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Math Help - Optimization Problem...

  1. #1
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    Optimization Problem...

    Consider a window the shape of which is a rectangle of height h surmounted a triangle having a height T that is 1.5 times the width w of the rectangle (as shown in the figure below).

    If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.

    h=?
    w=?
    Last edited by pakman134; March 22nd 2008 at 06:25 PM.
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  2. #2
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    We can let T=\frac{3}{2}w; \;\ \frac{2}{3}T=w

    A_{rectangle}=h(\frac{2}{3}T)

    A_{triangle}=\frac{1}{3}T^{2}

    Total \;\ area=h(\frac{2}{3}T)+\frac{1}{3}T^{2}...[1]

    Therefore, the perimeter can be expressed as

    P=2h+\frac{2T}{3}+2\sqrt{(\frac{T}{3})^{2}+T^{2}}...[2]

    Continue by expressing P in terms of T alone. Then, differentiate, set to 0 and solve for T. You can also express it in terms of w instead of T.
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  3. #3
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    hm...

    thanks!
    Last edited by pakman134; March 22nd 2008 at 06:59 PM. Reason: figured it part of it...
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