Distance Calculation Problem
Problem Statement:
A man goes out at 16:42 and arrives at a post box, 6 km away, at 17:30. He walked part of
the way at 5km/h and then, realizing the time, he ran the rest of the way at 10 km/h. How
far did he have to run?
Solution:
To determine how far the man had to run, we need to calculate the distance he walked and
the distance he ran based on the given speeds and times.
Let's denote:
- \( d_w \) = distance walked (in km)
- \( d_r \) = distance ran (in km)
- \( t_w \) = time spent walking (in hours)
- \( t_r \) = time spent running (in hours)
We know the following:
1. The total distance is 6 km.
\[ d_w + d_r = 6 \]
2. The total time is from 16:42 to 17:30, which is 48 minutes or \( \frac{48}{60} = 0.8 \)
hours.
\[ t_w + t_r = 0.8 \]
3. The walking speed is 5 km/h and the running speed is 10 km/h.
\[ d_w = 5 t_w \]
\[ d_r = 10 t_r \]
Using the equations for distance:
\[ d_w = 5 t_w \]
\[ d_r = 10 t_r \]
Substitute \( d_w \) and \( d_r \) into the total distance equation:
\[ 5 t_w + 10 t_r = 6 \]
We also have the total time equation:
\[ t_w + t_r = 0.8 \]
We can solve these two equations simultaneously. First, solve the total time equation for \(
t_w \):
\[ t_w = 0.8 - t_r \]
Substitute \( t_w \) into the distance equation:
\[ 5 (0.8 - t_r) + 10 t_r = 6 \]
Simplify and solve for \( t_r \):
\[ 4 - 5 t_r + 10 t_r = 6 \]
\[ 4 + 5 t_r = 6 \]
\[ 5 t_r = 2 \]
\[ t_r = 0.4 \]
Now, calculate \( d_r \):
\[ d_r = 10 t_r = 10 \times 0.4 = 4 \text{ km} \]
So, the man had to run 4 km.