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Two vehicles, a bus and a matatu, are traveling towards each other on a straight road connecting towns A and B, which are 840 km apart. The bus starts from town A at 11:45 a.m. traveling at a speed of

Mathematics

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3 parts
a

4:51 p.m.\text{4:51 p.m.}

b

408 km\text{408 km}

c

160 km\text{160 km}

Step-by-step solution

6 steps

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 ×\times Head start time
Distance = 80 km/h×1.5 h=120 km80 \text{ km/h} \times 1.5 \text{ h} = 120 \text{ 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 km840 \text{ km} - 120 \text{ km} = 720 \text{ 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/h80 \text{ km/h} + 120 \text{ km/h} = 200 \text{ km/h}.

Part a· 1 step

The time of day when the two vehicles met.

  1. 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 hours720 \text{ km} / 200 \text{ km/h} = 3.6 \text{ hours}.
    Convert 3.6 hours to hours and minutes: 3 hours+(0.6×60) minutes=3 hours 36 minutes3 \text{ hours} + (0.6 \times 60) \text{ minutes} = 3 \text{ hours } 36 \text{ 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.\boxed{\text{4:51 p.m.}}.

Part b

408 km\text{408 km}

How far from town A they met?

  1. 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 hours1.5 \text{ hours} + 3.6 \text{ hours} = 5.1 \text{ hours}.
    Distance from town A = Speed of bus ×\times Total time bus traveled
    Distance from town A = 80 km/h×5.1 h=408 km80 \text{ km/h} \times 5.1 \text{ h} = 408 \text{ km}.

    They met 408 km\boxed{\text{408 km}} from town A.

Part c

160 km\text{160 km}

How far outside town B the bus was when the Matatu reached town A.

  1. 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 hours840 \text{ km} / 120 \text{ km/h} = 7 \text{ hours}.

  2. 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.

  3. Calculate the distance covered by the bus during this time.

    Distance covered by bus = Speed of bus ×\times Total time bus traveled
    Distance covered by bus = 80 km/h×8.5 h=680 km80 \text{ km/h} \times 8.5 \text{ h} = 680 \text{ km}.

  4. 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 km840 \text{ km} - 680 \text{ km} = 160 \text{ km}.

    The bus was 160 km\boxed{\text{160 km}} outside town B (meaning 160 km away from town B, still approaching it).

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