(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, L0, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, L0, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] BOXMAX :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We consider an extension of the problem of constructing a box by cutting four equal squares from a rectangle and folding to one in which the cutouts can be more general quadrilateral introducing another variable. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: BOXMAX Full title: Maximize volume of a box made from cutting corners off flat sheet and folding along resulting edges. Last Revision Date: 31 May 1996. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@Rose-Hulman.Edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] We are given an 8 cm by 10 cm rectangular sheet of material and told to construct an open-topped box of maximum volume. :[font = subsection; inactive; preserveAspect] The simple case is one in which we make square cuts from each corner and fold to make a rectangular box. :[font = subsection; inactive; preserveAspect] Consider the more general case, in which we removecongruent quadrilaterals (wedges) from each corner of the sheet and fold the resulting figure so that the cut edges meet. :[font = subsection; inactive; preserveAspect] How does the maximum volume of the more general box compare with that of the rectangular box? :[font = subsection; inactive; preserveAspect] Hint: Place the sheet in the first quadrant with one vertex at the origin. Draw a line from the origin, at an angle of Pi/4 (toward the point (1, 1)) of length L. Call the endpoint of this line (a, a). Now we will make our cuts from (b, 0) and (0, b) to (a, a), removing a quadrilateral from the corner. We remove corresponding quadrilaterals from the other corners. :[font = subsection; inactive; preserveAspect; endGroup] Note that if b>"] ;[o] -ContourGraphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Thus we go after a maximum near a = 2 and y = b. We set the partial derivatives with respect to a and b equal to zero. :[font = input; preserveAspect] va[a_,b_] = D[v[a,b],a]; vb[a_,b_] = D[v[a,b],b]; :[font = input; preserveAspect; startGroup] solc = FindRoot[{va[a,b]==0,vb[a,b]==0},{a,2},{b,1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {a -> 2.047887004642715, b -> 1.085813528596048} ;[o] {a -> 2.04789, b -> 1.08581} :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Thus, ae, the square cut edge is 2.04789 cm. :[font = input; preserveAspect; startGroup] ae = a/.solc[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2.047887004642715 ;[o] 2.04789 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] And, be, the back cut is 1.08581 cm. :[font = input; preserveAspect; startGroup] be = b/.solc[[2]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1.085813528596048 ;[o] 1.08581 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] The box then has a base of dimensions :[font = input; preserveAspect; startGroup] baseLength = 10 - 2 ae :[font = output; output; inactive; preserveAspect; endGroup] 5.90422599071457 ;[o] 5.90423 :[font = input; preserveAspect; startGroup] baseWidth = 8 - 2 ae :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 3.904225990714569 ;[o] 3.90423 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] and a top of dimensions :[font = input; preserveAspect; startGroup] topLength = baseLength + 2 be :[font = output; output; inactive; preserveAspect; endGroup] 8.07585304790667 ;[o] 8.07585 :[font = input; preserveAspect; startGroup] topWidth = baseWidth + 2 be :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 6.075853047906666 ;[o] 6.07585 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] And hence the maximum volume is 61.2466 cm^3 as seen below. :[font = input; preserveAspect; startGroup] v[ae,be] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 61.24664284470312 ;[o] 61.2466 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We now draw a picture of the figure - presuming we lay the flat sheet on the x-y plane with one vertex at the origin and the 8 cm side along the x- and the 10 cm side along the y-axis. Pi' are the points in the flat sheet which we cut into and form the bottom vertices of the box and the Qi's are the resulting top vertices. :[font = input; preserveAspect] P1 = {ae,ae,0}; P2 = {ae,10-ae,0}; P3 = {8-ae,10-ae,0}; P4 = {8-ae,ae,0}; Q1 = {ae-be,ae-be,Sqrt[ae^2 - be^2]}; Q2 = {ae-be,10-ae+be,Sqrt[ae^2 - be^2]}; Q3 = {8-ae+be,10-ae+be,Sqrt[ae^2 - be^2]}; Q4 = {8-ae+be,ae-be,Sqrt[ae^2 - be^2]}; :[font = input; preserveAspect; startGroup] picc = Show[Graphics3D[{PointSize[.02], Point[P1], Point[P2], Point[P3], Point[P4],Point[Q1], Point[Q2], Point[Q3], Point[Q4], Thickness[.01], Line[{P1,P2}], Line[{P2,P3}], Line[{P3,P4}], Line[{P4,P1}], Line[{Q1,Q2}], Line[{Q2,Q3}], Line[{Q3,Q4}], Line[{Q4,Q1}], Line[{P1,Q1}], Line[{P2,Q2}], Line[{P3,Q3}], Line[{P4,Q4}] }],Boxed->False] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = subsection; inactive; preserveAspect] We examine the square cut case when we cut a square piece out of each corner :[font = subsection; inactive; preserveAspect; startGroup] We set up the volume for a square cut of a by a from each of the corners. :[font = input; preserveAspect; startGroup] rv[a_] = a (10 - 2a) (8-2a) :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (8 - 2*a)*(10 - 2*a)*a ;[o] (8 - 2 a) (10 - 2 a) a :[font = subsection; inactive; preserveAspect; startGroup] And maximize this volume. :[font = input; preserveAspect; startGroup] sols = Solve[rv'[a]==0,a]//N :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{a -> 1.472474768348053}, {a -> 4.527525231651947}} ;[o] {{a -> 1.47247}, {a -> 4.52753}} :[font = subsection; inactive; preserveAspect; startGroup] Thus we should cut in an edge of length 1.47247. :[font = input; preserveAspect; startGroup] ae = a/.sols[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1.472474768348053 ;[o] 1.47247 :[font = subsection; inactive; preserveAspect; startGroup] In this case we do not make a back cut. :[font = input; preserveAspect; endGroup] be = 0; :[font = subsection; inactive; preserveAspect; startGroup] And hence the maximum volume is 52.5138 cm^3 as seen below. :[font = input; preserveAspect; startGroup] rv[ae] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 52.51380432416969 ;[o] 52.5138 :[font = subsection; inactive; preserveAspect] A comparision between the back cut method and the square cut method shows that the former gives greater volume. :[font = subsection; inactive; preserveAspect] We note that we indeed get a greater volume by permitting the back cut for the volume in that case is 61.2466 cm^3. :[font = subsection; inactive; preserveAspect; startGroup] We now draw a picture of the figure resulting from the simple case - presuming we lay the flat sheet on the x-y plane with one vertex at the origin and the 8 cm side along the x- and the 10 cm side along the y-axis. Pi' are the points in the flat sheet which we cut into and form the bottom vertices of the box and the Qi's are the resulting top vertices. :[font = input; preserveAspect] P1 = {ae,ae,0}; P2 = {ae,10-ae,0}; P3 = {8-ae,10-ae,0}; P4 = {8-ae,ae,0}; Q1 = {ae-be,ae-be,Sqrt[ae^2 - be^2]}; Q2 = {ae-be,10-ae+be,Sqrt[ae^2 - be^2]}; Q3 = {8-ae+be,10-ae+be,Sqrt[ae^2 - be^2]}; Q4 = {8-ae+be,ae-be,Sqrt[ae^2 - be^2]}; :[font = input; preserveAspect; startGroup] pics = Show[Graphics3D[{PointSize[.02], Point[P1], Point[P2], Point[P3], Point[P4],Point[Q1], Point[Q2], Point[Q3], Point[Q4], Thickness[.01], Line[{P1,P2}], Line[{P2,P3}], Line[{P3,P4}], Line[{P4,P1}], Line[{Q1,Q2}], Line[{Q2,Q3}], Line[{Q3,Q4}], Line[{Q4,Q1}], Line[{P1,Q1}], Line[{P2,Q2}], Line[{P3,Q3}], Line[{P4,Q4}] }],Boxed->False] :[font = message; output; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "pics" is similar to existing symbol "picc". ;[o] General::spell1: Possible spelling error: new symbol name "pics" is similar to existing symbol "picc". :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = subsection; inactive; preserveAspect; startGroup] And finally we draw both figures together to compare them. :[font = input; preserveAspect; startGroup] Show[picc,pics,Axes->True] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] It is probably wise to make several cut out models for students. This will help them visualize what the cuts mean and the folding really is. Make some exaggerated cases, indeed, some cuts which do not even give rise to solids. Have some of the models ready for destruction, writing on them, and drawing lines and labeling variables on them. Students will tend to assume quantities incorrectly, the height of the formed box being the most common . ^*)