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.
ScanSolve AI Answer
Step 1: Calculate the head start time and distance for the bus. Bus leaves town A at 11:45 a.m. Matatu leaves town B at 1:15 p.m. (which is 13:15 in 24-hour format). Head start time for the bus = 13:15 - 11:45 = 1 hour 30 minutes = 1.5 hours. Distance covered by the bus during its head start = Speed of bus × Head start time Distance = 80 km/h × 1.5 h = 120 km. Step 2: Calculate the remaining distance and relative speed when both vehicles are traveling. Total distance between towns A and B = 840 km. Remaining distance when Matatu starts = 840 km - 120 km = 720 km. The vehicles are moving towards each other, so their relative speed is the sum of their speeds. Relative speed = Speed of bus + Speed of Matatu Relative speed = 80 km/h + 120 km/h = 200 km/h. a) The time of day when the two vehicles met. Step 3: Calculate the time taken for them to meet after the Matatu started. Time to meet = Remaining distance / Relative speed Time to meet = 720 km / 200 km/h = 3.6 hours. Convert 3.6 hours to hours and minutes: 3 hours + (0.6 × 60) minutes = 3 hours 36 minutes. The Matatu started at 1:15 p.m. Meeting time = 1:15 p.m. + 3 hours 36 minutes = 4:51 p.m. The time of day when the two vehicles met is 4:51 p.m.. b) How far from town A they met? Step 4: Calculate the total time the bus traveled until they met. Total time bus traveled = Head start time + Time after Matatu started Total time bus traveled = 1.5 hours + 3.6 hours = 5.1 hours. Distance from town A = Speed of bus × Total time bus traveled Distance from town A = 80 km/h × 5.1 h = 408 km. They met 408 km from town A. c) How far outside town B the bus was when the Matatu reached town A. Step 5: Calculate the time it took for the Matatu to reach town A. Time for Matatu to reach town A = Total distance / Speed of Matatu Time = 840 km / 120 km/h = 7 hours. Step 6: Calculate the total time the bus traveled until the Matatu reached town A. The Matatu started at 1:15 p.m. and traveled for 7 hours, so it reached town A at 1:15 p.m. + 7 hours = 8:15 p.m. The bus started at 11:45 a.m. Total time bus traveled = 8:15 p.m. - 11:45 a.m. In 24-hour format: 20:15 - 11:45 = 8 hours 30 minutes = 8.5 hours. Step 7: Calculate the distance covered by the bus during this time. Distance covered by bus = Speed of bus × Total time bus traveled Distance covered by bus = 80 km/h × 8.5 h = 680 km. Step 8: Determine how far outside town B the bus was. The bus started from town A and traveled 680 km towards town B. The total distance between A and B is 840 km. Since 680 km < 840 km, the bus had not yet reached town B. Distance from town B = Total distance - Distance covered by bus Distance from town B = 840 km - 680 km = 160 km. The bus was 160 km outside town B (meaning 160 km away from town B, still approaching it). That's 2 down. 3 left today — send the next one.