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Math Help - Surface area as function of height of box

  1. #1
    Member realintegerz's Avatar
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    Surface area as function of height of box

    A box with a square base (sides of length x) and an open top must have a volume of 32,000 cubic centimeters. Express the surface area as a function of the height of the box



    This is all I got...

    l x w x h = 32,000

    x = w, h

    h = 32,000/x^2 since Volume = x^2(h)
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  2. #2
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    Hello, realintegerz!

    Did you make a sketch?


    A box with a square base (sides of length x) and an open top
    must have a volume of 32,000 cubic centimeters.
    Express the surface area as a function of the height of the box.
    Code:
             *-------*
            /|      /|
           / |     / |
          *-------*  |
          |       |  |
          |       |  |
         h|       |  *
          |       | /
          |       |/ x
          *-------*
              x

    The volume is: . x^2h \:=\:32,\!000 \quad\Rightarrow\quad x^2 \:=\:\frac{32,\!000}{h} \quad\Rightarrow\quad x \:=\:\frac{80\sqrt{5}}{\sqrt{h}} .[1]


    The surface area is comprised of:

    . . a square base with area: x^2

    . . four side panels, each with area: xh

    The total surface area is: . A \;=\;x^2 + 4xh


    Substitute [1]: . A \;=\;\frac{32,\!000}{h} + 4\left(\frac{80\sqrt{5}}{\sqrt{h}}\right)h


    Therefore: . \boxed{A \;=\;\frac{32,\!000}{h} + 320\sqrt{5h}}

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