mathss

If x thousand dollars is spent on labor and y thousand dollars is spent on euqpment,The output of a certain factory will be Q(x,y)=2x^2+y^2+6y units.If 120,000 dollars is available,how should this be allocated between labor and equipment
to generate the largest possible output

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asked by tolasa
  1. for max Q, we need
    ∂Q/∂x=0 and ∂Q∂y=0
    ∂Q/∂x = 4x
    ∂Q/∂y = 2y+6
    so, there's a max/min at (0,-3).
    Since the 2nd partials are both positive, it's a minimum.

    So, we need to maximize
    2x^2+y^2+6y subject to
    x+y = 120

    Q(0,120) = 15120
    Q(120,0) = 28800
    Q(119,1)

    Clearly, since 2x^2 decreases faster than y^2+6y increases for each dollar moved from x to y, the max is Q(120,0).

    Doesn't seem right, since some equipment must be used.

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    posted by Steve

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