(*^ ::[ 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, "Times"; ; 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; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] BOXMAKE :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We seek to determine first the maximum volume box cut from a fixed planar region and then the minimum cost box cut from comparable region. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect] FileName: BOXMAKE :[font = subsection; inactive; preserveAspect] Full title: Determine the process for manufacturing the most profitable enclosed box given associated costs and revenue parameters. :[font = subsection; inactive; preserveAspect] Last Revision Date: 18 November 1996. :[font = subsection; inactive; preserveAspect] 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. :[font = subsection; inactive; preserveAspect] Contacts: 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. :[font = subsection; inactive; preserveAspect; endGroup] 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 consider the manufacturing process in which we need to make an enclosed, rectangular box from a flat (8.5" by 11" rectangle) sheet of metal. The box will have a rectangular bottom :[font = subsection; inactive; preserveAspect; startGroup] Teacher's Note: This description gives the variable formulation and may be omitted in favor of a diagram to be labeled. :[font = subsubsection; inactive; preserveAspect; endGroup] The basic approach will be to make a cross shaped flat piece consisting of a y" (vertical) by z" (horizontal) rectangle at the center with x"(horizontal) by y" (vertical) rectangles on both sides and x" by z" rectangles on top and bottom of the center rectangle. Finally a y" by z" rectangle will be centrally located at the bottom of the bottom x" by y" rectangle and will serve as a cover for the enclosed box. :[font = subsection; inactive; preserveAspect] (1) Find the dimensions of the box which will make the volume of the enclosed, rectangular box maximum. :[font = subsection; inactive; preserveAspect; startGroup] (2) Economic considerations in manufacturing process. :[font = subsubsection; inactive; preserveAspect] We have an incentive in forming the box to make a large contained volume but we also have costs associated with this manufacturing process. :[font = subsubsection; inactive; preserveAspect] We can charge $0.10/in^3 for the box. :[font = subsubsection; inactive; preserveAspect] The costs of materials is $0.015/square inch when new and any scrap left over from the process will be generating $0.004/square inch. :[font = subsubsection; inactive; preserveAspect] We note that on the box we need to file and glue the eight vertical edges at a cost of $0.01/linear inch, file all exposed edges (the three edges of the open top and the open flap itself) at a cost of $0.02/linear inch, and insert hinges in the open flap edge. The latter is a fixed cost of $0.40 no matter what geometry we use. :[font = subsubsection; inactive; preserveAspect; endGroup] Now, find the dimensions of the enclosed box which will make profit maximum. Recall Profit = Revenue - Cost. :[font = subsection; inactive; preserveAspect; endGroup] (3) Compare the outcomes of (1) and (2). :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Optimization - volume and revenue, geometry. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Formation of function from description, optimization of a function of one variable. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This problem is a somewhat standard optimization problem and with a diagram the students can formulate the geometric and economic optimization function in 10 minutes, the latter being a bit more difficult as they have to assign the various costs to specific pieces. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect] We would expect the student to formulate an objective function - in the first problem one of volume and in the second problem one of profit = revenue - cost. :[font = subsubsection; inactive; preserveAspect; endGroup] Further we would expect the student to be able to optimize such objective functions and compare the results. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] This problem demands careful reading, extraction of a drawing from a description if no drawing is provided, and a step-by-step accounting for all costs involved. Such care in accounting for terms in the objective function will be useful throughout future modeling activities. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect] This IS an extension of the standard maximum volume box out of a rectangle - only this time we require a top and are concerned with economics. :[font = subsubsection; inactive; preserveAspect] One could require double thickness of the bottom, using a piece from the unused material and costing for the glueing of the two pieces. :[font = subsubsection; inactive; preserveAspect; endGroup] One could discuss marginal increases in say total cost, for marginal increases in any component's cost. :[font = subsection; inactive; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] We appreciate the suggestions and corrections offered by Paul H. Bouknecht who teaches AP Calculus at Eau Gallie High School in Melbourne FL. :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (1) We label the edges of the box. :[font = subsubsection; inactive; preserveAspect; startGroup] We write down the constraints of the sheet :[font = input; preserveAspect; startGroup] ht = 2 x + z == 8.5 :[font = output; output; inactive; preserveAspect; endGroup] 2*x + z == 8.5 ;[o] 2 x + z == 8.5 :[font = input; preserveAspect; startGroup] len = 2 x + 2 y == 11 :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2*x + 2*y == 11 ;[o] 2 x + 2 y == 11 :[font = subsubsection; inactive; preserveAspect; startGroup] And we determine the sides y and z as functions of x. :[font = input; preserveAspect; startGroup] yxlen = y/.Solve[len,y][[1]] :[font = output; output; inactive; preserveAspect; endGroup] (11 - 2*x)/2 ;[o] 11 - 2 x -------- 2 :[font = input; preserveAspect; startGroup] zxht = z/.Solve[ht,z][[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 8.5 - 2.*x ;[o] 8.5 - 2. x :[font = subsubsection; inactive; preserveAspect; startGroup] Thus we have the volume as x y z, but since y and z are functions of x we have the volume as a function of x alone, vol(x). :[font = input; preserveAspect; startGroup] vol[x_] = x yxlen zxht :[font = output; output; inactive; preserveAspect; endGroup; endGroup] ((11 - 2*x)*(8.5 - 2.*x)*x)/2 ;[o] (11 - 2 x) (8.5 - 2. x) x ------------------------- 2 :[font = subsubsection; inactive; preserveAspect; startGroup] We plot vol(x) to see if there is a maximum volume. It appears to be around x = 1.5 in. :[font = input; preserveAspect; startGroup] Plot[vol[x],{x,0, 8.5/2}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We determine the dimensions x = 1.59 in, y = yxlen = 3.91 in, and z = zxht = 5.33 in. Also we determine the maximum volume vol = 33.07 in^3 when vol'(x) = 0. :[font = input; preserveAspect; startGroup] maxvol = {x, yxlen,zxht,vol[x]}/. Solve[vol'[x]==0,x][[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] {1.585417970380151, 3.914582029619849, 5.329164059239698, 33.07411749495332} ;[o] {1.58542, 3.91458, 5.32916, 33.0741} :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (2) We consider the problem of maximizing profit. We recall the data. :[font = subsubsection; inactive; preserveAspect] We can charge $0.10/in^3 for the box. :[font = subsubsection; inactive; preserveAspect] The costs of materials is $0.015/square inch when new and any scrap left over from the process will be generating $0.004/square inch. :[font = subsubsection; inactive; preserveAspect] We note that we need to file and glue the eight vertical edges (x) at a cost of $0.01/linear inch, file all exposed edges (the three edges (z + 2 x) of the open top and the three edges of the open flap itself (z + 2x)) at a cost of $0.02/linear inch, and insert small hinges in the open flap edge which is connected to the box top. The latter is a fixed cost of $0.40 no matter what geometry we use. :[font = subsubsection; inactive; preserveAspect; startGroup] We represent the cost using the x, y, and z variables - solved for x. :[font = input; preserveAspect; startGroup] BoxCost[x_] = .01 8 x + .02 (2 z + 4 y) + .015 8.5 11 + .4/.{y->yxlen,z->zxht} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1.8025 + 0.02*(2*(11 - 2*x) + 2*(8.5 - 2.*x)) + 0.08*x ;[o] 1.8025 + 0.02 (2 (11 - 2 x) + 2 (8.5 - 2. x)) + 0.08 x :[font = subsubsection; inactive; preserveAspect; startGroup] And finally we compute the revenue for our manufacturing process. :[font = input; preserveAspect; startGroup] BoxRevenue[x_] = Expand[.10 vol[x] + y .004 (8.5 11 - (4 x^2 + 2 x y))- BoxCost[x]/.{y->yxlen,z->zxht}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] -0.5255000000000003 + 4.139000000000002*x - 1.95*x^2 + 0.208*x^3 ;[o] 2 3 -0.5255 + 4.139 x - 1.95 x + 0.208 x :[font = subsubsection; inactive; preserveAspect; startGroup] We plot the revenue as a function of variable x. We note the possibilities of negative revenue in our feasible region 0 <= x <= 8.5) - we never got negative volume in this region. :[font = input; preserveAspect; startGroup] Plot[BoxRevenue[x],{x,0, 8.5/2}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We determine the dimensions x = 1.35 in, y = yxlen = 4.14 in, and z = zxht = 5.78 in. Also we determine the maximum volume vol = 32.60 in^3 when vol'(x) = 0. Our maximum revenue is $2.02 per box. :[font = input; preserveAspect; startGroup] max = {x, yxlen, zxht,vol[x], BoxRevenue[x]}/.Solve[BoxRevenue'[x]==0,x][[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {1.355081307097081, 4.144918692902919, 5.789837385805838, 32.51979029925658, 2.020061543301322} ;[o] {1.35508, 4.14492, 5.78984, 32.5198, 2.02006} :[font = subsubsection; inactive; preserveAspect; startGroup] In comparison with our simpler volume only Problem (1) we note that for maximum profit x decreases by .23 in, y increases by.2 in, and z increases by .4 in while the volume decreases by 0.56 in^3. We repeat the answers from the Problem (1) below. :[font = input; preserveAspect; startGroup] max = {x, xlen,xht,vol[x]}/.Solve[vol'[x]==0,x][[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {1.585417970380151, 3.914582029619849, 5.329164059239698, 33.07411749495332} ;[o] {1.58542, 3.91458, 5.32916, 33.0741} :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] We note that the volume for our most economical shape has decreased and is not the maximum volume possible. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] We got mixed up in our own problem formulation as was pointed out to us by Paul H. Bouknecht of Melbourne FL. Thus a good diagram and careful accounting of costs is essential. ^*)