This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — 23. a) i) The time when the car caught up with the bus. Step 1: Determine the departure times and speeds. Bus: Departure time = 7:00 am Speed (S_B) = 80 km/hr Car: Departure time = 7:00 am + 1.25 hours = 8:15 am Speed (S_C) = 100 km/hr Step 2: Let t be the time in hours the bus has traveled when the car catches up. The car travels for (t - 1.25) hours. Distance covered by bus (D_B) = S_B × t = 80t Distance covered by car (D_C) = S_C × (t - 1.25) = 100(t - 1.25) Step 3: Equate the distances to find t. When the car catches up, D_B = D_C. 80t = 100(t - 1.25) 80t = 100t - 125 125 = 100t - 80t 125 = 20t t = (125)/(20) = (25)/(4) = 6.25 hours Step 4: Convert the time t into hours and minutes and add to the bus's departure time. 6.25 hours = 6 hours and 0.25 × 60 minutes = 6 hours and 15 minutes. Time of catch-up = 7:00 am + 6 hours 15 minutes = 1:15 pm. The time when the car caught up with the bus is 1:15 pm. ii) The distance from Migori when the car caught up with the bus. Step 1: Use the time t from part (a)(i) and the bus's speed to find the distance. Distance = S_B × t Distance = 80 km/hr × 6.25 hours Distance = 500 km. The distance from Migori when the car caught up with the bus is 500 km. b) A Matatu left Kisumu for Migori at 10:10 a.m and moved at an average speed of 100km/hr. if the Matatu met the bus at the same time as the car, find the time when the Matatu arrived in Migori. Step 1: Determine the meeting time of the Matatu and the bus. The Matatu met the bus at the same time the car caught up with the bus, which is 1:15 pm. Step 2: Calculate the distance the bus traveled from Migori to the meeting point. Bus departure time = 7:00 am Meeting time = 1:15 pm Time bus traveled = 1:15 pm - 7:00 am = 6 hours 15 minutes = 6.25 hours. Distance from Migori to meeting point (D_B\_meet) = S_B × time bus traveled = 80 km/hr × 6.25 hours = 500 km. Step 3: Calculate the distance the Matatu traveled from Kisumu to the meeting point. Matatu departure time = 10:10 am Meeting time = 1:15 pm Time Matatu traveled = 1:15 pm - 10:10 am. 1:15 pm = 13:15 (24-hour format) 10:10 am = 10:10 (24-hour format) Time Matatu traveled = (13 hours 15 minutes) - (10 hours 10 minutes) = 3 hours 5 minutes. Convert 5 minutes to hours: (5)/(60) = (1)/(12) hours. Time Matatu traveled = 3 + (1)/(12) = (36+1)/(12) = (37)/(12) hours. Matatu speed (S_M) = 100 km/hr. Distance from Kisumu to meeting point (D_M\_meet) = S_M × time Matatu traveled = 100 km/hr × (37)/(12) hours = (3700)/(12) = (925)/(3) km. Step 4: Calculate the total distance between Migori and Kisumu. Total distance = D_B\_meet + D_M\_meet = 500 km + (925)/(3) km Total distance = (1500)/(3) + (925)/(3) = (2425)/(3) km. Step 5: Calculate the total time the Matatu takes to travel from Kisumu to Migori. Time = Total distanceMatatu speed = 2425/3 km100 km/hr = (2425)/(300) hours. Simplify the fraction: (2425)/(300) = (97)/(12) hours. Convert to hours and minutes: (97)/(12) = 8 with a remainder of 1. So, 8 hours and (1)/(12) hours. $(1)/(12) hours = (1)/(12)