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A rectangular plot is enclosed by 100 m of fencing. Let the length of one side of the rectangle be \(x\) metres.
  1. Show that the area, \(A \text{ m}^2\), of the rectangle is given by the function \(A(x) = 50x - x^2\).
  2. Find the dimensions of the rectangle that maximize its area, and show that this occurs when the rectangle is a square.

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