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Math Help - greatest area inscribed in a circle

  1. #1
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    greatest area inscribed in a circle

    find the rectangle of greatest area that can be inscribed in the circle x^2+y^2=36
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  2. #2
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    e^(i*pi)'s Avatar
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    The diagonal of the rectangle will be the diameter and I believe the shape is a square
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  3. #3
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    We can always set up a coordinate system so that the origin is at the center of the circle and the x-axis is parallel to one side of the rectangle. In that coordinate system, the circle has equation x^2+ y^2= R^2 for some R (the radius of the circle). If we call the corner of the square in the first quadrant (x_0, y_0), then, because the vertex lies oh the circle, x_0^2+ y_0^2= R^2 so y_0= \sqrt{R^2- x_0^2.

    The lengths of the sides of the rectangle are, by symmetry, just twice those x and y values so the area of the rectangle is (2x_0)(2y_0)= 4x_0\sqrt{R^2- x_0^2. Replace x_0 by x and find the value of x that maximizes A= 4x(R^2- x^2)^{1/2}.
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