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Math Help - Help please?

  1. #1
    Junior Member
    Joined
    Aug 2007
    From
    Houston, TX
    Posts
    46

    Help please?

    how would you do this....

    Find the largest area of a rectangle inside a circle with a radius of 4.
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    It should be a square.

    Draw the figure on paper, as I cannot show it here.
    Let x = width of the rectangle
    And y = the length
    Draw any diagonal. This diagonal is a diameter of the circle so it is 2r long.

    Area of rectangle, A = xy

    By Pythagorean theorem, x^2 +y^2 = (2r)^2
    x^2 +y^2 = 4r^2
    y^2 = 4r^2 -x^2
    y = sqrt(4r^2 -x^2)

    So,
    A = x*sqrt(4r^2 -x^2)
    Differentiate both sides with respect to x, (r is a constant),
    dA/dx = x[-2x / 2sqrt(4r^2 -x^2)] +sqrt(4r^2 -x^2)
    Set that to zero,
    0 = x(-x) +[sqrt(4r^2 -x^2)]^2
    0 = -x^2 +4r^2 -x^2
    2x^2 = 4r^2
    x^2 = 2r^2
    x = r*sqrt(2) ------***
    So,
    y = sqrt[4r^2 -2r^2] = sqrt[2r^2] = r*sqrt(2) also.

    Therefore, the largest area of a rectangle in a circle whose radius is 4 is
    A = 2sqrt(2) *2sqrt(2) = 4*2 = 8 sq.units ------------answer.
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  3. #3
    Newbie
    Joined
    Oct 2007
    Posts
    22
    Area is not 8 sq units...

    However ticbol is correct about it being a square.
    Here's a simple way of doing the question.
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