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A man leaves his work at 5:30 PM and heads toward home, which is 5 km away. He walks at [tex]\(5 \, \text{km/h}\)[/tex] until he receives a call from his wife. She misses him and wants to see him as soon as possible. He runs the rest of the way at [tex]\(12 \, \text{km/h}\)[/tex] and arrives home at 6:09 PM.

(a) Fill in the following table and set up a system of linear equations that you could use to help you solve for the distance walked and the distance ran.

1. Chart:
[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline
& Distance (km) & Speed (km/h) & Time (h) \\
\hline
Walks & & & \\
\hline
Runs & & & \\
\hline
\end{tabular}
\][/tex]

2. System of Equations:

(b) How far did he run?

(c) For how many minutes did he walk?



Answer :

GinnyAnswer
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].

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