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Consider a semicircle of radius \(20\) cm. A rectangle is inscribed in the semicircle, with its base lying on the diameter.
  1. Show that the area of the rectangle can be written as \(A(x)=2x\sqrt{400-x^2}\), where \(x\) is half the base.
  2. Find the value of \(x\) that maximizes \(A(x)\), and determine the maximum area of the rectangle (verification with the second derivative test is optional).

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