To begin with, let's identify and break down the information provided:
- Distance between Embu and Nairobi: 240 km
- Speed of the first motorist: 90 km/h
- Speed of the second motorist: 100 km/h
- Departure time of the first motorist: 8:00 AM
- Departure time of the second motorist: 8:30 AM
We want to find out how far they meet from Nairobi.
1. Calculate the time each motorist is traveling before they meet:
Let's denote [tex]\( t \)[/tex] as the time in hours after 8:00 AM when the two motorists meet.
- The first motorist travels at 90 km/h, so in [tex]\( t \)[/tex] hours, the distance covered by the first motorist is [tex]\( 90t \)[/tex] km.
- The second motorist starts at 8:30 AM, so he has been traveling for [tex]\( t - 0.5 \)[/tex] hours when they meet. Therefore, the distance covered by the second motorist is [tex]\( 100(t - 0.5) \)[/tex] km.
2. Set up an equation to represent the total distance:
Since the total distance between Embu and Nairobi is 240 km, we can write the equation:
[tex]\[
90t + 100(t - 0.5) = 240
\][/tex]
3. Simplify the equation:
[tex]\[
90t + 100t - 50 = 240
\][/tex]
[tex]\[
190t - 50 = 240
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
190t = 290
\][/tex]
[tex]\[
t = \frac{290}{190}
\][/tex]
[tex]\[
t \approx 1.5263 \text{ hours}
\][/tex]
So, the two motorists meet approximately 1.5263 hours after the first motorist started his journey at 8:00 AM.
5. Calculate the distance traveled by the second motorist:
Since the second motorist started traveling at 8:30 AM and the two meet 1.5263 hours after 8:00 AM:
[tex]\[
\text{Travel time for the second motorist} = 1.5263 - 0.5 \approx 1.0263 \text{ hours}
\][/tex]
The distance covered by the second motorist is given by:
[tex]\[
\text{Distance from Nairobi} = 100 \times 1.0263 = 102.6316 \text{ km}
\][/tex]
Conclusion: The two motorists meet approximately 102.63 km from Nairobi.
