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Math Help - Maximum Area

  1. #1
    Member McScruffy's Avatar
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    Maximum Area

    I'm having some trouble developing an equation to differentiate for this problem. Any help would be appreciated.

    Consider a symmetric cross inscribed in a circle with radius r. Determine the value x, where x is the length from the center of the cross to one of its ends, such that the value of x maximizes the area of the cross.

    Thanks
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by McScruffy View Post
    I'm having some trouble developing an equation to differentiate for this problem. Any help would be appreciated.

    Consider a symmetric cross inscribed in a circle with radius r. Determine the value x, where x is the length from the center of the cross to one of its ends, such that the value of x maximizes the area of the cross.

    Thanks

    You need an area function in terms of x only.
    Let the equation of the circle be:

    x^2 + y^2 = r^2

    and let x=k be the coordinate from the center to the end on the right.
    The y coordinate in the first quadrant is  y =\sqrt{r^2 - k^2} .

    Split the cross into two parts:

    The horizontal band including the shared square with the vertical band.
    This horizontal band has length 2k and height 2y, so its area is:

    A = 4 y k = 4 \sqrt{r^2-k^2} k

    The upper half of the vertical band has area given by 2y*height where height is k - y.

    B = 2y (k-y) = 2 \sqrt{r^2-k^2} (k - \sqrt{r^2-k^2})

    Combine A and 2B (note: it's 2B because you need the lower half of the vertical band as well), then do calculus with it.

    Does this help?
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  3. #3
    Member McScruffy's Avatar
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    Yes that helps a great deal. That's kind of the idea that I was thinking about, I just couldn't seem to get it worked out into a reasonable equation. But that makes a lot of sense.
    Thanks.
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