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Math Help - Calc III

  1. #1
    Junior Member
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    Unhappy Calc III

    Determine the dimensions of a rectangular box, without a top, having volume V ft³, which requires the least amount of material for its construction.

    I have no idea where to begin.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by tdat1979 View Post
    Determine the dimensions of a rectangular box, without a top, having volume V ft³, which requires the least amount of material for its construction.

    I have no idea where to begin.
    Conceptually, given a fixed volume V, you want to minimize (exterior) surface area.

    Let x=width, y=length, and z=height

    V=xyz
    S=xy+2xz+2yz (because there is no top)

    You want to minimize S.

    This looks like a good time to use Lagrange multipliers.

    \nabla S=\langle y+2z,x+2z,2x+2y\rangle
    \nabla V=\langle yz,xz,xy\rangle

    ( \nabla denotes the gradient.)

    So you need to solve \langle y+2z,x+2z,2x+2y\rangle=\lambda\langle yz,xz,xy\rangle given that xyz=V. (Remember that V is a constant.)
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