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March 24, 2017

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An airline will provide accommodations for a minimum of 2000 first-class, 1500 tourist, and 2400 economy-class passengers. Airplane P-1 costs $12,000 per miles to operate and can accommodate 40 first-class, 40 tourist, and 12 economy-class passengers. Airplane Q-2 costs $10,000 per miles to operate and can accommodate 80 first-class, 30tourist, and 40 economy-class passengers. How many of each type of plane should be used to minimize the operating cost?

  • Algebra 2 (Linear Programming) - ,

    C = 12000 p + 10000 q

    40 p + 80 q >/= 2000
    40 p + 30 q >/= 1500
    12 p + 40 q >/= 2400

    or
    p + 2 q >/= 50
    4 p + 3 q >/= 150
    3 p + 10 q >/= 60

    Graph those lines
    You will see that the First two lines are way up there to the right and above the third. Therefore only the first two need be investigated
    They intersect at:
    4 p + 8 q = 200
    4 p + 3 q = 150
    -----------------
    5 q = 50
    q = 10
    p = 30 so (p,q) at (30,10)
    also axis intersections at (0,50) and (50,0)
    test those three for cost
    12p + 10 q or 6 p + 5 q

    at intersection (30,10)
    6*30+5*10 =230

    at (0,50)
    6*0 + 5*50 = 250

    at (50,0)
    6*50 + 0 = 300

    so min cost at (30,10)

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