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Sagot :
Let's break down the problem step-by-step to find the answers and fill in the required chart and system of equations.
### Given Data:
1. The total distance to his home is 5 km.
2. He walks at a speed of 5 km/h.
3. He runs at a speed of 12 km/h.
4. He leaves at 5:30 PM and arrives home at 6:09 PM, which means the total time taken is 39 minutes.
### Part (a)
#### Filling in the Chart:
Let's denote:
- [tex]\( d_w \)[/tex]: Distance walked (in km)
- [tex]\( d_r \)[/tex]: Distance ran (in km)
- [tex]\( t_w \)[/tex]: Time spent walking (in hours)
- [tex]\( t_r \)[/tex]: Time spent running (in hours)
Since the man walks and runs, we split the journey into two parts. We'll need to convert the total time into hours for easier calculation.
[tex]\[ \text{Total time in hours} = \frac{39}{60} = 0.65 \, \text{hours} \][/tex]
We are also given the speeds:
- For walking: 5 km/h
- For running: 12 km/h
Using these symbols, we can fill in the chart partially:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & \text{Distance (km)} & \text{Speed (km/h)} & \text{Time (h)} \\ \hline \text{Walks} & d_w & 5 & t_w \\ \hline \text{Runs} & d_r & 12 & t_r \\ \hline \end{tabular} \][/tex]
#### System of Equations:
We can set up the following equations based on the given data:
1. Total distance equation:
[tex]\[ d_w + d_r = 5 \][/tex]
2. Total time equation:
[tex]\[ t_w + t_r = 0.65 \][/tex]
3. Distance equation when walking:
[tex]\[ d_w = 5t_w \][/tex]
4. Distance equation when running:
[tex]\[ d_r = 12t_r \][/tex]
### Part (b) How far did he run?
Given the numerical solution, we have:
- Distance walked [tex]\( d_w = 2 \)[/tex] km
- Distance ran [tex]\( d_r = 3 \)[/tex] km
Therefore,
[tex]\[ \text{Distance ran} = 3 \, \text{km} \][/tex]
### Part (c) For how many minutes did he walk?
Given the numerical solution, we also have:
- Time spent walking [tex]\( t_w = 0.4 \, \text{hours} \)[/tex]
To convert this to minutes:
[tex]\[ t_w \, (\text{minutes}) = 0.4 \times 60 = 24 \, \text{minutes} \][/tex]
### Summary:
(a)
- Chart:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & \text{Distance (km)} & \text{Speed (km/h)} & \text{Time (h)} \\ \hline \text{Walks} & 2 & 5 & 0.4 \\ \hline \text{Runs} & 3 & 12 & 0.25 \\ \hline \end{tabular} \][/tex]
- System of Equations:
[tex]\[ \begin{cases} d_w + d_r = 5 \\ t_w + t_r = 0.65 \\ d_w = 5t_w \\ d_r = 12t_r \end{cases} \][/tex]
(b)
He ran for [tex]\( 3 \, \text{km} \)[/tex].
(c)
He walked for [tex]\( 24 \, \text{minutes} \)[/tex].
### Given Data:
1. The total distance to his home is 5 km.
2. He walks at a speed of 5 km/h.
3. He runs at a speed of 12 km/h.
4. He leaves at 5:30 PM and arrives home at 6:09 PM, which means the total time taken is 39 minutes.
### Part (a)
#### Filling in the Chart:
Let's denote:
- [tex]\( d_w \)[/tex]: Distance walked (in km)
- [tex]\( d_r \)[/tex]: Distance ran (in km)
- [tex]\( t_w \)[/tex]: Time spent walking (in hours)
- [tex]\( t_r \)[/tex]: Time spent running (in hours)
Since the man walks and runs, we split the journey into two parts. We'll need to convert the total time into hours for easier calculation.
[tex]\[ \text{Total time in hours} = \frac{39}{60} = 0.65 \, \text{hours} \][/tex]
We are also given the speeds:
- For walking: 5 km/h
- For running: 12 km/h
Using these symbols, we can fill in the chart partially:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & \text{Distance (km)} & \text{Speed (km/h)} & \text{Time (h)} \\ \hline \text{Walks} & d_w & 5 & t_w \\ \hline \text{Runs} & d_r & 12 & t_r \\ \hline \end{tabular} \][/tex]
#### System of Equations:
We can set up the following equations based on the given data:
1. Total distance equation:
[tex]\[ d_w + d_r = 5 \][/tex]
2. Total time equation:
[tex]\[ t_w + t_r = 0.65 \][/tex]
3. Distance equation when walking:
[tex]\[ d_w = 5t_w \][/tex]
4. Distance equation when running:
[tex]\[ d_r = 12t_r \][/tex]
### Part (b) How far did he run?
Given the numerical solution, we have:
- Distance walked [tex]\( d_w = 2 \)[/tex] km
- Distance ran [tex]\( d_r = 3 \)[/tex] km
Therefore,
[tex]\[ \text{Distance ran} = 3 \, \text{km} \][/tex]
### Part (c) For how many minutes did he walk?
Given the numerical solution, we also have:
- Time spent walking [tex]\( t_w = 0.4 \, \text{hours} \)[/tex]
To convert this to minutes:
[tex]\[ t_w \, (\text{minutes}) = 0.4 \times 60 = 24 \, \text{minutes} \][/tex]
### Summary:
(a)
- Chart:
[tex]\[ \begin{tabular}{|l|l|l|l|} \hline & \text{Distance (km)} & \text{Speed (km/h)} & \text{Time (h)} \\ \hline \text{Walks} & 2 & 5 & 0.4 \\ \hline \text{Runs} & 3 & 12 & 0.25 \\ \hline \end{tabular} \][/tex]
- System of Equations:
[tex]\[ \begin{cases} d_w + d_r = 5 \\ t_w + t_r = 0.65 \\ d_w = 5t_w \\ d_r = 12t_r \end{cases} \][/tex]
(b)
He ran for [tex]\( 3 \, \text{km} \)[/tex].
(c)
He walked for [tex]\( 24 \, \text{minutes} \)[/tex].
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