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A rectangle is bounded by the x-axis and the semicircle y= �ã(25-x^2). What length and width should the rectangle have so that the area is maximum?

Place the semi-circle so it's center coincides with the origin. If you go a distance x to the right the max height is sqrt(25-x^2). Since x is only half the length of one side, the dimensions are 2x and sqrt(25-x^2), thus the area is
A=2x*sqrt(25-x^2)
Find dA/dx, set to 0 and solve for x.

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