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March 26, 2017

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A cardboard box without a top is to have volume 500000 cubic cm. Find the dimensions which minimize the amount of material used. List them in ascending order.

  • calc 3 - ,

    A.
    You can assume symmetry between length and width, which reduces to the width(=length) and the height.
    Furthermore, one of the two can be eliminated from the volume relation:
    w^2h=500000
    So the minimization problem is reduced to one single dimension as in elementary calculus.

    B.
    The same results can be obtained by calculating the area of material required:
    A=2h(b+w)+bw + L(bwh-500000)
    the second term introduces the Lagrange multiplier.
    Take partial derivatives with respect to w,b,h and L and solve for each variable from the 4 equations.
    This method should give the same results as in part A.

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