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$CellContext`xx$$) Tan[$CellContext`\[Alpha]] - $CellContext`y2]; $CellContext`c = ( 47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}, { 0, $CellContext`w - $CellContext`s}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}, {$CellContext`lgt, \ $CellContext`y2}, {0, $CellContext`w - $CellContext`s}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, $CellContext`y2 + $CellContext`h}, {$CellContext`x1, \ $CellContext`y1}, {$CellContext`lgt, 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$CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {$CellContext`z, 0}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`z, 0}}]; Grid[{{ Plot[ Piecewise[{{(47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2], $CellContext`u < $CellContext`kp}, {(1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^ Rational[ 1, 2], $CellContext`u >= $CellContext`kp}}], {$CellContext`u, 0, 8.5}, PlotRange -> {9, 13}, PlotStyle -> Black, AxesOrigin -> {0, 9}, ImageSize -> 0.7 {400, 350}, AxesLabel -> {DemonstrationsTools`x, "crease length"}, Axes -> True, Epilog -> {{ PointSize[0.02], Point[{$CellContext`xx$$, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}]}, AxesOrigin -> {0, 9}, {Green, Thickness[0.01], Line[{{0, 0}, {0, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}}]}, {Blue, Thickness[0.015], Line[{{0, 9}, {$CellContext`xx$$, 9}}]}}], Spacer[3], Graphics[{$CellContext`paper, $CellContext`lines, \ $CellContext`margin, {White, Polygon[$CellContext`wp]}, { Lighter[Yellow, 0.6], Polygon[$CellContext`ly]}, { Dashing[0.01], Thin, Black, Line[$CellContext`dg]}, {LightBrown, Thickness[0.005], Line[$CellContext`b1]}, {LightBrown, Thickness[0.005], Line[$CellContext`b2]}, {Blue, Thickness[0.004], Line[{{0, $CellContext`w}, {$CellContext`xx$$, \ $CellContext`w}}]}, { PointSize[0.015], Point[{$CellContext`xx$$, $CellContext`w}]}, {Green, Thickness[0.005], Line[$CellContext`cr]}}, AspectRatio -> Automatic, PlotRange -> {{0, 10}, {0, 9}}, ImageSize -> 0.55 {550, 425}, PlotRangePadding -> 1.7, PlotLabel -> Row[{ Style["x", Italic], " = ", NumberForm[ N[$CellContext`xx$$], {4, 2}], "\ncrease length = ", NumberForm[ N[ ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]], {5, 2}]}]]}}]), "Specifications" :> {{{$CellContext`xx$$, 2, "distance x"}, 0, 8.5, Appearance -> "Labeled"}}, "Options" :> {TrackedSymbols -> True}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{649., {165., 170.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`kp = 3.6375, $CellContext`s = 4.014705882352941, $CellContext`\[Alpha] = 0.46218133439179415`, $CellContext`x1 = 12.517213114754098`, $CellContext`lgt = 11.75, $CellContext`y1 = 3.2606557377049183`, $CellContext`w = 8.5, $CellContext`y2 = 1.7205882352941182`, $CellContext`h = 1.9222689075630242`, $CellContext`c = (47 Sqrt[289 + 4 $CellContext`u^2])/ 68, $CellContext`yp = {{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}, $CellContext`wp = {{0, 4.485294117647059}, {0, 0}, { 11.75, 0}, {11.75, 1.7205882352941182`}, { 0, 4.485294117647059}}, $CellContext`ly = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}, {0, 4.485294117647059}}, $CellContext`dg = {{ 0, 4.485294117647059}, {0, 0}, {11.75, 0}, {11.75, 1.7205882352941182`}}, $CellContext`b1 = {{0, 4.485294117647059}, { 0, 8.5}, {11.75, 8.5}, {11.75, 3.6428571428571423`}, { 12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}}, $CellContext`b2 = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}}, $CellContext`cr = {{0, 4.485294117647059}, { 11.75, 1.7205882352941182`}}, $CellContext`z = 8.892388535031847, Attributes[PlotRange] = {ReadProtected}, $CellContext`paper = { RGBColor[1., 1., 0.6], Polygon[{{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}]}, $CellContext`lines = {{{ Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.75, 0}, {10.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.25, 0}, {10.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.75, 0}, {9.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.25, 0}, {9.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.75, 0}, {8.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.25, 0}, {8.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.75, 0}, {7.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.25, 0}, {7.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.75, 0}, {6.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.25, 0}, {6.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.75, 0}, {5.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.25, 0}, {5.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.75, 0}, {4.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.25, 0}, {4.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.75, 0}, {3.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.25, 0}, {3.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.75, 0}, {2.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.25, 0}, {2.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.75, 0}, {1.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.25, 0}, {1.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.75, 0}, {0.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.25, 0}, {0.25, 8.5}}]}}}, $CellContext`margin = { RGBColor[1, 0, 0], Line[{{0, 7.5}, {11.75, 7.5}}]}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->449632576] }, {2}]] }, Open ]], Cell[CellGroupData[{ Cell["", "ManipulateCaptionSection"], Cell[TextData[{ "A sheet of legal paper is 8.5 by 11.75 inches long. The lower-left corner \ is brought to a point ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], " inches from the upper-left corner and folded to form a crease. This \ Demonstration shows a graph of the crease length as a function of ", Cell[BoxData[ FormBox["x", TraditionalForm]], "InlineMath"], " and a diagram of the folded paper. By moving the slider, you can estimate \ the length of the shortest and longest possible crease. The exact shortest \ possible crease (", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"51", " ", SqrtBox[ RowBox[{"3", " "}]]}], "8"], "\[TildeTilde]", "11.04"}], TraditionalForm]]], ") and the exact longest possible crease (", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox["47", "68"], SqrtBox[ RowBox[{ FractionBox["47", "2"], RowBox[{"(", RowBox[{"47", "-", RowBox[{"9", SqrtBox["13"]}]}], ")"}]}]]}], "\[TildeTilde]", "12.78"}], TraditionalForm]]], ") may be determined by using ", StyleBox["Mathematica", FontSlant->"Italic"], " or traditional calculus techniques." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.445437819565866*^9, 3.445437822503291*^9}, { 3.4454388289618998`*^9, 3.4454388311962175`*^9}, {3.445877202554592*^9, 3.4458773236006927`*^9}, {3.446221389541106*^9, 3.446221406150587*^9}}, CellID->381215968] }, Open ]], Cell[CellGroupData[{ Cell["", "ThumbnailSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`xx$$ = 2, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`xx$$], 2, "distance x"}, 0, 8.5}}, Typeset`size$$ = { 602., {120., 125.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`xx$16445$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`xx$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`xx$$, $CellContext`xx$16445$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`kp = 3.6375; $CellContext`s = (1/68) (289. - 4 $CellContext`xx$$^2); If[$CellContext`xx$$ < $CellContext`kp, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`x1 = \ $CellContext`xx$$ + $CellContext`lgt Cos[$CellContext`\[Alpha]]; $CellContext`y1 = $CellContext`w - \ $CellContext`lgt Sin[$CellContext`\[Alpha]]; $CellContext`y2 = $CellContext`w - \ $CellContext`s - $CellContext`lgt Cot[(Pi - $CellContext`\[Alpha])/2]; If[$CellContext`\[Alpha] == 0, $CellContext`h = 4.25, $CellContext`h = $CellContext`w - ($CellContext`lgt - \ $CellContext`xx$$) Tan[$CellContext`\[Alpha]] - $CellContext`y2]; $CellContext`c = ( 47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}, { 0, $CellContext`w - $CellContext`s}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}, {$CellContext`lgt, \ $CellContext`y2}, {0, $CellContext`w - $CellContext`s}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, $CellContext`y2 + $CellContext`h}, {$CellContext`x1, \ $CellContext`y1}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`lgt, \ $CellContext`y2}}, If[$CellContext`xx$$ < $CellContext`w, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`z = \ $CellContext`w Csc[$CellContext`\[Alpha]], $CellContext`z = $CellContext`w]; \ $CellContext`c = (1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {$CellContext`z, 0}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`z, 0}}]; Grid[{{ Plot[ Piecewise[{{(47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2], $CellContext`u < $CellContext`kp}, {(1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^ Rational[ 1, 2], $CellContext`u >= $CellContext`kp}}], {$CellContext`u, 0, 8.5}, PlotRange -> {9, 13}, PlotStyle -> Black, AxesOrigin -> {0, 9}, ImageSize -> 0.7 {400, 350}, AxesLabel -> {DemonstrationsTools`x, "crease length"}, Axes -> True, Epilog -> {{ PointSize[0.02], Point[{$CellContext`xx$$, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}]}, AxesOrigin -> {0, 9}, {Green, Thickness[0.01], Line[{{0, 0}, {0, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}}]}, {Blue, Thickness[0.015], Line[{{0, 9}, {$CellContext`xx$$, 9}}]}}], Spacer[3], Graphics[{$CellContext`paper, $CellContext`lines, \ $CellContext`margin, {White, Polygon[$CellContext`wp]}, { Lighter[Yellow, 0.6], Polygon[$CellContext`ly]}, { Dashing[0.01], Thin, Black, Line[$CellContext`dg]}, {LightBrown, Thickness[0.005], Line[$CellContext`b1]}, {LightBrown, Thickness[0.005], Line[$CellContext`b2]}, {Blue, Thickness[0.004], Line[{{0, $CellContext`w}, {$CellContext`xx$$, \ $CellContext`w}}]}, { PointSize[0.015], Point[{$CellContext`xx$$, $CellContext`w}]}, {Green, Thickness[0.005], Line[$CellContext`cr]}}, AspectRatio -> Automatic, PlotRange -> {{0, 10}, {0, 9}}, ImageSize -> 0.55 {550, 425}, PlotRangePadding -> 1.7, PlotLabel -> Row[{ Style["x", Italic], " = ", NumberForm[ N[$CellContext`xx$$], {4, 2}], "\ncrease length = ", NumberForm[ N[ ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]], {5, 2}]}]]}}]), "Specifications" :> {{{$CellContext`xx$$, 2, "distance x"}, 0, 8.5, Appearance -> "Labeled"}}, "Options" :> {TrackedSymbols -> True}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{649., {165., 170.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`kp = 3.6375, $CellContext`s = 4.014705882352941, $CellContext`\[Alpha] = 0.46218133439179415`, $CellContext`x1 = 12.517213114754098`, $CellContext`lgt = 11.75, $CellContext`y1 = 3.2606557377049183`, $CellContext`w = 8.5, $CellContext`y2 = 1.7205882352941182`, $CellContext`h = 1.9222689075630242`, $CellContext`c = (47 Sqrt[289 + 4 $CellContext`u^2])/ 68, $CellContext`yp = {{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}, $CellContext`wp = {{0, 4.485294117647059}, {0, 0}, { 11.75, 0}, {11.75, 1.7205882352941182`}, { 0, 4.485294117647059}}, $CellContext`ly = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}, {0, 4.485294117647059}}, $CellContext`dg = {{ 0, 4.485294117647059}, {0, 0}, {11.75, 0}, {11.75, 1.7205882352941182`}}, $CellContext`b1 = {{0, 4.485294117647059}, { 0, 8.5}, {11.75, 8.5}, {11.75, 3.6428571428571423`}, { 12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}}, $CellContext`b2 = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}}, $CellContext`cr = {{0, 4.485294117647059}, { 11.75, 1.7205882352941182`}}, $CellContext`z = 8.892388535031847, Attributes[PlotRange] = {ReadProtected}, $CellContext`paper = { RGBColor[1., 1., 0.6], Polygon[{{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}]}, $CellContext`lines = {{{ Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.75, 0}, {10.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.25, 0}, {10.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.75, 0}, {9.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.25, 0}, {9.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.75, 0}, {8.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.25, 0}, {8.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.75, 0}, {7.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.25, 0}, {7.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.75, 0}, {6.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.25, 0}, {6.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.75, 0}, {5.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.25, 0}, {5.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.75, 0}, {4.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.25, 0}, {4.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.75, 0}, {3.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.25, 0}, {3.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.75, 0}, {2.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.25, 0}, {2.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.75, 0}, {1.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.25, 0}, {1.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.75, 0}, {0.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.25, 0}, {0.25, 8.5}}]}}}, $CellContext`margin = { RGBColor[1, 0, 0], Line[{{0, 7.5}, {11.75, 7.5}}]}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->842378525] }, Open ]], Cell[CellGroupData[{ Cell["", "SnapshotsSection"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`xx$$ = 0.4, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`xx$$], 0.4, "distance x"}, 0, 8.5}}, Typeset`size$$ = {602., {120., 125.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`xx$16494$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`xx$$ = 0.4}, "ControllerVariables" :> { Hold[$CellContext`xx$$, $CellContext`xx$16494$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`kp = 3.6375; $CellContext`s = (1/68) (289. - 4 $CellContext`xx$$^2); If[$CellContext`xx$$ < $CellContext`kp, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`x1 = \ $CellContext`xx$$ + $CellContext`lgt Cos[$CellContext`\[Alpha]]; $CellContext`y1 = $CellContext`w - \ $CellContext`lgt Sin[$CellContext`\[Alpha]]; $CellContext`y2 = $CellContext`w - \ $CellContext`s - $CellContext`lgt Cot[(Pi - $CellContext`\[Alpha])/2]; If[$CellContext`\[Alpha] == 0, $CellContext`h = 4.25, $CellContext`h = $CellContext`w - ($CellContext`lgt - \ $CellContext`xx$$) Tan[$CellContext`\[Alpha]] - $CellContext`y2]; $CellContext`c = ( 47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}, { 0, $CellContext`w - $CellContext`s}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}, {$CellContext`lgt, \ $CellContext`y2}, {0, $CellContext`w - $CellContext`s}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, $CellContext`y2 + $CellContext`h}, {$CellContext`x1, \ $CellContext`y1}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`lgt, \ $CellContext`y2}}, If[$CellContext`xx$$ < $CellContext`w, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`z = \ $CellContext`w Csc[$CellContext`\[Alpha]], $CellContext`z = $CellContext`w]; \ $CellContext`c = (1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {$CellContext`z, 0}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`z, 0}}]; Grid[{{ Plot[ Piecewise[{{(47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2], $CellContext`u < $CellContext`kp}, {(1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^ Rational[ 1, 2], $CellContext`u >= $CellContext`kp}}], {$CellContext`u, 0, 8.5}, PlotRange -> {9, 13}, PlotStyle -> Black, AxesOrigin -> {0, 9}, ImageSize -> 0.7 {400, 350}, AxesLabel -> {DemonstrationsTools`x, "crease length"}, Axes -> True, Epilog -> {{ PointSize[0.02], Point[{$CellContext`xx$$, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}]}, AxesOrigin -> {0, 9}, {Green, Thickness[0.01], Line[{{0, 0}, {0, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}}]}, {Blue, Thickness[0.015], Line[{{0, 9}, {$CellContext`xx$$, 9}}]}}], Spacer[3], Graphics[{$CellContext`paper, $CellContext`lines, \ $CellContext`margin, {White, Polygon[$CellContext`wp]}, { Lighter[Yellow, 0.6], Polygon[$CellContext`ly]}, { Dashing[0.01], Thin, Black, Line[$CellContext`dg]}, {LightBrown, Thickness[0.005], Line[$CellContext`b1]}, {LightBrown, Thickness[0.005], Line[$CellContext`b2]}, {Blue, Thickness[0.004], Line[{{0, $CellContext`w}, {$CellContext`xx$$, \ $CellContext`w}}]}, { PointSize[0.015], Point[{$CellContext`xx$$, $CellContext`w}]}, {Green, Thickness[0.005], Line[$CellContext`cr]}}, AspectRatio -> Automatic, PlotRange -> {{0, 10}, {0, 9}}, ImageSize -> 0.55 {550, 425}, PlotRangePadding -> 1.7, PlotLabel -> Row[{ Style["x", Italic], " = ", NumberForm[ N[$CellContext`xx$$], {4, 2}], "\ncrease length = ", NumberForm[ N[ ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]], {5, 2}]}]]}}]), "Specifications" :> {{{$CellContext`xx$$, 0.4, "distance x"}, 0, 8.5, Appearance -> "Labeled"}}, "Options" :> {TrackedSymbols -> True}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{649., {165., 170.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`kp = 3.6375, $CellContext`s = 4.014705882352941, $CellContext`\[Alpha] = 0.46218133439179415`, $CellContext`x1 = 12.517213114754098`, $CellContext`lgt = 11.75, $CellContext`y1 = 3.2606557377049183`, $CellContext`w = 8.5, $CellContext`y2 = 1.7205882352941182`, $CellContext`h = 1.9222689075630242`, $CellContext`c = (47 Sqrt[289 + 4 $CellContext`u^2])/ 68, $CellContext`yp = {{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}, $CellContext`wp = {{0, 4.485294117647059}, {0, 0}, { 11.75, 0}, {11.75, 1.7205882352941182`}, { 0, 4.485294117647059}}, $CellContext`ly = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}, {0, 4.485294117647059}}, $CellContext`dg = {{ 0, 4.485294117647059}, {0, 0}, {11.75, 0}, {11.75, 1.7205882352941182`}}, $CellContext`b1 = {{0, 4.485294117647059}, { 0, 8.5}, {11.75, 8.5}, {11.75, 3.6428571428571423`}, { 12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}}, $CellContext`b2 = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}}, $CellContext`cr = {{0, 4.485294117647059}, { 11.75, 1.7205882352941182`}}, $CellContext`z = 8.892388535031847, Attributes[PlotRange] = {ReadProtected}, $CellContext`paper = { RGBColor[1., 1., 0.6], Polygon[{{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}]}, $CellContext`lines = {{{ Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.75, 0}, {10.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.25, 0}, {10.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.75, 0}, {9.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.25, 0}, {9.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.75, 0}, {8.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.25, 0}, {8.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.75, 0}, {7.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.25, 0}, {7.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.75, 0}, {6.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.25, 0}, {6.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.75, 0}, {5.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.25, 0}, {5.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.75, 0}, {4.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.25, 0}, {4.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.75, 0}, {3.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.25, 0}, {3.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.75, 0}, {2.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.25, 0}, {2.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.75, 0}, {1.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.25, 0}, {1.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.75, 0}, {0.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.25, 0}, {0.25, 8.5}}]}}}, $CellContext`margin = { RGBColor[1, 0, 0], Line[{{0, 7.5}, {11.75, 7.5}}]}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->300478340], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`xx$$ = 3.68, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`xx$$], 3.68, "distance x"}, 0, 8.5}}, Typeset`size$$ = {602., {120., 125.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`xx$16543$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`xx$$ = 3.68}, "ControllerVariables" :> { Hold[$CellContext`xx$$, $CellContext`xx$16543$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`kp = 3.6375; $CellContext`s = (1/68) (289. - 4 $CellContext`xx$$^2); If[$CellContext`xx$$ < $CellContext`kp, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`x1 = \ $CellContext`xx$$ + $CellContext`lgt Cos[$CellContext`\[Alpha]]; $CellContext`y1 = $CellContext`w - \ $CellContext`lgt Sin[$CellContext`\[Alpha]]; $CellContext`y2 = $CellContext`w - \ $CellContext`s - $CellContext`lgt Cot[(Pi - $CellContext`\[Alpha])/2]; If[$CellContext`\[Alpha] == 0, $CellContext`h = 4.25, $CellContext`h = $CellContext`w - ($CellContext`lgt - \ $CellContext`xx$$) Tan[$CellContext`\[Alpha]] - $CellContext`y2]; $CellContext`c = ( 47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}, { 0, $CellContext`w - $CellContext`s}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}, {$CellContext`lgt, \ $CellContext`y2}, {0, $CellContext`w - $CellContext`s}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, $CellContext`y2 + $CellContext`h}, {$CellContext`x1, \ $CellContext`y1}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`lgt, \ $CellContext`y2}}, If[$CellContext`xx$$ < $CellContext`w, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`z = \ $CellContext`w Csc[$CellContext`\[Alpha]], $CellContext`z = $CellContext`w]; \ $CellContext`c = (1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {$CellContext`z, 0}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`z, 0}}]; Grid[{{ Plot[ Piecewise[{{(47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2], $CellContext`u < $CellContext`kp}, {(1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^ Rational[ 1, 2], $CellContext`u >= $CellContext`kp}}], {$CellContext`u, 0, 8.5}, PlotRange -> {9, 13}, PlotStyle -> Black, AxesOrigin -> {0, 9}, ImageSize -> 0.7 {400, 350}, AxesLabel -> {DemonstrationsTools`x, "crease length"}, Axes -> True, Epilog -> {{ PointSize[0.02], Point[{$CellContext`xx$$, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}]}, AxesOrigin -> {0, 9}, {Green, Thickness[0.01], Line[{{0, 0}, {0, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}}]}, {Blue, Thickness[0.015], Line[{{0, 9}, {$CellContext`xx$$, 9}}]}}], Spacer[3], Graphics[{$CellContext`paper, $CellContext`lines, \ $CellContext`margin, {White, Polygon[$CellContext`wp]}, { Lighter[Yellow, 0.6], Polygon[$CellContext`ly]}, { Dashing[0.01], Thin, Black, Line[$CellContext`dg]}, {LightBrown, Thickness[0.005], Line[$CellContext`b1]}, {LightBrown, Thickness[0.005], Line[$CellContext`b2]}, {Blue, Thickness[0.004], Line[{{0, $CellContext`w}, {$CellContext`xx$$, \ $CellContext`w}}]}, { PointSize[0.015], Point[{$CellContext`xx$$, $CellContext`w}]}, {Green, Thickness[0.005], Line[$CellContext`cr]}}, AspectRatio -> Automatic, PlotRange -> {{0, 10}, {0, 9}}, ImageSize -> 0.55 {550, 425}, PlotRangePadding -> 1.7, PlotLabel -> Row[{ Style["x", Italic], " = ", NumberForm[ N[$CellContext`xx$$], {4, 2}], "\ncrease length = ", NumberForm[ N[ ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]], {5, 2}]}]]}}]), "Specifications" :> {{{$CellContext`xx$$, 3.68, "distance x"}, 0, 8.5, Appearance -> "Labeled"}}, "Options" :> {TrackedSymbols -> True}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{649., {165., 170.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`kp = 3.6375, $CellContext`s = 4.014705882352941, $CellContext`\[Alpha] = 0.46218133439179415`, $CellContext`x1 = 12.517213114754098`, $CellContext`lgt = 11.75, $CellContext`y1 = 3.2606557377049183`, $CellContext`w = 8.5, $CellContext`y2 = 1.7205882352941182`, $CellContext`h = 1.9222689075630242`, $CellContext`c = (47 Sqrt[289 + 4 $CellContext`u^2])/ 68, $CellContext`yp = {{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}, $CellContext`wp = {{0, 4.485294117647059}, {0, 0}, { 11.75, 0}, {11.75, 1.7205882352941182`}, { 0, 4.485294117647059}}, $CellContext`ly = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}, {0, 4.485294117647059}}, $CellContext`dg = {{ 0, 4.485294117647059}, {0, 0}, {11.75, 0}, {11.75, 1.7205882352941182`}}, $CellContext`b1 = {{0, 4.485294117647059}, { 0, 8.5}, {11.75, 8.5}, {11.75, 3.6428571428571423`}, { 12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}}, $CellContext`b2 = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}}, $CellContext`cr = {{0, 4.485294117647059}, { 11.75, 1.7205882352941182`}}, $CellContext`z = 8.892388535031847, Attributes[PlotRange] = {ReadProtected}, $CellContext`paper = { RGBColor[1., 1., 0.6], Polygon[{{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}]}, $CellContext`lines = {{{ Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.75, 0}, {10.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.25, 0}, {10.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.75, 0}, {9.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.25, 0}, {9.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.75, 0}, {8.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.25, 0}, {8.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.75, 0}, {7.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.25, 0}, {7.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.75, 0}, {6.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{6.25, 0}, {6.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.75, 0}, {5.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{5.25, 0}, {5.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.75, 0}, {4.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{4.25, 0}, {4.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.75, 0}, {3.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{3.25, 0}, {3.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.75, 0}, {2.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{2.25, 0}, {2.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.75, 0}, {1.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{1.25, 0}, {1.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.75, 0}, {0.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{0.25, 0}, {0.25, 8.5}}]}}}, $CellContext`margin = { RGBColor[1, 0, 0], Line[{{0, 7.5}, {11.75, 7.5}}]}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->133913347], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`xx$$ = 6.28, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`xx$$], 6.28, "distance x"}, 0, 8.5}}, Typeset`size$$ = {602., {120., 125.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`xx$16592$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`xx$$ = 6.28}, "ControllerVariables" :> { Hold[$CellContext`xx$$, $CellContext`xx$16592$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`kp = 3.6375; $CellContext`s = (1/68) (289. - 4 $CellContext`xx$$^2); If[$CellContext`xx$$ < $CellContext`kp, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`x1 = \ $CellContext`xx$$ + $CellContext`lgt Cos[$CellContext`\[Alpha]]; $CellContext`y1 = $CellContext`w - \ $CellContext`lgt Sin[$CellContext`\[Alpha]]; $CellContext`y2 = $CellContext`w - \ $CellContext`s - $CellContext`lgt Cot[(Pi - $CellContext`\[Alpha])/2]; If[$CellContext`\[Alpha] == 0, $CellContext`h = 4.25, $CellContext`h = $CellContext`w - ($CellContext`lgt - \ $CellContext`xx$$) Tan[$CellContext`\[Alpha]] - $CellContext`y2]; $CellContext`c = ( 47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}, { 0, $CellContext`w - $CellContext`s}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}, {$CellContext`lgt, \ $CellContext`y2}, {0, $CellContext`w - $CellContext`s}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`lgt, 0}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, $CellContext`y2 + $CellContext`h}, {$CellContext`x1, \ $CellContext`y1}, {$CellContext`lgt, $CellContext`y2}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`x1, $CellContext`y1}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`lgt, \ $CellContext`y2}}, If[$CellContext`xx$$ < $CellContext`w, $CellContext`\[Alpha] = ArcTan[$CellContext`xx$$/$CellContext`s]; $CellContext`z = \ $CellContext`w Csc[$CellContext`\[Alpha]], $CellContext`z = $CellContext`w]; \ $CellContext`c = (1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^Rational[1, 2]; $CellContext`yp = {{0, 0}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {0, 0}}; $CellContext`wp = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`ly = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`dg = {{ 0, $CellContext`w - $CellContext`s}, {0, 0}, {$CellContext`z, 0}}; $CellContext`b1 = {{ 0, $CellContext`w - $CellContext`s}, { 0, $CellContext`w}, {$CellContext`lgt, $CellContext`w}, \ {$CellContext`lgt, 0}, {$CellContext`z, 0}}; $CellContext`b2 = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`xx$$, \ $CellContext`w}, {$CellContext`z, 0}}; $CellContext`cr = {{ 0, $CellContext`w - $CellContext`s}, {$CellContext`z, 0}}]; Grid[{{ Plot[ Piecewise[{{(47/68) (289 + 4 $CellContext`u^2)^ Rational[1, 2], $CellContext`u < $CellContext`kp}, {(1/ 2) (($CellContext`w^2 + $CellContext`u^2)^3/($CellContext`w^2 \ $CellContext`u^2))^ Rational[ 1, 2], $CellContext`u >= $CellContext`kp}}], {$CellContext`u, 0, 8.5}, PlotRange -> {9, 13}, PlotStyle -> Black, AxesOrigin -> {0, 9}, ImageSize -> 0.7 {400, 350}, AxesLabel -> {DemonstrationsTools`x, "crease length"}, Axes -> True, Epilog -> {{ PointSize[0.02], Point[{$CellContext`xx$$, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}]}, AxesOrigin -> {0, 9}, {Green, Thickness[0.01], Line[{{0, 0}, {0, ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]}}]}, {Blue, Thickness[0.015], Line[{{0, 9}, {$CellContext`xx$$, 9}}]}}], Spacer[3], Graphics[{$CellContext`paper, $CellContext`lines, \ $CellContext`margin, {White, Polygon[$CellContext`wp]}, { Lighter[Yellow, 0.6], Polygon[$CellContext`ly]}, { Dashing[0.01], Thin, Black, Line[$CellContext`dg]}, {LightBrown, Thickness[0.005], Line[$CellContext`b1]}, {LightBrown, Thickness[0.005], Line[$CellContext`b2]}, {Blue, Thickness[0.004], Line[{{0, $CellContext`w}, {$CellContext`xx$$, \ $CellContext`w}}]}, { PointSize[0.015], Point[{$CellContext`xx$$, $CellContext`w}]}, {Green, Thickness[0.005], Line[$CellContext`cr]}}, AspectRatio -> Automatic, PlotRange -> {{0, 10}, {0, 9}}, ImageSize -> 0.55 {550, 425}, PlotRangePadding -> 1.7, PlotLabel -> Row[{ Style["x", Italic], " = ", NumberForm[ N[$CellContext`xx$$], {4, 2}], "\ncrease length = ", NumberForm[ N[ ReplaceAll[$CellContext`c, $CellContext`u -> \ $CellContext`xx$$]], {5, 2}]}]]}}]), "Specifications" :> {{{$CellContext`xx$$, 6.28, "distance x"}, 0, 8.5, Appearance -> "Labeled"}}, "Options" :> {TrackedSymbols -> True}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{649., {165., 170.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`kp = 3.6375, $CellContext`s = 4.014705882352941, $CellContext`\[Alpha] = 0.46218133439179415`, $CellContext`x1 = 12.517213114754098`, $CellContext`lgt = 11.75, $CellContext`y1 = 3.2606557377049183`, $CellContext`w = 8.5, $CellContext`y2 = 1.7205882352941182`, $CellContext`h = 1.9222689075630242`, $CellContext`c = (47 Sqrt[289 + 4 $CellContext`u^2])/ 68, $CellContext`yp = {{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}, $CellContext`wp = {{0, 4.485294117647059}, {0, 0}, { 11.75, 0}, {11.75, 1.7205882352941182`}, { 0, 4.485294117647059}}, $CellContext`ly = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}, {0, 4.485294117647059}}, $CellContext`dg = {{ 0, 4.485294117647059}, {0, 0}, {11.75, 0}, {11.75, 1.7205882352941182`}}, $CellContext`b1 = {{0, 4.485294117647059}, { 0, 8.5}, {11.75, 8.5}, {11.75, 3.6428571428571423`}, { 12.517213114754098`, 3.2606557377049183`}, {11.75, 1.7205882352941182`}}, $CellContext`b2 = {{0, 4.485294117647059}, { 2, 8.5}, {12.517213114754098`, 3.2606557377049183`}}, $CellContext`cr = {{0, 4.485294117647059}, { 11.75, 1.7205882352941182`}}, $CellContext`z = 8.892388535031847, Attributes[PlotRange] = {ReadProtected}, $CellContext`paper = { RGBColor[1., 1., 0.6], Polygon[{{0, 0}, {0, 8.5}, {11.75, 8.5}, {11.75, 0}, {0, 0}}]}, $CellContext`lines = {{{ Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.75, 0}, {10.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{10.25, 0}, {10.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.75, 0}, {9.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{9.25, 0}, {9.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.75, 0}, {8.75, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{8.25, 0}, {8.25, 8.5}}]}, { Thickness[0.001], RGBColor[0, 0, 1], Line[{{7.75, 0}, {7.75, 8.5}}]}, { 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