Tin Box with Maximum Volume
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PROBLEM: A piece of sheet tin three feet square is to be made into a rectangular box open at the top, by cutting out equal squares from the corners and bending up the sides of the resulting piece parallel with the edges. Among all such boxes, to find the box of greatest volume. This is the problem J. L. Walsh used in his 1947 Classroom Note in The American Mathematical Monthly to illustrate his rigorous analysis of maximum-minimum problems. A version of the problem appears in many calculus books. Let the tin sheet have dimensions  , with  , and suppose a square with side  is cut from each corner. The volume of the resulting box is  ,  . Since  is continuous on  , it has a maximum value there. Let the maximum occur when  . Since  and  for  ,  , the maximum cannot occur where  or  . If  , then  for  and  near  . And if  , then  for  and  near  . Therefore,  . When  ,  . The box of greatest volume is obtained by cutting six-inch squares from each corner. Which tin sheets have integer dimensions and  rational? It is easier to make the biggest box for such sheets, and the resulting calculus problems are easier to solve. This Demonstration allows you to choose a triple  } and set  and  for a multiplier  , for relatively prime  and  with  and rational optimal cut  . These are the triples:  |
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 Former Harvard Professor Walsh expanded on his classroom note in a short pamphlet, concerned with finding global maximum and minimum values. He argued that the second derivative is unnecessary in searching for global extrema. These principles are sufficient, for continuous piecewise differentiable functions: Extreme Value theorem: If  is continuous on a closed bounded interval  , it has a (global) maximum and a (global) minimum there. Increasing (Decreasing) at a Point theorem: If   ), there is a number  such that if  , then  (  ) and if  , then   ). Critical Point Theorem: If  has a maximum (minimum) on a bounded or unbounded open interval  at  , where  , then either  or  does not exist. Increasing (Decreasing) on an Interval theorem: If  (  ) for  in a bounded or unbounded open interval  , then  is strictly increasing (decreasing) on the interval. J. L. Walsh, "A Rigorous Treatment of the First Maximum Problem in the Calculus", The American Mathematical Monthly, 54(1), 1947 pp. 35–36. J. L. Walsh, A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, Boston: Heath, 1962. 
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