Question

1. the distance between two towns a and b is 360km. a minibus left town a at 8.15a.m and travelled towards b at an average speed of 90km/hr. a matatu left town b at 10.35a.m on the same day and travelled towards a at an average speed of 110km/hr. (4mk)

a) i) how far from a did they meet?
ii) at what time did the two vehicles meet? (2mk)
b) a motorist left his home at 10.30a.m on the same day and travelled at an average speed of 100km/hr. he arrived at b at the same time as minibus. calculate the distance from b to his home. (4mk)

Explanation:

Step1: Calculate the distance the minibus travels before the matatu starts

The time from 8:15 a.m. to 10:35 a.m. is \(2\) hours and \(20\) minutes, which is \(\frac{7}{3}\) hours.
Using the formula \(d = vt\) (where \(d\) is distance, \(v\) is speed, and \(t\) is time), the distance the minibus travels in \(\frac{7}{3}\) hours at a speed of \(90\) km/hr is \(d_1=90\times\frac{7}{3}=210\) km.

Step2: Calculate the remaining distance and the time until they meet

The remaining distance between the two vehicles when the matatu starts is \(360 - 210=150\) km.
The relative speed of the two vehicles (since they are moving towards each other) is \(90 + 110 = 200\) km/hr.
Using the formula \(t=\frac{d}{v}\), the time until they meet after 10:35 a.m. is \(t=\frac{150}{200}=0.75\) hours (or \(45\) minutes).

Step3: Calculate the distance from A when they meet

The minibus travels an additional distance of \(90\times0.75 = 67.5\) km after 10:35 a.m.
The total distance from A is \(210+67.5 = 277.5\) km.

Step4: Calculate the meeting time

They meet \(45\) minutes after 10:35 a.m., so the meeting time is 11:20 a.m.

Step5: Calculate the time the minibus takes to reach B

The total distance from A to B is \(360\) km. The minibus speed is \(90\) km/hr. The total time for the minibus to reach B is \(t_{total}=\frac{360}{90}=4\) hours. It started at 8:15 a.m., so it reaches B at 12:15 p.m.
The motorist starts at 10:30 a.m. and reaches B at 12:15 p.m., so his travel time is \(1\) hour and \(45\) minutes (\(\frac{7}{4}\) hours).
Using \(d = vt\), with \(v = 100\) km/hr, the distance from his home to B is \(d=100\times\frac{7}{4}=175\) km.

Answer:

a) i) \(277.5\) km
ii) 11:20 a.m.
b) \(175\) km