(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 46332, 1225]*) (*NotebookOutlinePosition[ 46967, 1247]*) (* CellTagsIndexPosition[ 46923, 1243]*) (*WindowFrame->Normal*) Notebook[{ Cell["Need to assume various quantities are real and positive", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \($Assumptions = {$Assumptions, m > 0, H > 0, y > 0}\)], "Input"], Cell[BoxData[ \({True, m > 0, H > 0, y > 0}\)], "Output"] }, Open ]], Cell["Define our function x(y)", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x = Sqrt[m*y]\)], "Input"], Cell[BoxData[ \(\@\(m\ y\)\)], "Output"] }, Open ]], Cell["Define the integrand for the surface area integral...", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(intgrnd = Simplify[x*Sqrt[1 + D[x, y]^2]]\)], "Input"], Cell[BoxData[ \(1\/2\ \@\(m\ \((m + 4\ y)\)\)\)], "Output"] }, Open ]], Cell["...and do the integral.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(surfarea = Simplify[Integrate[2*Pi*intgrnd, {y, 0, H}]]\)], "Input"], Cell[BoxData[ \(1\/6\ \((\(-m\^2\) + 4\ H\ \@\(m\ \((4\ H + m)\)\) + \@\(m\^3\ \((4\ H + m)\)\))\)\ \ \[Pi]\)], "Output"] }, Open ]], Cell["\<\ Substitute in the value of H (as a function of m), to get the \ bowl's surface area\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bowlarea = Simplify[surfarea /. {H \[Rule] Sqrt[1000/\((m*Pi)\)]}]\)], "Input"], Cell[BoxData[ \(1\/6\ \((\(-m\^2\) + \@\(m\^4 + 40\ m\^\(5/2\)\ \@\(10\/\[Pi]\)\) + 40\ \@\(\(400\ \@10\)\/\(\@m\ \[Pi]\^\(3/2\)\) + \(10\ \ m\)\/\[Pi]\))\)\ \[Pi]\)], "Output"] }, Open ]], Cell["See what the cost is for m=2 (example done in class)", "Text"], Cell[CellGroupData[{ 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{176.938, 0}} -> {3.46275, 248.006, 0.0161578, \ 0.0210598}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Yes, there's definitely an optimal choice of m that gives the given \ volume for the least surface area (and, therefore, least cost). Let's find \ that optimal value. We can try to find where d(Area)/dm = 0...\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dbdm\ = \ Simplify[D[bowlarea, m]]\)], "Input"], Cell[BoxData[ \(1\/6\ \[Pi]\ \((\(-2\)\ m + \(2\ \((m\^3 + 25\ m\^\(3/2\)\ \ \@\(10\/\[Pi]\))\)\)\/\@\(m\^4 + 40\ m\^\(5/2\)\ \@\(10\/\[Pi]\)\) + \(20\ \ \((\(-200\) + m\^\(3/2\)\ \@\(10\ \[Pi]\))\)\)\/\(m\^\(5/4\)\ \@\(40\ \@10 + \ m\^\(3/2\)\ \@\[Pi]\)\ \[Pi]\^\(3/4\)\))\)\)], "Output"] }, Open ]], Cell["\<\ ...yikes! Maybe not. Oh well, let's just get the answer \ numerically:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMinimum[bowlarea, {m, 6}]\)], "Input"], Cell[BoxData[ \({248.2228500509618`, {m \[Rule] 6.195533876160505`}}\)], "Output"] }, Open ]], Cell["\<\ So x = y^2/6.2 is the function that gives a 1 liter parabolic bowl \ from the least amount of material.\ \>", "Text"] }, FrontEndVersion->"5.1 for Macintosh", ScreenRectangle->{{0, 1280}, {0, 828}}, WindowSize->{520, 706}, WindowMargins->{{4, Automatic}, {Automatic, 4}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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