Maximum Area Field with a Corner Wall
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A farmer wants to construct a rectangular field with largest area using  feet of fencing and an existing SW corner wall that extends  feet east and  feet north. How should he construct the field if he can attach the ends of the fence anywhere he wants to the corner wall? When  , this is the problem of maximizing a rectangle with a given perimeter. When  and  or  and  , it is equivalent to the problem of maximizing the area of a field when one side is a river. If  and  are the lengths of the sides of the field,  and there are four cases: These constraints define  piecewise for  . The maximum value of  can occur where  or is discontinuous. The green rectangle has the largest area for given  ,  , and  . Note that it can be a square, twice as long as wide, or something in between. In the in-between cases, one side either has length  or length  . In these cases, the maximum occurs where  is discontinuous. Check "set  to make  largest" to study the shape of the largest rectangle as  ,  , or  are varied. The various cases (and maximum shapes) are: When  , (e.g.,  ,  ),  (square),  (side  ),  (twice as long as wide),  (side  ), and  (square). When  , (e.g.,  ,  ):  (square),  (side  ),  (side  ), and  (square). When  , (e.g.,  ,  ):  (square),  (side  ),  (twice as long as wide),  (side  ), and  (square). When  , (e.g.,  ,  ):  (square),  (side  ),  (side  ), and  (square). In the cases  , (e.g.,  ,  ) or  (e.g.,  ,  ), there is just one twice as long as wide maximum, when  , and  is discontinuous when  is a maximum. |
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 This problem is a generalization of a problem of J. L. Walsh, former Professor of Mathematics at Harvard: A farmer wishes to enclose a rectangular field of largest area. He has already erected a straight fence of length 100 feet, and has at his disposal 200 additional feet of fencing. What is the largest rectangular field he can fence in, by making use of all or part of the fence already standing? J. L. Walsh, A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, Heath, 1962.
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