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Math Help - Linear Programming

  1. #1
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    Linear Programming

    I understand the problems that give various constraints and ask to maximize or minimize something, but not this type.

    A chem manufacturer wants to rent a fleet of 24 tank cars with a combined carrying capacity of 504,000 gallons. Tank cars with three capacities are available: 7000 gallons, 14,000 gallons, 28,000 gallons. How many of each should be leased?

    Set x, y and z as the three different capacity cars.

    x+y+z=24
    7000x+14000y+28000z=504000

    There are various answers. For example it can be 18 z's, or 17 z's and 2 y's, or 17z's and 4 x's.

    The answer asks to express it as:

    x=_t+_ y=_t+_ and z=t where _<=t<=_

    that's something times t plus something, etc. where something <=t<=something
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  2. #2
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    Re: Linear Programming

    Determined the answer.

    7000 gallon car = x1
    14000 gallon car = x2
    28000 gallon car = x3




    Two equations from the given information:


    x1+x2+x3=24 The number of the combined tank cars must be 24


    7000x1+14000x2+28000x3 = 504000 Their combined capacity must be 504000


    There are two equations with three unknowns. That can't be solved unless we select a value for one variable.


    Call x3=t. Now solve for the other two variables in terms of t.


    x1+x2+t=24
    x1=24-x2-t


    7000x1+14000x2+28000t=504000
    7000(24-x2-t)+14000x2+28000t=504000
    168000-7000x2-7000t+14000x2+28000t=504000
    7000x2+21000t=336000
    7000x2=336000-21000t


    x2=48-3t= -3t + 48


    x1+48-3t+t=24
    x1-2t=-24
    x1=2t-24




    To find the constraints on t, set each equation (x1 and x2 in terms of t) >= 0. Then solve for t.


    x1 = 2t-24 >= 0
    2t>=24
    t>=12


    x2=-3t+48>=0
    -3t>=48
    t<=16


    So t is in the range 12<=t<=16


    t>=0, but that's redundant.


    So the answer is B.


    x1 = 2t-24
    x2 = -3t+48
    x3=t
    12<=t<=16
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  3. #3
    MHF Contributor
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    Re: Linear Programming

    Only 3 solutions if at least one of each:
    2,9,13
    4,6,14
    6,3,15
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