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Math Help - Norman window problem

  1. #1
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    Norman window problem

    A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 38 ft, express the area A of the window as a function of the width x of the window.


    What I have is A = pi(x)/2+xh

    I know it gives you the 38 ft for a reason. And the fact its highlighted in red also hints that I should have used it somewhere.
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  2. #2
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    Hello, leftyguitarjoe!

    Your answer has the area in terms of both x and h.
    Your answer should have x only.


    A Norman window has the shape of a rectangle surmounted by a semicircle.
    If the perimeter of the window is 38 ft, express the area A of the window
    as a function of the width x of the window.
    Code:
                  * * *
              *           *
            *               *
           *                 *
    
          *                   *
          * - - - - * - - - - *
          |                   |
          |                   |
        h |                   | h
          |                   |
          |                   |
          * - - - - - - - - - *
                    x

    The semicircle has diameter x and radius \frac{x}{2}

    The circumference of the semicircle is: . \frac{1}{2} \times \pi x \:=\:\frac{\pi}{2}x

    The perimeter is: . P \;=\;\frac{\pi}{2}x + x + 2h \:=\:38 \quad\Rightarrow\quad 2h + \left(\frac{\pi+2}{2}\right)x \:=\:38

    . . 2h \:=\:38 - \left(\frac{\pi+2}{2}\right)x \quad\Rightarrow\quad h \:=\:19 - \left(\frac{\pi + 2}{4}\right)x .[1]



    The area of the semicircle is: . \frac{1}{2}\pi r^2 \:=\: \frac{\pi}{2}\left(\frac{x}{2}\right)^2 \:=\: \frac{\pi}{8}x^2

    The area of the rectangle is: . xh

    The total area is: . A \;=\;\frac{\pi}{8}x^2 + xh .[2]


    Substitute [1] into [2]: . A \;=\;\frac{\pi}{8}x^2 + x\left[19 - \left(\frac{\pi+2}{4}\right)x\right]

    . . which simplifies to: . A \;=\;19 x - \left(\frac{\pi +4}{8}\right)x^2


    But check my work . . . please!
    .
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