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- 2 $CellContext`x), $CellContext`dervolgraph[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Show[{ Plot[ $CellContext`dervol[$CellContext`x, $CellContext`a, \ $CellContext`b], {$CellContext`x, 0, Max[$CellContext`a/2, $CellContext`b/2]}, PlotStyle -> Thick, PlotLabel -> Row[{ Derivative[1][ Style["V", Italic]], " = ", Factor[ $CellContext`dervol[ Style["x", Italic], $CellContext`a, $CellContext`b]]}]], Graphics[{Thick, Line[{{$CellContext`x0, $CellContext`dervol[$CellContext`x0, $CellContext`a, \ $CellContext`b]}, {$CellContext`x0, 0}}]}]}, ImageSize -> {180, 150}], $CellContext`dervol[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a - 2 $CellContext`x0) ($CellContext`b - 2 $CellContext`x0) - 2 ($CellContext`a - 2 $CellContext`x0) $CellContext`x0 - 2 ($CellContext`b - 2 $CellContext`x0) $CellContext`x0, Attributes[Derivative] = {NHoldAll, ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.6350345452918077`*^9, 3.63503461489913*^9}, CellID->452409411] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "ManipulateCaptionSection"], Cell["\<\ Problem: A piece of sheet tin three feet square is to be made into a \ rectangular box open at the top by cutting out equal squares from the corners \ and bending up the sides of the resulting piece parallel with the edges. \ Among all such boxes, to find the box of greatest volume.\ \>", "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457122690275372*^9, 3.457122690763093*^9}, { 3.457122747476899*^9, 3.457122823352478*^9}, {3.4571228619058123`*^9, 3.457122861908148*^9}, {3.457167813430203*^9, 3.457167915817277*^9}, 3.457616395349894*^9, {3.505768289289219*^9, 3.505768289906096*^9}, { 3.505768542950172*^9, 3.505768545536627*^9}}, CellID->1664929489], Cell[TextData[{ "This is the problem J. L. Walsh used in his 1947 Classroom Note in ", StyleBox["The American Mathematical Monthly", FontSlant->"Italic"], " to illustrate a rigorous analysis of maximum-minimum problems. A version \ of the problem appears in many calculus books and in Walsh\[CloseCurlyQuote]s \ 1962 booklet." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457166995267165*^9, 3.4571671389196568`*^9}, { 3.457167950921612*^9, 3.457167962382772*^9}, {3.45716800601679*^9, 3.457168020374899*^9}, {3.4571680646199303`*^9, 3.4571681152283707`*^9}, { 3.457168171062812*^9, 3.457168172630525*^9}, {3.457168205533012*^9, 3.457168279875889*^9}, {3.457168311699409*^9, 3.457168433699457*^9}, { 3.457168513968523*^9, 3.457168513971055*^9}, {3.45726873863899*^9, 3.457268742280245*^9}, {3.4572775110849953`*^9, 3.457277519368601*^9}, 3.457616395376897*^9, {3.505650485729063*^9, 3.505650509804379*^9}, { 3.5056511364774237`*^9, 3.505651137213272*^9}, {3.505672349004697*^9, 3.505672356609212*^9}}, CellID->2140012416], Cell[TextData[{ "Let the tin sheet have dimensions ", Cell[BoxData[ FormBox[ RowBox[{"a", "\[Times]", "b"}], TraditionalForm]], "InlineMath"], ", with ", Cell[BoxData[ FormBox[ RowBox[{"a", "\[GreaterEqual]", "b"}], TraditionalForm]], "InlineMath"], ", and suppose a square with side ", Cell[BoxData[ FormBox[ StyleBox["x", FontSlant->"Italic"], TraditionalForm]], "InlineMath"], " is cut from each corner. The volume of the resulting box is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "(", "x", ")"}], "=", RowBox[{"x", " ", RowBox[{"(", RowBox[{"a", "-", RowBox[{"2", "x"}]}], ")"}], " ", RowBox[{"(", RowBox[{"b", "-", RowBox[{"2", "x"}]}], ")"}]}]}], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "x", "\[LessEqual]", FractionBox["b", "2"]}], TraditionalForm]], "InlineMath"], "." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457168526540379*^9, 3.457168624618685*^9}, { 3.457168678416874*^9, 3.457168909854722*^9}, {3.457169003890749*^9, 3.4571690190939083`*^9}, {3.457174480510919*^9, 3.457174593067091*^9}, { 3.4571917206438313`*^9, 3.457191724268429*^9}, {3.457191772494803*^9, 3.4571918082282047`*^9}, {3.457266653990438*^9, 3.4572666802954683`*^9}, { 3.457616395390898*^9, 3.4576163954048996`*^9}, {3.4576165404604034`*^9, 3.4576165848558426`*^9}, {3.4576167084962053`*^9, 3.457616709113267*^9}, { 3.505651438764181*^9, 3.505651438769601*^9}}, CellID->1967566256], Cell[TextData[{ "Walsh\[CloseCurlyQuote]s rigorous analysis uses the extreme value theorem: \ A continuous function on a closed bounded interval has minimum and maximum \ values, and the critical point theorem: If a function has an extreme value at \ an interior point of an interval, its derivative at the point is either zero \ or does not exist.", " ", "A proof of the extreme value theorem is best left to an advanced calculus \ course, but the critical point theorem depends only on the definition of \ derivative. An extreme value cannot occur where ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "x", ")"}]}], ">", "0"}], TraditionalForm]], "InlineMath"], " or ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "x", ")"}]}], "<", "0"}], TraditionalForm]], "InlineMath"], "." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.505651439754455*^9, 3.5056515803466873`*^9}, { 3.5056516150067043`*^9, 3.505651748016068*^9}, {3.5056517902756653`*^9, 3.505651792178816*^9}, {3.505652336437151*^9, 3.505652341453673*^9}, { 3.505652807036601*^9, 3.505652899489553*^9}, {3.5056529295908127`*^9, 3.505653064569333*^9}, {3.505672414098523*^9, 3.505672429328299*^9}, { 3.505768355561392*^9, 3.505768378803491*^9}, {3.505768448835424*^9, 3.505768469984511*^9}, {3.505769801900053*^9, 3.505769808173191*^9}}, CellID->664152553], Cell[TextData[{ "Since ", Cell[BoxData[ FormBox[ RowBox[{"V", "(", "x", ")"}], TraditionalForm]], "InlineMath"], " is continuous on ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "x", "\[LessEqual]", FractionBox["b", "2"]}], TraditionalForm]], "InlineMath"], ", it has a maximum value there, by the extreme value theorem." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457177861864368*^9, 3.4571778699980907`*^9}, { 3.457177998269594*^9, 3.457178291380425*^9}, {3.457178329871707*^9, 3.457178336632114*^9}, {3.4571911607327414`*^9, 3.457191178600738*^9}, { 3.4571916282674637`*^9, 3.457191671536769*^9}, {3.4576165968760448`*^9, 3.45761659802816*^9}, {3.50565056277496*^9, 3.50565059381625*^9}, { 3.505650702045308*^9, 3.505650729742672*^9}, {3.505650761577005*^9, 3.5056508583924427`*^9}, {3.5056509689693727`*^9, 3.505651047251265*^9}, { 3.505651848917671*^9, 3.505651878529497*^9}, {3.505651939370537*^9, 3.505651942402478*^9}, {3.5056523167520647`*^9, 3.505652324958787*^9}, { 3.5056523782098513`*^9, 3.505652379686256*^9}, 3.505672457517397*^9, { 3.505768404879929*^9, 3.5057684112700367`*^9}}, CellID->642308845], Cell[TextData[{ "Let the maximum occur at ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SubscriptBox["x", "0"]}], TraditionalForm]], "InlineMath"], "." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457177861864368*^9, 3.4571778699980907`*^9}, { 3.457177998269594*^9, 3.457178291380425*^9}, {3.457178329871707*^9, 3.457178336632114*^9}, {3.4571911607327414`*^9, 3.457191178600738*^9}, { 3.4571916282674637`*^9, 3.457191671536769*^9}, {3.4576165968760448`*^9, 3.45761659802816*^9}, {3.50565056277496*^9, 3.50565059381625*^9}, { 3.505650702045308*^9, 3.505650729742672*^9}, {3.505650761577005*^9, 3.5056508583924427`*^9}, {3.5056509689693727`*^9, 3.505651047251265*^9}, { 3.505651848917671*^9, 3.505651878529497*^9}, {3.505651939370537*^9, 3.505651942402478*^9}, {3.5056523167520647`*^9, 3.505652324958787*^9}, { 3.5056523782098513`*^9, 3.505652379680093*^9}}, CellID->1709867028], Cell[TextData[{ "Since ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "(", "0", ")"}], "=", RowBox[{ RowBox[{"V", "(", FractionBox["b", "2"], ")"}], "=", "0"}]}], TraditionalForm]], "InlineMath"], ", and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "(", "x", ")"}], ">", "0"}], TraditionalForm]], "InlineMath"], " when ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", "x", "<", "b"}], TraditionalForm]], "InlineMath"], ", we have ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", SubscriptBox["x", "0"], "<", FractionBox["b", "2"]}], TraditionalForm]], "InlineMath"], "." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.457177861864368*^9, 3.4571778699980907`*^9}, { 3.457177998269594*^9, 3.457178291380425*^9}, {3.457178329871707*^9, 3.457178336632114*^9}, {3.4571911607327414`*^9, 3.457191178600738*^9}, { 3.4571916282674637`*^9, 3.457191671536769*^9}, {3.4576165968760448`*^9, 3.45761659802816*^9}, {3.50565056277496*^9, 3.50565059381625*^9}, { 3.505650702045308*^9, 3.505650729742672*^9}, {3.505650761577005*^9, 3.5056508583924427`*^9}, {3.5056509689693727`*^9, 3.505651047251265*^9}, { 3.505651848917671*^9, 3.5056518727373238`*^9}, {3.5056519948930607`*^9, 3.505652015602528*^9}, {3.505652149995742*^9, 3.505652151020052*^9}, { 3.505652184390472*^9, 3.505652184396142*^9}}, CellID->1939444812], Cell[TextData[{ "Since ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "'"}], RowBox[{"(", "x", ")"}]}], TraditionalForm]], "InlineMath"], " exists for ", Cell[BoxData[ FormBox[ RowBox[{"0", "<", "x", "<", FractionBox["b", "2"]}], TraditionalForm]], "InlineMath"], ", the ", "critical point theorem ", "implies ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"V", "'"}], RowBox[{"(", SubscriptBox["x", "0"], ")"}]}], "=", "0"}], TraditionalForm]], "InlineMath"], "." }], "ManipulateCaption", CellChangeTimes->{ 3.35696210375764*^9, {3.5056521868534517`*^9, 3.505652295666013*^9}, { 3.505652330077969*^9, 3.50565235895406*^9}, {3.505652447984173*^9, 3.5056524666716623`*^9}, {3.505768486332793*^9, 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3]}, {40, 33, 6}, {48, 13, 3}, {48, 35, Rational[20, 3]}, {55, 16, Rational[11, 3]}, {55, 39, Rational[15, 2]}, {65, 9, Rational[13, 6]}, {65, 56, 10}, {77, 32, 7}, {77, 45, Rational[55, 6]}, {80, 17, 4}, {80, 63, Rational[35, 3]}, {91, 40, Rational[26, 3]}, {91, 51, Rational[21, 2]}, {96, 11, Rational[8, 3]}, {96, 85, 15}, {99, 19, Rational[9, 2]}, {99, 80, Rational[44, 3]}}}, {{ Hold[$CellContext`q$$], 1, ""}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, { Hold[ PaneSelector[{True -> Column[{"Choose {a, b, optimal x}: ", Manipulate`Place[1], "Choose dimension multiplier: ", Manipulate`Place[2]}], False -> ""}, Dynamic[$CellContext`rationalbox$$]]], Manipulate`Dump`ThisIsNotAControl}}, Typeset`size$$ = { 420., {201.84375, 207.15625}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`a$12620$$ = 0, $CellContext`b$12621$$ = 0, $CellContext`x0$12622$$ = 0, $CellContext`rationalbox$12623$$ = False, $CellContext`pick$12624$$ = 0, $CellContext`q$12625$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 2, StandardForm, "Variables" :> {$CellContext`a$$ = 8, $CellContext`b$$ = 6, $CellContext`pick$$ = {1, 1, Rational[1, 6]}, $CellContext`q$$ = 1, $CellContext`rationalbox$$ = False, $CellContext`x0$$ = 1.131}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$12620$$, 0], Hold[$CellContext`b$$, $CellContext`b$12621$$, 0], Hold[$CellContext`x0$$, $CellContext`x0$12622$$, 0], Hold[$CellContext`rationalbox$$, $CellContext`rationalbox$12623$$, False], Hold[$CellContext`pick$$, $CellContext`pick$12624$$, 0], Hold[$CellContext`q$$, $CellContext`q$12625$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`rationalbox$$, $CellContext`a$$ = $CellContext`q$$ Part[$CellContext`pick$$, 1]; $CellContext`b$$ = $CellContext`q$$ Part[$CellContext`pick$$, 2], $CellContext`pick$$ = {1, 1, 1/6}]; If[$CellContext`x0$$ > Min[$CellContext`a$$/2, $CellContext`b$$/2], $CellContext`x0$$ = Min[0.2 $CellContext`a$$, 0.2 $CellContext`b$$]]; Column[{ Text[ TraditionalForm[ Row[{"Tin box holds the most when ", $CellContext`x, " = ", $CellContext`crit[$CellContext`a$$, $CellContext`b$$], "."}]]], Grid[{{ $CellContext`markedtin[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$], $CellContext`tray3D[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$]}, { $CellContext`volgraph[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$], $CellContext`dervolgraph[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$]}}, Frame -> All]}, ItemSize -> {Automatic, 4}, Alignment -> {Center, Center}]), "Specifications" :> {{{$CellContext`a$$, 8}, 1, 100, 1, Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`b$$, 6}, 1, 100, 1, Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`x0$$, 1.131, "Vary the height x"}, 0, Dynamic[ Min[$CellContext`a$$/2, $CellContext`b$$/2]], Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`rationalbox$$, False, "Show integer dimensions a and b with rational height for biggest \ box:"}, {False, True}, ControlPlacement -> Top}, {{$CellContext`pick$$, {1, 1, Rational[1, 6]}, ""}, {{1, 1, Rational[1, 6]}, {8, 3, Rational[2, 3]}, {8, 5, 1}, {15, 7, Rational[3, 2]}, {15, 8, Rational[5, 3]}, {21, 5, Rational[7, 6]}, {21, 16, 3}, {35, 11, Rational[5, 2]}, {35, 24, Rational[14, 3]}, {40, 7, Rational[5, 3]}, {40, 33, 6}, {48, 13, 3}, {48, 35, Rational[20, 3]}, {55, 16, Rational[11, 3]}, {55, 39, Rational[15, 2]}, {65, 9, Rational[13, 6]}, {65, 56, 10}, {77, 32, 7}, {77, 45, Rational[55, 6]}, {80, 17, 4}, {80, 63, Rational[35, 3]}, {91, 40, Rational[26, 3]}, {91, 51, Rational[21, 2]}, {96, 11, Rational[8, 3]}, {96, 85, 15}, {99, 19, Rational[9, 2]}, {99, 80, Rational[44, 3]}}, ControlPlacement -> 1}, {{$CellContext`q$$, 1, ""}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, ControlPlacement -> 2}, PaneSelector[{True -> Column[{"Choose {a, b, optimal x}: ", Manipulate`Place[1], "Choose dimension multiplier: ", Manipulate`Place[2]}], False -> ""}, Dynamic[$CellContext`rationalbox$$]]}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{713., {284., 291.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`crit[ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a + $CellContext`b - Sqrt[$CellContext`a^2 - $CellContext`a $CellContext`b + \ $CellContext`b^2])/6, $CellContext`markedtin[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Graphics[{Thick, Line[{{0, 0}, {$CellContext`a, 0}, {$CellContext`a, $CellContext`b}, { 0, $CellContext`b}, {0, 0}}], Dashed, Line[{{{$CellContext`x, 0}, {$CellContext`x, $CellContext`x}, { 0, $CellContext`x}}, {{$CellContext`a - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x}, \ {$CellContext`a, $CellContext`x}}, {{$CellContext`a - $CellContext`x, \ $CellContext`b}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x}, {$CellContext`a, $CellContext`b - $CellContext`x}}, {{ 0, $CellContext`b - $CellContext`x}, {$CellContext`x, \ $CellContext`b - $CellContext`x}, {$CellContext`x, $CellContext`b}}}]}, ImageSize -> {200, 200}], $CellContext`tray3D[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Graphics3D[{ Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}], {Thick, Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`x, $CellContext`b - $CellContext`x, $CellContext`x}, \ {$CellContext`x, $CellContext`b - $CellContext`x, 0}}], Dashed, Line[{{$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}]}, {Thick, Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`a - $CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`a - $CellContext`x, $CellContext`x, 0}}], Dashed, Line[{{$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}]}, {Thick, Polygon[{{$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, \ $CellContext`x}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`a - $CellContext`x, \ $CellContext`b - $CellContext`x, 0}}], Dashed, Line[{{$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}}]}, { Thick, Polygon[{{$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`x, $CellContext`b - \ $CellContext`x, 0}}], Dashed, Line[{{$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}}]}, {Thick, Line[{{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, 0, 0}, {$CellContext`a - $CellContext`x, 0, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a, $CellContext`x, 0}, {$CellContext`a, $CellContext`b - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b, 0}, {$CellContext`x, $CellContext`b, 0}, {$CellContext`x, $CellContext`b - $CellContext`x, 0}, { 0, $CellContext`b - $CellContext`x, 0}, { 0, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}}]}}, PlotRange -> {{0, $CellContext`a}, {0, $CellContext`b}, {0, Min[$CellContext`a/2, $CellContext`b/2]}}, ImageSize -> {200, 200}], Attributes[PlotRange] = {ReadProtected}, $CellContext`volgraph[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Show[{ Plot[ $CellContext`vol[$CellContext`x, $CellContext`a, $CellContext`b], \ {$CellContext`x, 0, Max[$CellContext`a/2, $CellContext`b/2]}, PlotStyle -> Thick, PlotLabel -> Row[{ Style["V", Italic], " = ", $CellContext`vol[ Style["x", Italic], $CellContext`a, $CellContext`b]}]], Graphics[{Thick, Line[{{$CellContext`x0, $CellContext`vol[$CellContext`x0, $CellContext`a, \ $CellContext`b]}, {$CellContext`x0, 0}}]}]}, ImageSize -> {180, 150}], $CellContext`vol[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := ($CellContext`x ($CellContext`a - 2 $CellContext`x)) ($CellContext`b - 2 $CellContext`x), $CellContext`dervolgraph[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Show[{ Plot[ $CellContext`dervol[$CellContext`x, $CellContext`a, \ $CellContext`b], {$CellContext`x, 0, Max[$CellContext`a/2, $CellContext`b/2]}, PlotStyle -> Thick, PlotLabel -> Row[{ Derivative[1][ Style["V", Italic]], " = ", Factor[ $CellContext`dervol[ Style["x", Italic], $CellContext`a, $CellContext`b]]}]], Graphics[{Thick, Line[{{$CellContext`x0, $CellContext`dervol[$CellContext`x0, $CellContext`a, \ $CellContext`b]}, {$CellContext`x0, 0}}]}]}, ImageSize -> {180, 150}], $CellContext`dervol[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a - 2 $CellContext`x0) ($CellContext`b - 2 $CellContext`x0) - ( 2 ($CellContext`a - 2 $CellContext`x0)) $CellContext`x0 - ( 2 ($CellContext`b - 2 $CellContext`x0)) $CellContext`x0, Attributes[Derivative] = {NHoldAll, ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->1669357358], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 8, $CellContext`b$$ = 5, $CellContext`pick$$ = {8, 5, 1}, $CellContext`q$$ = 1, $CellContext`rationalbox$$ = True, $CellContext`x0$$ = 1.5, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`a$$], 8}, 1, 100, 1}, {{ Hold[$CellContext`b$$], 5}, 1, 100, 1}, {{ Hold[$CellContext`x0$$], 1.5, "Vary the height x"}, 0, Dynamic[ Min[$CellContext`a$$/2, $CellContext`b$$/2]]}, {{ Hold[$CellContext`rationalbox$$], True, "Show integer dimensions a and b with rational height for biggest \ box:"}, {False, True}}, {{ Hold[$CellContext`pick$$], {8, 5, 1}, ""}, {{1, 1, Rational[1, 6]}, {8, 3, Rational[2, 3]}, {8, 5, 1}, {15, 7, Rational[3, 2]}, {15, 8, Rational[5, 3]}, {21, 5, Rational[7, 6]}, {21, 16, 3}, {35, 11, Rational[5, 2]}, {35, 24, Rational[14, 3]}, {40, 7, Rational[5, 3]}, {40, 33, 6}, {48, 13, 3}, {48, 35, Rational[20, 3]}, {55, 16, Rational[11, 3]}, {55, 39, Rational[15, 2]}, {65, 9, Rational[13, 6]}, {65, 56, 10}, {77, 32, 7}, {77, 45, Rational[55, 6]}, {80, 17, 4}, {80, 63, Rational[35, 3]}, {91, 40, Rational[26, 3]}, {91, 51, Rational[21, 2]}, {96, 11, Rational[8, 3]}, {96, 85, 15}, {99, 19, Rational[9, 2]}, {99, 80, Rational[44, 3]}}}, {{ Hold[$CellContext`q$$], 1, ""}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}}, { Hold[ PaneSelector[{True -> Column[{"Choose {a, b, optimal x}: ", Manipulate`Place[1], "Choose dimension multiplier: ", Manipulate`Place[2]}], False -> ""}, Dynamic[$CellContext`rationalbox$$]]], Manipulate`Dump`ThisIsNotAControl}}, Typeset`size$$ = { 420., {201.84375, 207.15625}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`a$12704$$ = 0, $CellContext`b$12705$$ = 0, $CellContext`x0$12706$$ = 0, $CellContext`rationalbox$12707$$ = False, $CellContext`pick$12708$$ = 0, $CellContext`q$12709$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 2, StandardForm, "Variables" :> {$CellContext`a$$ = 8, $CellContext`b$$ = 5, $CellContext`pick$$ = {8, 5, 1}, $CellContext`q$$ = 1, $CellContext`rationalbox$$ = True, $CellContext`x0$$ = 1.5}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$12704$$, 0], Hold[$CellContext`b$$, $CellContext`b$12705$$, 0], Hold[$CellContext`x0$$, $CellContext`x0$12706$$, 0], Hold[$CellContext`rationalbox$$, $CellContext`rationalbox$12707$$, False], Hold[$CellContext`pick$$, $CellContext`pick$12708$$, 0], Hold[$CellContext`q$$, $CellContext`q$12709$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ( If[$CellContext`rationalbox$$, $CellContext`a$$ = $CellContext`q$$ Part[$CellContext`pick$$, 1]; $CellContext`b$$ = $CellContext`q$$ Part[$CellContext`pick$$, 2], $CellContext`pick$$ = {1, 1, 1/6}]; If[$CellContext`x0$$ > Min[$CellContext`a$$/2, $CellContext`b$$/2], $CellContext`x0$$ = Min[0.2 $CellContext`a$$, 0.2 $CellContext`b$$]]; Column[{ Text[ TraditionalForm[ Row[{"Tin box holds the most when ", $CellContext`x, " = ", $CellContext`crit[$CellContext`a$$, $CellContext`b$$], "."}]]], Grid[{{ $CellContext`markedtin[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$], $CellContext`tray3D[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$]}, { $CellContext`volgraph[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$], $CellContext`dervolgraph[$CellContext`x0$$, $CellContext`a$$, \ $CellContext`b$$]}}, Frame -> All]}, ItemSize -> {Automatic, 4}, Alignment -> {Center, Center}]), "Specifications" :> {{{$CellContext`a$$, 8}, 1, 100, 1, Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`b$$, 5}, 1, 100, 1, Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`x0$$, 1.5, "Vary the height x"}, 0, Dynamic[ Min[$CellContext`a$$/2, $CellContext`b$$/2]], Appearance -> "Labeled", ControlPlacement -> Top}, {{$CellContext`rationalbox$$, True, "Show integer dimensions a and b with rational height for biggest \ box:"}, {False, True}, ControlPlacement -> Top}, {{$CellContext`pick$$, {8, 5, 1}, ""}, {{1, 1, Rational[1, 6]}, {8, 3, Rational[2, 3]}, {8, 5, 1}, {15, 7, Rational[3, 2]}, {15, 8, Rational[5, 3]}, {21, 5, Rational[7, 6]}, {21, 16, 3}, {35, 11, Rational[5, 2]}, {35, 24, Rational[14, 3]}, {40, 7, Rational[5, 3]}, {40, 33, 6}, {48, 13, 3}, {48, 35, Rational[20, 3]}, {55, 16, Rational[11, 3]}, {55, 39, Rational[15, 2]}, {65, 9, Rational[13, 6]}, {65, 56, 10}, {77, 32, 7}, {77, 45, Rational[55, 6]}, {80, 17, 4}, {80, 63, Rational[35, 3]}, {91, 40, Rational[26, 3]}, {91, 51, Rational[21, 2]}, {96, 11, Rational[8, 3]}, {96, 85, 15}, {99, 19, Rational[9, 2]}, {99, 80, Rational[44, 3]}}, ControlPlacement -> 1}, {{$CellContext`q$$, 1, ""}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, ControlPlacement -> 2}, PaneSelector[{True -> Column[{"Choose {a, b, optimal x}: ", Manipulate`Place[1], "Choose dimension multiplier: ", Manipulate`Place[2]}], False -> ""}, Dynamic[$CellContext`rationalbox$$]]}, "Options" :> { ControlPlacement -> Left, AutorunSequencing -> {1, 2, 3, 4}}, "DefaultOptions" :> {ControllerLinking -> True}], ImageSizeCache->{713., {284., 291.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`crit[ Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a + $CellContext`b - Sqrt[$CellContext`a^2 - $CellContext`a $CellContext`b + \ $CellContext`b^2])/6, $CellContext`markedtin[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Graphics[{Thick, Line[{{0, 0}, {$CellContext`a, 0}, {$CellContext`a, $CellContext`b}, { 0, $CellContext`b}, {0, 0}}], Dashed, Line[{{{$CellContext`x, 0}, {$CellContext`x, $CellContext`x}, { 0, $CellContext`x}}, {{$CellContext`a - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x}, \ {$CellContext`a, $CellContext`x}}, {{$CellContext`a - $CellContext`x, \ $CellContext`b}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x}, {$CellContext`a, $CellContext`b - $CellContext`x}}, {{ 0, $CellContext`b - $CellContext`x}, {$CellContext`x, \ $CellContext`b - $CellContext`x}, {$CellContext`x, $CellContext`b}}}]}, ImageSize -> {200, 200}], $CellContext`tray3D[ Pattern[$CellContext`x, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Graphics3D[{ Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}], {Thick, Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`x, $CellContext`b - $CellContext`x, $CellContext`x}, \ {$CellContext`x, $CellContext`b - $CellContext`x, 0}}], Dashed, Line[{{$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}]}, {Thick, Polygon[{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`a - $CellContext`x, $CellContext`x, $CellContext`x}, \ {$CellContext`a - $CellContext`x, $CellContext`x, 0}}], Dashed, Line[{{$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}]}, {Thick, Polygon[{{$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, \ $CellContext`x}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`a - $CellContext`x, \ $CellContext`b - $CellContext`x, 0}}], Dashed, Line[{{$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}}]}, { Thick, Polygon[{{$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`x, $CellContext`b - \ $CellContext`x, $CellContext`x}, {$CellContext`x, $CellContext`b - \ $CellContext`x, 0}}], Dashed, Line[{{$CellContext`x, $CellContext`b - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}}]}, {Thick, Line[{{{$CellContext`x, $CellContext`x, 0}, {$CellContext`x, 0, 0}, {$CellContext`a - $CellContext`x, 0, 0}, {$CellContext`a - $CellContext`x, $CellContext`x, 0}, {$CellContext`a, $CellContext`x, 0}, {$CellContext`a, $CellContext`b - $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b - \ $CellContext`x, 0}, {$CellContext`a - $CellContext`x, $CellContext`b, 0}, {$CellContext`x, $CellContext`b, 0}, {$CellContext`x, $CellContext`b - $CellContext`x, 0}, { 0, $CellContext`b - $CellContext`x, 0}, { 0, $CellContext`x, 0}, {$CellContext`x, $CellContext`x, 0}}}]}}, PlotRange -> {{0, $CellContext`a}, {0, $CellContext`b}, {0, Min[$CellContext`a/2, $CellContext`b/2]}}, ImageSize -> {200, 200}], Attributes[PlotRange] = {ReadProtected}, $CellContext`volgraph[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] := Show[{ Plot[ $CellContext`vol[$CellContext`x, $CellContext`a, $CellContext`b], \ {$CellContext`x, 0, Max[$CellContext`a/2, $CellContext`b/2]}, PlotStyle -> Thick, PlotLabel -> Row[{ Style["V", Italic], " = ", $CellContext`vol[ Style["x", Italic], $CellContext`a, $CellContext`b]}]], Graphics[{Thick, Line[{{$CellContext`x0, 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Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a - 2 $CellContext`x0) ($CellContext`b - 2 $CellContext`x0) - ( 2 ($CellContext`a - 2 $CellContext`x0)) $CellContext`x0 - ( 2 ($CellContext`b - 2 $CellContext`x0)) $CellContext`x0, Attributes[Derivative] = {NHoldAll, ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->15267851], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 45, $CellContext`b$$ = 24, $CellContext`pick$$ = {15, 8, Rational[5, 3]}, $CellContext`q$$ = 3, $CellContext`rationalbox$$ = True, $CellContext`x0$$ = 4.295999999999999, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", 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$CellContext`b], {$CellContext`x, 0, Max[$CellContext`a/2, $CellContext`b/2]}, PlotStyle -> Thick, PlotLabel -> Row[{ Derivative[1][ Style["V", Italic]], " = ", Factor[ $CellContext`dervol[ Style["x", Italic], $CellContext`a, $CellContext`b]]}]], Graphics[{Thick, Line[{{$CellContext`x0, $CellContext`dervol[$CellContext`x0, $CellContext`a, \ $CellContext`b]}, {$CellContext`x0, 0}}]}]}, ImageSize -> {180, 150}], $CellContext`dervol[ Pattern[$CellContext`x0, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]] = ($CellContext`a - 2 $CellContext`x0) ($CellContext`b - 2 $CellContext`x0) - ( 2 ($CellContext`a - 2 $CellContext`x0)) $CellContext`x0 - ( 2 ($CellContext`b - 2 $CellContext`x0)) $CellContext`x0, Attributes[Derivative] = {NHoldAll, ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], 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