
4. For Alonzo, plot points (1,1), (2,2), (3,2), and (4,4).
### Step 2: Calculating the Average Speed of Each Runner
The average speed of each runner is calculated as the total distance traveled divided by the total time taken.
#### Sally's Average Speed:
[tex]\[ \text{Total Distance} = 8 \text{ meters} - 2 \text{ meters} = 6 \text{ meters} \][/tex]
[tex]\[ \text{Total Time} = 4 \text{ seconds} - 1 \text{ second} = 3 \text{ seconds} \][/tex]
[tex]\[ \text{Sally's Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6 \text{ meters}}{3 \text{ seconds}} = 2.0 \text{ meters per second} \][/tex]
#### Alonzo's Average Speed:
[tex]\[ \text{Total Distance} = 4 \text{ meters} - 1 \text{ meter} = 3 \text{ meters} \][/tex]
[tex]\[ \text{Total Time} = 4 \text{ seconds} - 1 \text{ second} = 3 \text{ seconds} \][/tex]
[tex]\[ \text{Alonzo's Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{3 \text{ meters}}{3 \text{ seconds}} = 1.0 \text{ meters per second} \][/tex]
### Step 3: Identifying the Runner Who Stops Briefly
To determine if either runner stops, we look for intervals where their distance does not change.
By examining the data:
- For Sally, the distances steadily increase (2, 4, 6, 8 meters), indicating no stops.
- For Alonzo, the distance stays the same between 2 seconds and 3 seconds (both are 2 meters). Thus, Alonzo stops briefly between 2 and 3 seconds.
### Step 4: Identifying the Time Intervals Where Both Run at the Same Speed
We calculate the speed in each interval and compare them to find intervals where the speeds are equal.
#### Speeds in Each Interval:
For Sally:
- From 1s to 2s: Speed = [tex]\( \frac{4-2}{2-1} = 2 \text{ m/s} \)[/tex]
- From 2s to 3s: Speed = [tex]\( \frac{6-4}{3-2} = 2 \text{ m/s} \)[/tex]
- From 3s to 4s: Speed = [tex]\( \frac{8-6}{4-3} = 2 \text{ m/s} \)[/tex]
For Alonzo:
- From 1s to 2s: Speed = [tex]\( \frac{2-1}{2-1} = 1 \text{ m/s} \)[/tex]
- From 2s to 3s: Speed = [tex]\( \frac{2-2}{3-2} = 0 \text{ m/s} \)[/tex] (Alonzo stops)
- From 3s to 4s: Speed = [tex]\( \frac{4-2}{4-3} = 2 \text{ m/s} \)[/tex]
Both Sally and Alonzo run at the same speed (2 m/s) during the interval from 3 seconds to 4 seconds (3s to 4s).
### Final Summary:
- Sally's average speed: 2.0 meters per second
- Alonzo's average speed: 1.0 meters per second
- Runner who stops briefly: Alonzo, between 2 seconds and 3 seconds
- Time interval where both run at the same speed: From 3 seconds to 4 seconds
Thus, this completes the detailed step-by-step analysis of the problem.
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