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Math Help - Optimization w/ Surface Area

  1. #1
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    Exclamation Optimization w/ Surface Area

    I am given the dimensions of a box (h=14,w=10,l=3) I have to preserve the ratio of H:W, which is 7:5. L does't need to be porportional to anything. I am told I must maintain the H:W ratio and the volume. The volume I found to be 420 in.^3.
    The surface area equation is 2lw+2lh+2wh
    I need to minimize the amount of surface area on the box while maintaining the specified factors including the volume.
    I have tried many things including replacing lw, lh, wh, none have worked. Can someone please show me how to do this, I need to find:
    • Equation of SA(l) where SA(l)=aL^2+b/L^c
    • The minima

    If someone can please show me how to do it?
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  2. #2
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    Re: Optimization w/ Surface Area

    LHW = 420

    W = \frac{5H}{7}

    LH \cdot \frac{5H}{7} = 420

    H = \sqrt{\frac{7 \cdot 84}{L}} = 14\sqrt{\frac{3}{L}}

    W = 10\sqrt{\frac{3}{L}}


    A = 2(LH+LW+HW)

    A = 2\left(14\sqrt{3L} + 10\sqrt{3L} + \frac{420}{L^2} \right)

    A = 24\left(2\sqrt{3L} + \frac{35}{L^2} \right)
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