(*^ ::[ 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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The problem is posed in terms of equity of crust distribution for French bread. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: BREADCUT Full title: How to cut a hemispherical loaf of French bread so that each of 8 guests gets the same amount of crust. Last Revision Date: 27 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] Consider a hemispherical loaf of crusty French bread of radius 8". Alice and Bill just love the crust of the bread and are having 6 friends over for dinner. They wish to slice the bread so that each of the 8 dinner party members will get an equal amount of crust. :[font = subsection; inactive; preserveAspect] Alice suggests that they just cut the bread vertically (parallel to a plane going through the center) into pieces each one-eighth of the diameter wide. She claims that in this way each person will have the same amount of crust. :[font = subsection; inactive; preserveAspect] Bill says, "Just because the width of each piece is the same does not mean the amount of crust on each piece is the same." :[font = subsection; inactive; preserveAspect; endGroup] Formulate this problem mathematically and settle the argument. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Surface area of rotation. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Knowledge of arc length formula for function of one variable and surface area of rotation. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect] 15 minutes in one class, more if you permit discussion to wander or if you introduce physical objects and permit a great deal of conjecturing. :[font = subsubsection; inactive; preserveAspect] Students will debate the reasonableness of Alice's conjecture and be pressed to conjecture another approach, e.g., cutting wider sections at end or in the middle. :[font = subsubsection; inactive; preserveAspect; endGroup] Even more time can be used if you pose the problem as open ended without positing the cutting scheme offered above. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] We expect the students to argue about this in class and to really find the mathematical analysis as a proof of one side or the other. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Students will see :[font = subsubsection; inactive; preserveAspect; endGroup] Students will see how analysis confirms conjecture and settles debate. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students will improve their communication skills as they are forced to argue their point of view based on a mathematical model. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] One could request other subdivisions, e.g., in proportion 1:2:3:4:5:6:7:8. Or one could examine other shapes, e.g. ellipsoid to see how complicated the problem might really be if the geometry is different. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] We consider the problem of a hemisphere of radius 8" as a half of the shape of a rotated circle, x^2 + y^2 = 8^2, of radius 8" about the x axis. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] To obtain this we shall take one half of the volume obtained from rotating the top half of x^2 + y^2 = 8^2 about the x-axis. :[font = input; preserveAspect; endGroup] f[x_] = Sqrt[64 - x^2]; :[font = subsection; inactive; preserveAspect] We obtain the surface area of the slice of width a going from x value h to x value h + a. :[font = subsection; inactive; preserveAspect; startGroup] We do this by rotating an element of arc length Sqrt[1 + f'[x]^2] through a radius f[x] and taking one-half of this amount. :[font = input; Cclosed; preserveAspect; startGroup] va[h_] = 1/2 Integrate[2 Pi f[x] Sqrt[1 + f'[x]^2], {x,h,h+a}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (-16*h*((64 - h^2)^(-1))^(1/2)*(64 - h^2)^(1/2)*Pi + 2*(a + h)*(64 - (a + h)^2)^(1/2)* (-64/(-64 + (a + h)^2))^(1/2)*Pi)/2 ;[o] 1 2 (-16 h Sqrt[-------] Sqrt[64 - h ] Pi + 2 64 - h 2 2 (a + h) Sqrt[64 - (a + h) ] -64 Sqrt[--------------] Pi) / 2 2 -64 + (a + h) :[font = subsection; inactive; preserveAspect] In our case we are suggesting that we cut the total width of the 16" diameter hemisphere of bread into 8 equal width pieces of length a = 2". :[font = subsection; inactive; preserveAspect; startGroup] Thus we compute the surface area from h to h + 2. Here h is in the domain h = -8" to h = 6". :[font = input; Cclosed; preserveAspect; startGroup] party[h_] = va[h]/.{a->2} :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (-16*h*((64 - h^2)^(-1))^(1/2)*(64 - h^2)^(1/2)*Pi + 2*(2 + h)*(64 - (2 + h)^2)^(1/2)* (-64/(-64 + (2 + h)^2))^(1/2)*Pi)/2 ;[o] 1 2 (-16 h Sqrt[-------] Sqrt[64 - h ] Pi + 2 64 - h 2 2 (2 + h) Sqrt[64 - (2 + h) ] -64 Sqrt[--------------] Pi) / 2 2 -64 + (2 + h) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] We simplify this by noting the square root terms cancel appropriately. :[font = input; Cclosed; preserveAspect; startGroup] p[h_] = Expand[(- 16 h Pi + 2 (2 + h) 8 Pi)/2] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 16*Pi ;[o] 16 Pi :[font = subsection; inactive; preserveAspect; startGroup] And we see that no matter where we start cutting (h in the domain h = -8 to h = 6) when we cut a width of bread 2" long the surface area is always 16 Pi ~ 50.2655 in^2. The plot confirms this. :[font = input; Cclosed; preserveAspect; startGroup] 16 Pi//N :[font = output; output; inactive; preserveAspect; endGroup] 50.26548245743669 ;[o] 50.2655 :[font = input; preserveAspect; startGroup] Plot[p[h],{h,-8,6},PlotRange->{0,60}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect] This is an immediate application of the principle of rotating an element of arc length about an axis (half way, not a full rotation) to determine the surface area. :[font = subsection; inactive; preserveAspect] The shape rotated is the top half of a circle of radius 8" and we are taking horizontal slices of this surface area, by integrating over regular intervals. :[font = subsection; inactive; preserveAspect; endGroup; endGroup] When students confront the intuitive responses they have concerning their opinions they will readily go to analysis to "sovle the question once and for all." ^*)