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Math Help - Area of a Window

  1. #1
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    Area of a Window

    A Norman window is constucted by adjoining a semi-circle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

    I know the equations:
    A=xy+\frac {pi*x^2}{4}
    P=x+2y+\frac {pi*x}{2}=16
    (Don't know how to make the pi symbol in LaTeX, )

    I tried to solve it multiple times but each time I end with something not possible, such as 4+1=0(just an example). Any help is appreciated, thanks. Here is a drawing of the figure, sorry for my bad paint skills.
    Attached Thumbnails Attached Thumbnails Area of a Window-lols.png  
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  2. #2
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    Quote Originally Posted by lancelot854 View Post
    A Norman window is constucted by adjoining a semi-circle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

    I know the equations:
    A=xy+\frac {pi*x^2}{4}
    P=x+2y+\frac {pi*x}{2}=16
    (Don't know how to make the pi symbol in LaTeX, )

    I tried to solve it multiple times but each time I end with something not possible, such as 4+1=0(just an example). Any help is appreciated, thanks. Here is a drawing of the figure, sorry for my bad paint skills.
    area of the semicircle is \displaystyle \frac{\pi \left(\frac{x}{2}\right)^2}{2} = \frac{\pi x^2}{8}
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    Well, that's one mistake, but even so, I still get messed up along the way. -.-
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  4. #4
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    \displaystyle x\left(\frac{\pi + 2}{2}\right) + 2y = 16

    \displaystyle y = 8 - x\left(\frac{\pi + 2}{4}\right)

    \displaystyle A = 8x - x^2\left(\frac{\pi + 2}{4}\right) + \frac{\pi x^2}{8}

    \displaystyle A = 8x - x^2 \left(\frac{\pi + 2}{4} -\frac{\pi}{8} \right)

    \displaystyle A = 8x - x^2 \left(\frac{\pi + 4}{8}\right)

    A will be a maximum when

    \displaystyle x = \frac{32}{\pi + 4}
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  5. #5
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    Could I ask how you came to find \frac{\pi + 2}{2} and \frac{\pi + 2}{4}? Thanks for the help.

    EDIT: Nevermind, figured it out after a lot of thinking. :P
    Last edited by lancelot854; December 2nd 2010 at 06:22 PM.
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