To solve this problem, we need to determine the time at which the two motorists meet each other.
1. Define Variables and Known Information:
- The distance between Embu and Mairdal is 240 km.
- The first motorist leaves Embu at 8:00 AM and travels at an average speed of 90 km/hr.
- The second motorist leaves Nairobi at 8:30 AM and travels at an average speed of 100 km/hr.
2. Establish equations for distances traveled:
- Let [tex]\( t \)[/tex] be the time in hours after 8:00 AM when the two motorists meet.
- The distance covered by the first motorist when they meet is [tex]\( 90t \)[/tex] km, since distance = speed × time.
- The second motorist starts 0.5 hours later. Hence, the time traveled by the second motorist when they meet is [tex]\( t - 0.5 \)[/tex] hours.
- The distance covered by the second motorist when they meet is [tex]\( 100 \times (t - 0.5) \)[/tex] km.
3. Set up the equation considering the total distance:
- The sum of the distances covered by both motorists should equal the total distance between Embu and Mairdal.
- Therefore, we have the equation:
[tex]\[
90t + 100(t - 0.5) = 240
\][/tex]
4. Simplify and solve for t:
- First, expand and simplify the equation:
[tex]\[
90t + 100t - 50 = 240
\][/tex]
[tex]\[
190t - 50 = 240
\][/tex]
- Add 50 to both sides:
[tex]\[
190t = 290
\][/tex]
- Divide both sides by 190:
[tex]\[
t = \frac{290}{190} \approx 1.526
\][/tex]
5. Interpret the result:
- The time [tex]\( t \approx 1.526 \)[/tex] hours after 8:00 AM is when the two motorists meet.
- To convert this into a more understandable time, we break it down into hours and minutes.
- [tex]\( 0.526 \)[/tex] hours is approximately [tex]\( 0.526 \times 60 \approx 31.578 \)[/tex] minutes.
6. Determine the exact meeting time:
- Since 1.526 hours after 8:00 AM is approximately 1 hour and 31.578 minutes, the motorists meet at around 9:31 AM.
Therefore, the two motorists meet at approximately 9:31 AM.
