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A farmer fences off a rectangular field with a total of \(4000 \,\mathrm{m}\) of fencing. Since the field is located along a river, the farmer only needs to fence three of the four sides.
  1. Let \(x\) be the length of each side perpendicular to the river, and \(y\) the side parallel to the river. Write an expression for the area \(A\) of the field in terms of \(x\).
  2. Determine the dimensions of the field which maximize the area (verification with the second derivative test is optional).

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