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ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.5}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{609., {257., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->141444057] }, {2}]] }, Open ]], Cell[CellGroupData[{ Cell["", "ManipulateCaptionSection"], Cell[TextData[{ "This Demonstration illustrates two common types of max-min problem from a \ Calculus I course\[LongDash]those of finding the maximum volume and finding \ the maximum surface area of a geometric figure inscribed in a sphere. The \ figures available are a cylinder, a cone, and a cuboid with a square base. \ The sphere has radius 1. For the cylinder and the cone, ", StyleBox["r", FontSlant->"Italic"], " is the radius of the base. For the cuboid, ", StyleBox["r ", FontSlant->"Italic"], "is one-half the length of a side of the base. An option to show the \ projection of the sphere and the inscribed solid onto the ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], "-", Cell[BoxData[ FormBox["z", TraditionalForm]], "InlineMath"], " plane is included (see snapshot 5)." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.46058146515625*^9, 3.460581506390625*^9}, { 3.4607461028779335`*^9, 3.460746157783832*^9}, {3.460746220783429*^9, 3.4607462609081717`*^9}, {3.46099259719335*^9, 3.4609925989276915`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "ThumbnailSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.5, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.5}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], False, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = {560., {178., 184.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49328$$ = 0, $CellContext`obj$49329$$ = False, $CellContext`r$49330$$ = 0, $CellContext`disp$49331$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, \ $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.5}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49328$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49329$$, False], Hold[$CellContext`r$$, $CellContext`r$49330$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49331$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.5}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{609., {257., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->278711856] }, Open ]], Cell[CellGroupData[{ Cell["", "SnapshotsSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.654780879378743, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.654780879378743}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], False, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = {560., {178., 184.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49398$$ = 0, $CellContext`obj$49399$$ = False, $CellContext`r$49400$$ = 0, $CellContext`disp$49401$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, \ $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.654780879378743}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49398$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49399$$, False], Hold[$CellContext`r$$, $CellContext`r$49400$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49401$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.654780879378743}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{609., {257., 262.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->455082061], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.492, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.492}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], False, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49468$$ = 0, $CellContext`obj$49469$$ = False, $CellContext`r$49470$$ = 0, $CellContext`disp$49471$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cylinder, \ $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.492}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49468$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49469$$, False], Hold[$CellContext`r$$, $CellContext`r$49470$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49471$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cylinder, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.492}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->347755046], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cone, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.7071067811865475, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cone, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.7071067811865475}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], False, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49538$$ = 0, $CellContext`obj$49539$$ = False, $CellContext`r$49540$$ = 0, $CellContext`disp$49541$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cone, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`volume, $CellContext`r$$ = 0.7071067811865475}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49538$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49539$$, False], Hold[$CellContext`r$$, $CellContext`r$49540$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49541$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cone, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`volume, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.7071067811865475}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->19285775], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cuboid, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.5700000000000001, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cuboid, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.5700000000000001}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], False, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49608$$ = 0, $CellContext`obj$49609$$ = False, $CellContext`r$49610$$ = 0, $CellContext`disp$49611$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = False, $CellContext`fig$$ = $CellContext`cuboid, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.5700000000000001}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49608$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49609$$, False], Hold[$CellContext`r$$, $CellContext`r$49610$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49611$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cuboid, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.5700000000000001}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, False, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->331179878], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`disp$$ = True, $CellContext`fig$$ = $CellContext`cone, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.56, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`fig$$], $CellContext`cone, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{ Hold[$CellContext`obj$$], $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}}, {{ Hold[$CellContext`r$$], 0.56}, 0, Dynamic[$CellContext`maxr$$]}, {{ Hold[$CellContext`disp$$], True, "show projection"}, {False, True}}, {{ Hold[$CellContext`maxr$$], 1}}}, Typeset`size$$ = Automatic, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`fig$49678$$ = 0, $CellContext`obj$49679$$ = False, $CellContext`r$49680$$ = 0, $CellContext`disp$49681$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`disp$$ = True, $CellContext`fig$$ = $CellContext`cone, $CellContext`maxr$$ = 1, $CellContext`obj$$ = $CellContext`surface, $CellContext`r$$ = 0.56}, "ControllerVariables" :> { Hold[$CellContext`fig$$, $CellContext`fig$49678$$, 0], Hold[$CellContext`obj$$, $CellContext`obj$49679$$, False], Hold[$CellContext`r$$, $CellContext`r$49680$$, 0], Hold[$CellContext`disp$$, $CellContext`disp$49681$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`r$$ > $CellContext`maxr$$, $CellContext`r$$ = \ $CellContext`maxr$$]; $CellContext`theSphere = Graphics3D[{ Opacity[0.4], Sphere[]}]; $CellContext`theCircle = Graphics[ Circle[]]; $CellContext`theCylinder = Graphics3D[{Green, Cylinder[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, Sqrt[ 1 - $CellContext`r$$^2]}}, $CellContext`r$$]}]; \ $CellContext`volCylinder[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 2 Sqrt[1 - $CellContext`r^2]; $CellContext`surfCylinder[ Pattern[$CellContext`r, Blank[]]] := 2 Pi $CellContext`r^2 + 2 Pi $CellContext`r 2 Sqrt[1 - $CellContext`r^2]; $CellContext`projCylinder = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - $CellContext`r$$^2]}]}]; $CellContext`theCone = Graphics3D[{Green, Cone[{{0, 0, -Sqrt[1 - $CellContext`r$$^2]}, {0, 0, 1}}, $CellContext`r$$]}]; $CellContext`volCone[ Pattern[$CellContext`r, Blank[]]] := (1/3) Pi $CellContext`r^2 (1 + Sqrt[1 - $CellContext`r^2]); $CellContext`surfCone[ Pattern[$CellContext`r, Blank[]]] := Pi $CellContext`r^2 + Pi $CellContext`r Sqrt[$CellContext`r^2 + (1 + Sqrt[1 - $CellContext`r^2])^2]; $CellContext`projCone = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Polygon[{{-$CellContext`r$$, -Sqrt[1 - $CellContext`r$$^2]}, {0, 1}, {$CellContext`r$$, - Sqrt[1 - $CellContext`r$$^2]}}]}]; $CellContext`theCuboid = Graphics3D[{Green, Cuboid[{-$CellContext`r$$, -$CellContext`r$$, - Sqrt[ 1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, $CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; $CellContext`volCuboid[ Pattern[$CellContext`r, Blank[]]] := 4 $CellContext`r^2 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`surfCuboid[ Pattern[$CellContext`r, Blank[]]] := 8 $CellContext`r^2 + 4 2 $CellContext`r 2 Sqrt[1 - 2 $CellContext`r^2]; $CellContext`projCuboid = Graphics[{ EdgeForm[ Directive[Thick, Blue]], RGBColor[0.25, 0.75, 0.25], Rectangle[{-$CellContext`r$$, - Sqrt[1 - 2 $CellContext`r$$^2]}, {$CellContext`r$$, Sqrt[1 - 2 $CellContext`r$$^2]}]}]; Switch[$CellContext`fig$$, $CellContext`cylinder, \ $CellContext`theFigure = $CellContext`theCylinder; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCylinder; \ $CellContext`surfFunc = $CellContext`surfCylinder; $CellContext`proj = \ $CellContext`projCylinder, $CellContext`cone, $CellContext`theFigure = \ $CellContext`theCone; $CellContext`maxr$$ = 1; $CellContext`volFunc = $CellContext`volCone; \ $CellContext`surfFunc = $CellContext`surfCone; $CellContext`proj = \ $CellContext`projCone, $CellContext`cuboid, $CellContext`theFigure = \ $CellContext`theCuboid; $CellContext`maxr$$ = Sqrt[2]/2; If[$CellContext`r$$ > Sqrt[2]/2, $CellContext`r$$ = 0.25]; $CellContext`volFunc = $CellContext`volCuboid; \ $CellContext`surfFunc = $CellContext`surfCuboid; $CellContext`proj = \ $CellContext`projCuboid]; $CellContext`volPlot = Plot[ $CellContext`volFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"volume of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`surfPlot = Plot[ $CellContext`surfFunc[$CellContext`x], {$CellContext`x, 0, $CellContext`maxr$$}, PlotLabel -> Style[ Row[{"surface area of ", $CellContext`fig$$}], 18], ImageSize -> 250, ImagePadding -> 20]; $CellContext`projPlot = Show[$CellContext`theCircle, $CellContext`proj, Axes -> True, AxesLabel -> {$CellContext`x, $CellContext`z}, ImageSize -> 200]; $CellContext`pointOnVolCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`volFunc[$CellContext`r$$]}]}]; \ $CellContext`pointOnSurfCurve = Graphics[{ PointSize[Large], Green, Point[{$CellContext`r$$, $CellContext`surfFunc[$CellContext`r$$]}]}]; \ $CellContext`volDisplay = Show[$CellContext`volPlot, $CellContext`pointOnVolCurve, ImageSize -> 250]; $CellContext`surfDisplay = Show[$CellContext`surfPlot, $CellContext`pointOnSurfCurve, ImageSize -> 250]; Switch[$CellContext`obj$$, $CellContext`volume, \ $CellContext`displayFunc = $CellContext`volDisplay, $CellContext`surface, \ $CellContext`displayFunc = $CellContext`surfDisplay]; Switch[$CellContext`disp$$, False, $CellContext`theDisplay = $CellContext`displayFunc, True, $CellContext`theDisplay = $CellContext`projPlot]; Text[ Pane[ Row[{ Show[$CellContext`theSphere, $CellContext`theFigure, Axes -> True, AxesOrigin -> {0, 0, 0}, ImageSize -> 300], Column[{ Show[$CellContext`theDisplay], "", Style[ Row[{ Style["r", Italic], " = ", NumberForm[$CellContext`r$$, {6, 4}]}], 24], Style[ Row[{"volume = ", NumberForm[ $CellContext`volFunc[$CellContext`r$$], {6, 4}]}], 24], Style[ Row[{"surface area = ", NumberForm[ $CellContext`surfFunc[$CellContext`r$$], {6, 4}]}], 24]}]}], ImageSize -> {560, 360}]]), "Specifications" :> {{{$CellContext`fig$$, $CellContext`cone, "figure"}, {$CellContext`cylinder, $CellContext`cone, \ $CellContext`cuboid}}, {{$CellContext`obj$$, $CellContext`surface, "objective"}, {$CellContext`volume -> "volume", $CellContext`surface -> "surface area"}, Enabled -> Dynamic[ Not[$CellContext`disp$$]]}, {{$CellContext`r$$, 0.56}, 0, Dynamic[$CellContext`maxr$$]}, {{$CellContext`disp$$, True, "show projection"}, {False, True}}, {{$CellContext`maxr$$, 1}, ControlType -> None}}, "Options" :> { TrackedSymbols -> True, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->434134272] }, Open ]], Cell["", "DetailsSection"], Cell[CellGroupData[{ Cell["", "ControlSuggestionsSection"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[True], Cell[" Resize Images"]}], "\"Click inside an image to reveal its orange resize frame.\\nDrag any of \ the orange resize handles to resize the image.\""]], "ControlSuggestions", CellChangeTimes->{3.35696210375764*^9, 3.4607462886579943`*^9}, FontFamily->"Verdana", CellTags->"ResizeImages"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[True], Cell[" Rotate and Zoom in 3D"]}], RowBox[{ "\"Drag a 3D graphic to rotate it. Starting the drag near the center \ tumbles\\nthe graphic; starting near a corner turns it parallel to the plane \ of the screen.\\nHold down \"", FrameBox[ "Ctrl", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" (or \"", FrameBox[ "Cmd", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" on Mac) and drag up and down to zoom.\""}]]], "ControlSuggestions", CellChangeTimes->{3.35696210375764*^9, 3.460746290720481*^9}, FontFamily->"Verdana", CellTags->"RotateAndZoomIn3D"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Drag Locators"]}], RowBox[{"\"Drag any locator (\"", GraphicsBox[ LocatorBox[ Scaled[{0.5, 0.5}]], ImageSize -> 20], "\", etc.) to move it around.\""}]]], "ControlSuggestions", FontFamily->"Verdana", CellTags->"DragLocators"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Create and Delete Locators"]}], RowBox[{"\"Insert a new locator in the graphic by holding down the \"", FrameBox[ "Alt", Background -> GrayLevel[0.9], FrameMargins -> 2, FrameStyle -> GrayLevel[0.9]], "\" key\\nand clicking where you want it to be. 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3.4607462923923454`*^9}, FontFamily->"Verdana", CellTags->"SliderZoom"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[False], Cell[" Gamepad Controls"]}], "\"Control this Demonstration with a gamepad or other\\nhuman interface \ device connected to your computer.\""]], "ControlSuggestions", CellChangeTimes->{3.35696210375764*^9, 3.3895522232313623`*^9}, FontFamily->"Verdana", CellTags->"GamepadControls"], Cell[BoxData[ TooltipBox[ RowBox[{ CheckboxBox[True], Cell[" Automatic Animation"]}], RowBox[{"\"Animate a slider in this Demonstration by clicking the\"", AdjustmentBox[ Cell[ GraphicsData[ "CompressedBitmap", "eJzzTSzJSM1NLMlMTlRwL0osyMhMLlZwyy8CCjEzMjAwcIKwAgOI/R/IhBKc\n\ /4EAyGAG0f+nTZsGwgysIJIRKsWKLAXGIHFmEpUgLADxWUAkI24jZs+eTaEt\n\ IG+wQKRmzJgBlYf5lhEA30OqWA=="], "Graphics", ImageSize -> {9, 9}, ImageMargins -> 0, CellBaseline -> Baseline], BoxBaselineShift -> 0.1839080459770115, BoxMargins -> {{0., 0.}, {-0.1839080459770115, 0.1839080459770115}}], "\"button\\nnext to the 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