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Math Help - Math optimization

  1. #1
    rmn
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    Math optimization

    http://i207.photobucket.com/albums/b...EvErSe_/22.jpg

    I have absolutely no idea how to do this or even start it
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  2. #2
    Eater of Worlds
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    Let k be the amount of light admitted per unit area of glass.

    The total amount of light admitted by the whole window is:

    L=k(\text{area of clear glass})+\frac{1}{2}k(\text{area of colored glass})=2krh+\frac{1}{4}{\pi}kr^{2}

    But the perimeter is P=2h+2r+{\pi}r

    h=\frac{1}{2}(P-2r-{\pi}r)

    So,

    L=kr(P-2r-{\pi}r)+\frac{1}{4}{\pi}kr^{2}=k\left[Pr-(2+{\pi}-\frac{\pi}{4})r^{2}\right]=k\left[Pr-\frac{8+3\pi}{4}r^{2}\right]

    for 0<r<\frac{P}{2+{\pi}}

    Now, differentiate wrt r, set to 0 and solve for r.
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  3. #3
    rmn
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    sry what does L mean and isnt the rectangle area equal lw not rh?
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  4. #4
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    L stands for Light. I believe he used r because the length of the rectangle is 2*radius of the semicircle. h is just the height of the rectangle.
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