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Math Help - Rectangular Sheet of Metal Problem

  1. #1
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    Rectangular Sheet of Metal Problem

    A rectangular piece of sheet metal with an area of 1,600 in 2 is to be bent into a cylindrical length of stovepipe having a volume of 800 in 3. What are the dimensions of the sheet metal?
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  2. #2
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    Hello, tmac11522!

    A rectangular piece of sheet metal with an area of 1600 in²
    is to be bent into a cylindrical length of stovepipe having a volume of 800 in³.
    What are the dimensions of the sheet metal?
    Code:
          *-----------*
          |           |
        W |           |
          |           |
          *-----------*
                L

    The area of this rectangle is: . LW \:=\:1600 \quad\Rightarrow\quad W \:=\:\frac{1600}{L} .[1]
    This is the height h of the cylinder.

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


    The volume of the cylinder is 800: . \pi r^2h \:=\:800

    Substitute [1] and [2]: . \pi\left(\frac{L}{2\pi}\right)^2\left(\frac{1600}{  L}\right) \:=\: 800

    . . from which we get: . L \:=\:2\pi

    Substitute into [1]: . W \:=\:\frac{1600}{2\pi} \:=\:\frac{800}{\pi}


    Therefore, the dimensions are: . 2\pi \times \frac{800}{\pi} inches.

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