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Math Help - optimazation problem - cylinder

  1. #1
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    Post optimazation problem - cylinder

    hi,i wqas having a hard time solving this problem...any help would be great

    Consider a rectangle of perimeter of 12 inches. We obtain a cylinder by gluing vertical side to the other vertical side. Find the dimensions of the rectangle so that the cylinder has maximal volume.
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  2. #2
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    Hello, lsanger!

    Consider a rectangle of perimeter of 12 inches.
    We obtain a cylinder by gluing vertical side to the other vertical side.
    Find the dimensions of the rectangle so that the cylinder has maximal volume.

    We have an L\times W rectangle.

    Code:
          * - - - - - - - - *
          |                 |
          |                 |
        W |                 |
          |                 |
          |                 |
          * - - - - - - - - *
                   L

    The perimeter is 12: . 2L + 2W \:=\:12 \quad\Rightarrow\quad W \:=\:6-L .[1]



    It is "rolled" into a cylinder.
    The side view looks like this:

    Code:
          * - + - *
          |   :   |
          |   :   |
        h |   :   |
          |   :   |
          |   :   |
          * - + - *
                r

    The circumference of the circular base is L.
    . . 2\pi r \:=\:L \quad\Rightarrow\quad r \:=\:\frac{L}{2\pi} .[2]

    The height h of the cylinder is W. .[3]


    The volume of a cylinder is: . V \;=\;\pi r^2h

    Substitute [2] and [3]: . V \;=\;\pi\left(\frac{L}{2\pi}\right)^2W \;=\;\frac{1}{4\pi}L^2W

    Substitute [1]: . V \;=\;\frac{1}{4\pi}L^2(6-L) \quad\Rightarrow\quad V \;=\;\frac{1}{4\pi}(6L^2-L^3)


    And that is the function we must maximize.

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