(*^ ::[ 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. 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Developer: Lynn Kiaer, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA 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. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect; endGroup] How many cans of soda can fit in a standard piece of paper? That is, if an 8.5 by 11 inch piece of paper has squares at each corner cut off and the resulting outlined box is folded to create an open-topped box of maximum volume, how many cans of soda can be poured into the box? :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Optimization, estimation, units. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect; endGroup] This is a nice twist on a standard problem that has three interesting features. One, students who have not yet solved the problem generally guess around 1 or 2 cans, when the answer is a little over 3. And two, the students have to really think about units. Finally, the students can actually see (and drink!) the results. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Optimization and familiarity with units of volume. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This can be done in less than 30 minutes in class. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students will offer early estimate and confirm their estimate using an optimization model. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students will be better estimators. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] Do the same with other optimization volume issues. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Suggested by Matthias Kawski, Department of Mathematics, Arizona State University :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTIONS :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Of course, the easy solution to the standard box problem... :[font = subsubsection; inactive; preserveAspect; startGroup] If the height of the box is h, then the volume of the box is :[font = input; preserveAspect; endGroup] BoxVol[h_] = h (8.5 - 2h) (11 - 2 h); :[font = subsubsection; inactive; preserveAspect; startGroup] To find the height (and other dimensions) of the box of maximum volume, we can set the derivative equal to zero :[font = input; preserveAspect; startGroup] sol = Solve[BoxVol'[h]==0,h] :[font = output; output; inactive; preserveAspect; endGroup] {{h -> 1.585417970380151}, {h -> 4.914582029619849}} ;[o] {{h -> 1.58542}, {h -> 4.91458}} :[font = input; preserveAspect] ht = sol[[1,1,2]]; :[font = input; preserveAspect; startGroup] BoxVol[ht] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 66.14823498990665 ;[o] 66.1482 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] But this solution is not enormously useful when we turn to the problem of how much soda will fit in the resulting box. A twelve-ounce can holds 355 ml, or 355 cubic centimeters. An 8.5 by 11 inch piece of paper is approximately 21.5 by 28 cm. :[font = input; preserveAspect] MetricVol[h_] = h (21.5 - 2h) (28 - 2h); :[font = input; preserveAspect; startGroup] sol = Solve[MetricVol'[h]==0,h] :[font = output; output; inactive; preserveAspect; endGroup] {{h -> 4.019653284500981}, {h -> 12.48034671549902}} ;[o] {{h -> 4.01965}, {h -> 12.4803}} :[font = input; preserveAspect] ht = sol[[1,1,2]]; :[font = input; preserveAspect; startGroup] MetricVol[ht] :[font = output; output; inactive; preserveAspect; endGroup] 1080.019638102276 ;[o] 1080.02 :[font = input; preserveAspect; startGroup] MetricVol[ht]/355 :[font = output; output; inactive; preserveAspect; endGroup] 3.042308839724722 ;[o] 3.04231 :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Three cans and then some! :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] It is very useful in this problem to have the students guess how many cans-worth of soda will fit in the box. These estimation skills are often underutilized in the classroom. If the students work with `folder paper' and carefully score the edges to be folded with something that won't actually cut the paper, they can build the box and tape it together. And, if two students support the sides (because otherwise the sides would bow out), three cans of soda can be poured in (and quickly sipped out again using straws). ^*)