A farmer has an  by  foot corner wall and  feet of fence. He wants to use the fence to construct an  by  foot rectangular field using the corner wall for one corner, and part or all of its sides. What should  and  be to maximize the area  of the field? For the demonstration,  ,  ,  , and  . There are four cases relating  ,  ,  ,  and  , depending on whether  ,  and  ,  and  , or  . Vary  to see the possible field boundaries. The largest field is shown in green, and as a "double square" when it is a 2 by 1 or 1 by 2 rectangle. Study its shape as  (or  or  ) varies, in the cases  ,  and  . It can be a square, a 1x2 or 2x1 rectangle, or an "intermediate" rectangle. The possible shapes and a formula for the dimensions of the largest field are derived in the Details.
Maximizing the area of a rectangle with given perimeter, and maximizing the area of a rectangular field bordering a river with a given amount of fence, are special cases (  , and  ,  ). In these problems,  is maximum when the rectangle is a square, and a 2 by 1 rectangle, respectively. The case  =0 and  is a problem of both V. L. Klee, Jr., and J. L. Walsh. In this problem, the max is when  , where  does not exist. The "Corner Wall Problem" is interesting because there can be one, two, or three critical points, and the largest rectangle can be described geometrically. In general,  is a continuous, piecewise linear function of  , for  .  is zero at the endpoints and positive in between. A positive maximum exists. The maximum occurs where  or where  does not exist, i.e. where  or  . There can be up to three critical points. The maximum can be found by 1) determining the domain for  for each case, 2) determining whether  at a point interior to the domain for some case, 3) computing and comparing the values of  where  and at the endpoints for each case. The maximum is the largest of these values. This method is straightforward, given particular values of  ,  , and  , and is the method used in the Demonstration to find the maximum and where it occurs. But, it is not easy to carry this method out for arbitrary values of  ,  , and  . A formula for the dimensions of the maximum rectangle in terms of  ,  , and  can be determined as follows. Note the graph of  is concave downward because  when it exists. Thus there is a unique maximum. The maximum occurs either where  , or where  and  , when  does not exist. That is because, at these points,  is increasing from the left and decreasing to the right. Since  ,  . In the four cases,  ,  ,  , and  , and these formulas give the left and right derivatives when  or  . Suppose  . In cases 1 and 4, we see  is increased by making the rectangle more square. In cases 2 and 3, A is increased by making it more 2×1 or more 1×2. This tells us the largest rectangle is either a square, a 1×2 or 2×1 rectangle, or an "intermediate" rectangle with side  or side  . We first consider the non-boundary cases,  and  .  is maximum when  . Case 1:  and  :  . In this case  . A is max when  . Thus,  and  . Case 2:  and  :  . In this case,  .  is max when  . Thus,  and  . Case 3:  and  :  . In this case,  .  is max when  and  . Case 4:  and  :  . In this case,  . A is max when  . Thus,  and  . Now the boundary cases, where  or  . A is maximum when  and  . Case 12:  and  :  and  . In this case,  , and  is max when  and  . Case 34:  and  :  and  . In this case,  , and  is max when  and  . Case 13:  and  :  and  . In this case  , and  is max when  and  . Case 24:  and  :  and  . In this case  , and  is max when  and  . Case 1234:  and  :  and  . In this case,  , and  is max when  and  . Solving the inequalities in each of these cases, we determine when and where the maximum can occur. The inequalities were solved graphically. An example is given for each case.  is a maximum, with  , when (1)  , where  . E.g.,  ,  ,  . (2)  and  , where  and  . E.g.,  ,  ,  . (3)  and  , where  and  . E.g.,  ,  ,  . (4)  , where  . E.g.,  ,  ,  .  is a maximum, with  or  , when (12)  and  or  and  , where  and  . E.g.,  ,  ,  , and  ,  ,  . (34)  and  , or  and  , where  and  . E.g.,  ,  ,  , and  ,  and  . (13)  and  , or  and  , where  and  . E.g.,  ,  ,  , or  ,  ,  . (24)  and  , or  and  , where  and  . E.g.,  ,  ,  , or  ,  ,  . (1234)  and  , where  and  . E.g.,  ,  ,  . SUMMARY. We express these results as inequalities for  , given  and  : When  , the maximum rectangle is square with  if  , has  ,  if  , is 1x2 with  if  , has  ,  if  , and is square with  if  . E. g.,  ,  . The maximum rectangle is square if  , has  if  , is 1x2 if  , has  if  , and is square if  . When  , the maximum rectangle is square with  if  , has  ,  if  , has  ,  if  , and is square with  if  . E. g.,  ,  . The maximum rectangle is square if  , has  if  , has  if  , and is square if  . When  , the maximum rectangle is square with  if  , has  ,  if  , is 2x1 with  if  , has  ,  if  , and is square with  if  . E. g.,  ,  . The maximum rectangle is square if  , has  if  , is 2x1 if  , has  if  , and is square if  . When  , the maximum rectangle is square with  if  , has  ,  if  , has  ,  if  , and is square with  if  . E. g.,  ,  . The maximum rectangle is square if  , has  if  , has  if  , and is square if  . When  , and the wall a staight wall, the maximum rectangle is 2x1 with  ,  if  , has  ,  if  , and is a square with  if  . Thus, for the Walsh/Klee problem, where  and  , the maximum rectangle has  and  . This Demonstration can check, and also suggest these results. When the (green) maximum area rectangle is 1×2 or 2×1, it is drawn as two squares, to emphasize this fact. All possibilites can be examined by choosing  and  so  and  , and varying  . For reference, a function maxcornerformula[a,b,L], which implements the summary formula, is included in the Initializiation Code section. Changing its name to maxcorner will cause it to replace the function which finds the maximum rectangle by comparing values of A at critical points. 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. Also, C. O. Oakley credits the same problem to V. L. Klee, Jr., but when the fance makes use of all of the existing fence. C. O. Oakley, "End-Point Maxima and Minima, The American Mathematical Monthly, Vol. 54, No. 7, Part 1 (Aug. - Sep., 1947), pp. 407-409. Pierre Malraison studied the general case of the "Wall Problem" in a film and slides produced at a 1974 NSF Workshop at Carleton College, Computer Graphics for Learning Mathematics.
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