A Lamu bus left Mombasa for Nairobi at 8:30 a.m. traveling at 120 km/hr. A Coast bus left Nairobi for Mombasa at 8:00 a.m. traveling at 80 km/hr. The distance between Mombasa and Nairobi is 400 km. Wh

Here's the solution to the problem:

i) The time the Lamu Bus arrived in Nairobi.

Step 1: Calculate the time taken by the Lamu bus to travel from Mombasa to Nairobi.
The distance between Nairobi and Mombasa is $400 \text{ km}$.
The speed of the Lamu bus is $120 \text{ km/hr}$.

$$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$$ $$\text{Time taken by Lamu bus} = \frac{400 \text{ km}}{120 \text{ km/hr}} = \frac{10}{3} \text{ hours}$$

Convert the time to hours and minutes:

$$\frac{10}{3} \text{ hours} = 3 \frac{1}{3} \text{ hours} = 3 \text{ hours} + \left(\frac{1}{3} \times 60\right) \text{ minutes} = 3 \text{ hours } 20 \text{ minutes}$$

Step 2: Calculate the arrival time of the Lamu bus in Nairobi.
The Lamu bus left Mombasa at 8:30 a.m.

$$\text{Arrival time} = 8:30 \text{ a.m.} + 3 \text{ hours } 20 \text{ minutes} = 11:50 \text{ a.m.}$$

The Lamu Bus arrived in Nairobi at $\boxed{\text{11:50 a.m.}}$.

ii) The time the two buses met.

Step 1: Calculate the distance covered by the Coast bus before the Lamu bus started.
The Coast bus left Nairobi at 8:00 a.m.
The Lamu bus left Mombasa at 8:30 a.m.
Time difference = 30 minutes = 0.5 hours.
Speed of Coast bus = $80 \text{ km/hr}$.

$$\text{Distance covered by Coast bus in 0.5 hours} = 80 \text{ km/hr} \times 0.5 \text{ hr} = 40 \text{ km}$$

Step 2: Calculate the remaining distance between the buses at 8:30 a.m.
Total distance = $400 \text{ km}$.

$$\text{Remaining distance} = 400 \text{ km} - 40 \text{ km} = 360 \text{ km}$$

Step 3: Calculate the relative speed of the two buses.
The buses are moving towards each other.
Speed of Coast bus = $80 \text{ km/hr}$.
Speed of Lamu bus = $120 \text{ km/hr}$.

$$\text{Relative speed} = 80 \text{ km/hr} + 120 \text{ km/hr} = 200 \text{ km/hr}$$

Step 4: Calculate the time taken for the buses to meet from 8:30 a.m.

$$\text{Time to meet} = \frac{\text{Remaining distance}}{\text{Relative speed}} = \frac{360 \text{ km}}{200 \text{ km/hr}} = \frac{18}{10} \text{ hours} = 1.8 \text{ hours}$$

Convert 1.8 hours to hours and minutes:

$$1.8 \text{ hours} = 1 \text{ hour} + (0.8 \times 60) \text{ minutes} = 1 \text{ hour } 48 \text{ minutes}$$

Step 5: Determine the meeting time.
The buses started moving towards each other at 8:30 a.m.

$$\text{Meeting time} = 8:30 \text{ a.m.} + 1 \text{ hour } 48 \text{ minutes} = 10:18 \text{ a.m.}$$

The two buses met at $\boxed{\text{10:18 a.m.}}$.

iii) The distance from Nairobi to the point where the buses met.

Step 1: Calculate the total time the Coast bus traveled until it met the Lamu bus.
The Coast bus left Nairobi at 8:00 a.m. and met at 10:18 a.m.

$$\text{Total travel time for Coast bus} = 10:18 \text{ a.m.} - 8:00 \text{ a.m.} = 2 \text{ hours } 18 \text{ minutes}$$

Convert 18 minutes to hours: $18 \text{ minutes} = \frac{18}{60} \text{ hours} = 0.3 \text{ hours}$.

$$\text{Total travel time for Coast bus} = 2 + 0.3 = 2.3 \text{ hours}$$

Step 2: Calculate the distance covered by the Coast bus from Nairobi.
Speed of Coast bus = $80 \text{ km/hr}$.

$$\text{Distance from Nairobi} = \text{Speed} \times \text{Time} = 80 \text{ km/hr} \times 2.3 \text{ hours} = 184 \text{ km}$$

The distance from Nairobi to the point where the buses met is $\boxed{\text{184 km}}$.

iv) How far Coast Bus is from Mombasa when Lamu bus arrives in Nairobi.

Step 1: Determine the arrival time of the Lamu bus in Nairobi.
From part (i), the Lamu bus arrived in Nairobi at 11:50 a.m.

Step 2: Calculate the total time the Coast bus traveled until 11:50 a.m.
The Coast bus left Nairobi at 8:00 a.m.

$$\text{Total travel time for Coast bus} = 11:50 \text{ a.m.} - 8:00 \text{ a.m.} = 3 \text{ hours } 50 \text{ minutes}$$

Convert 50 minutes to hours: $50 \text{ minutes} = \frac{50}{60} \text{ hours} = \frac{5}{6} \text{ hours}$.

$$\text{Total travel time for Coast bus} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \text{ hours}$$

Step 3: Calculate the distance covered by the Coast bus from Nairobi.
Speed of Coast bus = $80 \text{ km/hr}$.

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