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A School is Preparing a Trip for 400 Students

Updated July 5, 2024

A school is preparing a trip for 400 students. The company providing the transportation has 10 buses with 50 seats each and 8 buses with 40 seats each, but only has 9 drivers available. The rental cost for a large bus is $800 and $600 for a small bus. Calculate how many buses of each type should be used for the trip to minimize the total cost.

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Quizzma TeamJuly 5, 2024

To transport 400 students at the least possible cost, the school should use 4 large buses and 5 small buses. The total cost will be $5200.

AnonymousJuly 5, 2024

Case-by-Case Analysis

  1. x=9,y=0x = 9, y = 0x=9,y=0

    50(9)+40(0)=450(≥400)OK50(9) + 40(0) = 450 \quad (\geq 400) \quad \text{OK}
    Cost=800(9)+600(0)=7200\text{Cost} = 800(9) + 600(0) = 7200

  2. x=8,y=1x = 8, y = 1x=8,y=1

    50(8)+40(1)=440(≥400)OK50(8) + 40(1) = 440 \quad (\geq 400) \quad \text{OK}
    Cost=800(8)+600(1)=6800\text{Cost} = 800(8) + 600(1) = 6800

  3. x=7,y=2x = 7, y = 2x=7,y=2

    50(7)+40(2)=430(≥400)OK50(7) + 40(2) = 430 \quad (\geq 400) \quad \text{OK}
    Cost=800(7)+600(2)=6400\text{Cost} = 800(7) + 600(2) = 6400

  4. x=6,y=3x = 6, y = 3x=6,y=3

    50(6)+40(3)=420(≥400)OK50(6) + 40(3) = 420 \quad (\geq 400) \quad \text{OK}
    Cost=800(6)+600(3)=6000\text{Cost} = 800(6) + 600(3) = 6000

  5. x=5,y=4x = 5, y = 4x=5,y=4

    50(5)+40(4)=410(≥400)OK50(5) + 40(4) = 410 \quad (\geq 400) \quad \text{OK}
    Cost=800(5)+600(4)=5600\text{Cost} = 800(5) + 600(4) = 5600

  6. x=4,y=5x = 4, y = 5x=4,y=5

    50(4)+40(5)=400(≥400)OK50(4) + 40(5) = 400 \quad (\geq 400) \quad \text{OK}
    Cost=800(4)+600(5)=5200\text{Cost} = 800(4) + 600(5) = 5200