(*^ ::[ 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, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 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; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; noKeepOnOnePage; preserveAspect; startGroup] VCRTAPE :[font = subtitle; inactive; preserveAspect] Tape thickness and speed. :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] It's easy to estimate how thick a piece of paper is by measuring a ream of 500 sheets. This (fairly easy) problem deals with estimating the tape thickness on a video cassette, and then asks for a function which represents the tape driver speed. The problem is accessible to precalculus students, but accuracy and modeling issues also make it an appropriate application of polar integration! :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] FileName: VCRTAPE :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Full title: Tape thickness and speed. :[font = subsection; inactive; preserveAspect] Last Update: 5/31/96 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Developer: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Klebanoff@rose-hulman.edu. FAX: 812-877-3198. :[font = subsection; inactive; noKeepOnOnePage; 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; noKeepOnOnePage; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] The following specifications were taken for a certain model of video cassette recorder. Tape Speed: 1.3125 inches per second Play Time: 2 hours Radius of tape spindle: 12 mm Outer radius of wound tape: 43 mm :[font = subsection; inactive; preserveAspect] 1) How thick is the tape? :[font = subsection; inactive; preserveAspect; endGroup] 2) Derive a function that gives the tape spindle driver speed (in rotations per minute -- rpm) as a function of the radius. Note that the tape speed is constant. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] Angular speed, circumference, arc length in polar coordinates. :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Algebra, circumference of a circle; (polar coordinates in calculus provides an optional approach.) :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Time allotment - time management :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Some students may need hints to Problem 2 such as: (1) what's the spindle speed when the radius is X? or Y? :[font = subsection; inactive; noKeepOnOnePage; preserveAspect] Future payoffs :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] A much harder problem is to determine the tape driver speed as a function of time. Offer as a challenge problem. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect; startGroup] 1) How thick is the tape? :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We first convert all unit lengths to inches and unit times to seconds. :[font = input; preserveAspect; startGroup] TapeSpeed = 1.3125 :[font = output; output; inactive; preserveAspect; endGroup] 1.3125 ;[o] 1.3125 :[font = input; preserveAspect; startGroup] PlayTime = 2 3600 :[font = output; output; inactive; preserveAspect; endGroup] 7200 ;[o] 7200 :[font = input; preserveAspect; startGroup] RadSpind = 12/25.4 :[font = output; output; inactive; preserveAspect; endGroup] 0.4724409448818898 ;[o] 0.472441 :[font = input; preserveAspect; startGroup] RadWound = 43/25.4 :[font = output; output; inactive; preserveAspect; endGroup] 1.692913385826772 ;[o] 1.69291 :[font = input; preserveAspect; startGroup] TapeLength = TapeSpeed PlayTime :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 9450. ;[o] 9450. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Approach 1: Model with cylindrical shells. :[font = text; inactive; preserveAspect] The number n, :[font = input; preserveAspect; startGroup] n = (RadWound - RadSpind)/Thick :[font = output; output; inactive; preserveAspect; endGroup] 1.220472440944882/Thick ;[o] 1.22047 ------- Thick :[font = text; inactive; preserveAspect] is the approximate number of shells required. We add up the circumference of each shell to solve for the thickness. Note that the first shell has circumference 2 p (RadSpind + Thick) and each shell around that has circumference given by 2 p (RadSpind + i*Thick) where i goes from 1 to n. Using the fact that Sum[i, {i, 1, n}] = n(n+1)/2, we can simplify the sum of circumferences Sum[2 p (RadSpind + i*Thick), {i, 1, n}] as 2 p (RadSpind n + Thick n(n+1)/2) = 2 p n (RadSpind + (n + 1) Thick/2) ;[s] 11:0,0;163,1;164,2;240,3;241,4;389,5;390,6;430,7;431,8;466,9;467,10;498,-1; 11:1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] thickness = Flatten[ Solve[2 Pi n(RadSpind + (n+1) Thick)/2 == TapeLength, Thick]]; :[font = input; preserveAspect; startGroup] TapeThickness1 = Thick /. thickness :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0.0006871586630389147 ;[o] 0.000687159 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Approach 2: Model as a spiral -- use polar coordinates. :[font = text; inactive; preserveAspect] As theta: 0 -> 2 p, we want the spiral to increase in radius by Thick. We also require that when theta = 0, that the radius is RadSpind. Finally, we want the length to be TapeLength when the radius is RadWound. ;[s] 3:0,0;17,1;18,2;212,-1; 3:1,11,8,Times,0,12,0,0,0;1,0,0,Symbol,0,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect] We know the total arc length. So, our goal is to determine the arc length of a spiral as a function of the spacing until it becomes the correct length. :[font = input; preserveAspect; startGroup] r[theta_] = RadSpind + theta Thick/(2 Pi) :[font = output; output; inactive; preserveAspect; endGroup] 0.4724409448818898 + (theta*Thick)/(2*Pi) ;[o] theta Thick 0.472441 + ----------- 2 Pi :[font = input; preserveAspect; startGroup] rprime = r'[theta] :[font = output; output; inactive; preserveAspect; endGroup] Thick/(2*Pi) ;[o] Thick ----- 2 Pi :[font = input; preserveAspect] PolarArcLength = Integrate[Sqrt[r[theta]^2 + rprime^2], {theta, 0, 2 Pi n}]; :[font = input; preserveAspect] thickness = FindRoot[PolarArcLength == TapeLength, {Thick, 0.0006}]; :[font = input; preserveAspect; startGroup] TapeThickness2 = Thick /. thickness :[font = output; output; inactive; preserveAspect; endGroup] 0.000878567266517569 ;[o] 0.000878567 :[font = input; preserveAspect; startGroup] m = n /. Thick -> TapeThickness2; PolarPlot[RadSpind + theta TapeThickness2/(2 Pi), {theta, 0, m 2 Pi}, PlotPoints -> 400] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] 2) Derive a function that gives the tape spindle driver speed (in rotations per minute -- rpm) as a function of the radius. :[font = text; inactive; preserveAspect] We require the tape speed to remain constant, so the spindle speed will have to vary with the radius of the tape: :[font = input; preserveAspect; startGroup] SpindleSpeed[radius_] := TapeSpeed/(2 Pi radius) Plot[SpindleSpeed[radius], {radius, RadSpind, RadWound}, AxesLabel -> {"radius of wound tape [in]", "rpm"}, PlotRange -> {{0, 2}, {0, 1}}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] Neither of the tape thickness models offered necessarily yields the true tap thickness. However, they both give good approximations. Also, giving two solutions provides a nice way to check our work. ^*)