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Thread: Calc III

  1. #1
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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 $\displaystyle V$, you want to minimize (exterior) surface area.

    Let $\displaystyle x=width$, $\displaystyle y=length$, and $\displaystyle z=height$

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

    You want to minimize $\displaystyle S$.

    This looks like a good time to use Lagrange multipliers.

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

    ($\displaystyle \nabla$ denotes the gradient.)

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