To solve this problem, let’s analyze the given data and determine when the two vehicles will meet.

Given Information

1. Distance between Kisumu and Nairobi: 360 km
2. Bus:
   • Leaves Kisumu for Nairobi at 9:30 a.m.
   • Average speed: 81 km/h
3. Matatu:
   • Leaves Nairobi for Kisumu at 10:10 a.m.
   • Average speed: 72 km/h

Steps to Solve

1. Calculate the time difference between the departure of the bus and the matatu:

   • The bus leaves at 9:30 a.m., and the matatu leaves at 10:10 a.m.
   • Time difference = 40 minutes (or $\frac{40}{60} = \frac{2}{3}$ hours).

2. Determine how far the bus has traveled before the matatu starts:

   • Distance = Speed × Time
   • $\text{Distance covered by bus in } \frac{2}{3} \text{ hours} = 81 \, \text{km/h} \times \frac{2}{3} = 54 \, \text{km}$.

   So, at 10:10 a.m., the bus is 54 km closer to Nairobi, leaving 306 km (360 - 54) between the bus and matatu.

3. Set up an equation to find the time taken for the two vehicles to meet after 10:10 a.m.:

   • Let $t$ be the time in hours after 10:10 a.m. when the two vehicles meet.
   • The combined distance they cover each hour when moving towards each other is the sum of their speeds: $81 + 72 = 153$ km/h.
   • The remaining distance between them at 10:10 a.m. is 306 km.

   Since Distance = Rate × Time, we set up the equation: $153t = 306$

4. Solve for $t$: $t = \frac{306}{153} = 2 \, \text{hours}$

5. Calculate the meeting time:

   • The matatu leaves at 10:10 a.m.
   • After 2 hours, the two vehicles will meet.
   • Meeting time = 10:10 a.m. + 2 hours = 12:10 p.m.

Answer

The two vehicles will meet at 12:10 p.m..

Would you like further details or have any questions?

Here are some related questions to explore:

1. What if the bus had a speed of 90 km/h instead of 81 km/h?
2. How long would it take for the vehicles to meet if they left at the same time?
3. What distance would each vehicle cover before meeting?
4. How would the meeting time change if the matatu’s speed was increased to 80 km/h?
5. What if the distance between the cities was different, say 400 km?

Tip: When solving problems involving two moving objects, remember to consider the relative speed if they are moving towards or away from each other.