Example 1: Related Rates

A Boat Approaching a Dock

Manipulate[
Column[{

Grid[{{
Column[{
Text@Grid[{{"speed at which rope is being pulled = 1 m/s"}}],
Text@Row[{"(solid line)"}],
"",
Text@Grid[{{"speed of boat approaching dock = ",
SetAccuracy[(((( v)^2)+1)^(1/2))/(v),4]," m/s"
}}],
Text@Row[{"(dashed line)"}]
}],
"  ",
Show[
Plot[(((( v)^2)+1)^(1/2))/(v),{v,0,8},AxesLabel->{"distance","speed"},PlotRange->{{0,8},{0,10}},AspectRatio->1],
Graphics[Line[{{v,0},{v,(((( v)^2)+1)^(1/2))/(v)}}]],
Graphics[Line[{{0,(((( v)^2)+1)^(1/2))/(v)},{v,(((( v)^2)+1)^(1/2))/(v)}}]],
ImageSize->{250,250},ImagePadding->{{25,40},{25,20}}]
}},ItemSize->{{20,4,20},Automatic},Alignment->Left],

Graphics[{
{Blue,Polygon[{{-5,-2},{-5,-1},{15,-1},{15,-2}}]},
{Hue[.5],Polygon[{{-5,-1},{-5,3},{15,3},{15,-1}}]},
{Red,Polygon[{{(8-v)-4.8,0},{(8-v)-4,-1},{(8-v)-1,-1},{(8-v)-.2,0}}]},{Red,Polygon[{{(8-v)-2.65,0},{(8-v)-2.65,2.5},{(8-v)-2.35,2.5},{(8-v)-2.35,0}}]},
{White,Polygon[{{(8-v)-2.65,2.3},{(8-v)-4.3,.6},{(8-v)-3.1,0}}]},
{White,Polygon[{{(8-v)-2.35,2.3},{(8-v)-.7,.6},{(8-v)-1.8,0}}]},
Line[{{(8-v)-4.6,-.2},{(8-v)-.4,-.2}}],
{Dashed,Line[{{(8-v),0},{8,0}}]},
Line[{{(8-v),0},{8,1}}],
{Brown,Polygon[{{8,-1},{8,1},{8.2,1},{8.2,-1}}]},
{Brown,Polygon[{{8,-1},{8,0},{8.5,0},{8.5,-1}}]},
{Brown,Polygon[{{8,0},{8,.2},{15,.2},{15,0}}]},
{Brown,Polygon[{{14.5,-1},{14.5,0},{15,0},{15,-1}}]}
},PlotRange->{{-5,15},{-2,3}},ImageSize->590]
}],
{{v,8,"distance of boat from dock (m): "},11,.01}
]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Related Rates: A Boat Approaching a Dock" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/RelatedRatesABoatApproachingADock/

Contributed by: Jamie Ding (Torrey Pines High School)

Example 2: Related Rates

Related Rates Clock

Manipulate[Column[{
Graphics[{
{Opacity[0.5],Circle[{0,0},8]},
{Dashed,Opacity[0.3],Circle[{0,0},4]},
{Thickness[0.008],Arrow[{{0,0},.8{minuteY[t],minuteX[t]}}]},
{Thickness[0.012],Arrow[{{0,0},{hourY[t],hourX[t]}}]},
{Dashed,Lighter[Red,t/120],Line[{.8{minuteY[t],minuteX[t]},{hourY[t],hourX[t]}}]},
Text[Style["12",12,Bold,"Label"],{0,7}],Text[Style["3",12,Bold,"Label"],{7,0}],Text[Style["6",12,Bold,"Label"],{0,-7}],Text[Style["9",12,Bold,"Label"],{-7,0}],
{Opacity[0.2],Line[{{7.5,0},{-7.5,0}}]},
{Opacity[0.2],Line[{{0,-7.5},{0,7.5}}]}
},
ImageSize->300,PlotRange->15],Style[Row[{"distance = ",ToString[Chop[SetAccuracy[Distance[t],3],.01]]}],12,"Label"],Style[Row[{"rate of change = ",ToString[Chop[SetAccuracy[rateDistance[t],3],.01]]}],12,"Label"]},Center],
{{t,5,"time in hours"},0,11.999999}
,Initialization:>{minuteX[t_] := 8 Cos[(2Pi) t];
minuteY[t_] := 8 Sin[(2Pi) t];
hourX[t_] := 4 Cos[(Pi/6)t];
hourY[t_] := 4 Sin[(Pi/6)t];
distanceX[t_] := hourX[t] - minuteX[t];
distanceY[t_] := hourY[t] - minuteY[t];Distance[t_] := (distanceX[t]^2 + distanceY[t]^2)^0.5;
rateDistance[t_] = D[Distance[t],t];}
]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Related Rates Clock" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/RelatedRatesClock/

Contributed by: Ryan Chuang and Jeffrey Tsao (Torrey Pines High School)

Example 3: Related Rates

A Snowball' s Rate of Change

Manipulate[Module[{nn},nn[x_]:=NumberForm[x,{4,2}];
Deploy[
Grid[{
{Graphics3D[Sphere[{0,0,0},Sqrt[(-t+400*Pi)/(4*Pi)]],PlotRange->10,Boxed->False,PlotLabel->"snowball",ImageSize->100],Show[ParametricPlot[{-t+400Pi,((-t+400Pi)/Pi)^0.5},{t,0,400Pi},AspectRatio->1/3,AxesOrigin->{0,0}],Graphics[{RGBColor[1,.21,0],Arrow[{{-t+400Pi,((-t+400Pi)/Pi)^0.5},{-t+400Pi,0}}]},PlotRange->400Pi],Graphics[{RGBColor[1,.21,0],Arrow[{{-t+400Pi,((-t+400Pi)/Pi)^0.5},{0,((-t+400Pi)/Pi)^0.5}}]}],ImageSize->175,
PlotLabel->"diameter versus surface area"]},
{Show[{
Plot[-t+400Pi,{t,0,400Pi}],
Graphics[{RGBColor[1,.21,0],Line[{{t,-t+400Pi},{t,0}}]}],
Graphics[{RGBColor[1,.21,0],Arrow[{{t,-t+400Pi},{0,-t+400Pi}}]}]},PlotLabel->Column[{"surface area S versus time",nn[-t+400Pi]}],PlotRange->{0,400Pi},ImageSize->175],
Show[{
Plot[((-t+400Pi)/Pi)^0.5,{t,0,400Pi}],
Graphics[{RGBColor[1,.21,0],Line[{{t,((-t+400Pi)/Pi)^0.5},{t,0}}]},PlotRange->400Pi],Graphics[{RGBColor[1,.21,0],Arrow[{{t,((-t+400Pi)/Pi)^0.5},{0,((-t+400Pi)/Pi)^0.5}}]}]},PlotLabel->Column[{"diameter D versus time",nn[((-t+400Pi)/Pi)^0.5]}],ImageSize->175,PlotRange->{{0,400Pi},{0,20}}]},{Show[{Plot[-1,{t,0,400Pi}],Graphics[{RGBColor[.67,.75,.15],Arrow[{{t,-1},{t,0}}]}]},PlotRange->{-2,2},PlotLabel->Column[{"(dS/dt)",nn@t}],ImageSize->175],
Show[{Plot[(-1/(2*Pi))*Sqrt[Pi/(400*Pi-t)],{t,0,410Pi}],Graphics[{RGBColor[.67,.75,.15],Arrow[{{t,(((-t+400Pi)/Pi)^0.5)/(2*(-t+400Pi))},{t,-.1}}]}],
Graphics[{RGBColor[.67,.75,.15],Arrow[{{t,(((-t+400Pi)/Pi)^0.5)/(2*(-t+400Pi))},{0,(((-t+400Pi)/Pi)^0.5)/(2*(-t+400Pi))}}]}]},PlotLabel->Column[{"(dD/dt)",nn[(((-t+400Pi)/Pi)^0.5)/(2*(-t+400Pi))]}],AxesOrigin->{0,-.035},PlotRange->{Automatic,{-.035,-.005}},ImageSize->175]}}]]],{{t,0,"time t"},0,400Pi-0.00001,Appearance->"Labeled"}]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"A Snowball's Rate of Change" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/ASnowballsRateOfChange/

Contributed by: Cindy Piao and Karen Ye (Torrey Pines High School)

Example 4: Related Rates

Related Rates : Triangle Angle and Area

Manipulate[
Column[{Row[{Text@Style["rate of change of area = ",Bold,Darker[Blue,.4]],10Cos[w].06}],Graphics[{{LightBlue,EdgeForm[Black],Disk[{0,0},1,{0,w}]},{Thick,Line[{{0,0},{5,0},{4Cos[w],4Sin[w]},{0,0}}]}},ImageSize->300,PlotRange->{{-1,6},{-1,5}}]}],{{w,π/8,"angle θ"},0,π/2,π/36,Appearance->"Labeled"}]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Related Rates: Triangle Angle and Area" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/RelatedRatesTriangleAngleAndArea/

Contributed by: Kevin Balch (Torrey Pines High School)

Example 5: Related Rates

Relation of Radius, Surface Area, and Volume of a Sphere

Manipulate[TraditionalForm@Text@Grid[{{Style[Column[{
Row[{"Take the derivative of "}],
Row[{V," = ",((4 π (r^3))/3)," to get ",HoldForm@D[V,t]," = ", 4Pi (r^2)HoldForm@D[r,t],"."}],
Row[{"Using ",HoldForm@D[V,t]," = ",1," and solving, ",HoldForm@D[r,t]," = ", (1/(4 π (r^2))),"."}],
"",
Row[{"Also,  ",A," = ", 4Pi (r^2),","}],
Row[{"so ",HoldForm@D[A,t]," = ", 8 Pi r HoldForm@D[r,t]," = ",((HoldForm[8 Pi r])/(HoldForm[4 Pi (r^2)]))," = ",((8 Pi r)/(4 Pi (r^2))),"."}],"",
Plot[{1,(1/((6^(2/3)) (π^(1/3)) (t^(2/3)))),((2 (2^(2/3)) ((((π/3)))^(1/3)))/(t^(1/3)))},{t,0,10},
Axes->{True,False},Epilog->Line[{{tt,0},{tt,1000}}],PlotStyle->Thickness[Medium],PlotLabel->Row[MapIndexed[Style[HoldForm[D[#1,t]],ColorData[1,First[#2]]]&,{V,r,A}],",   "],ImageSize->300]},Center],14],

Column[{Graphics[{ColorData[1,1],Rectangle[{0,0},{(((3 tt)/(4 π ))^(1/3)),.01}]},PlotRange->{{0,1.366},{0,.01}},Axes->{True,False},ImageSize->250,PlotLabel->Style["radius","Label"]],
Graphics[{ColorData[1,2],Rectangle[{0,0},{(Sqrt[2 π]),(1/(Sqrt[2 π]))}((36π (tt^2))^(1/6))]},PlotRange->{{0,11.9},{0,1.89}},ImageSize->250,Axes->True,PlotLabel->Style["surface area","Label"]],
Graphics3D[{ColorData[1,3],Specularity[White,100],Sphere[{0,0,0},(((3 tt)/(4 π ))^(1/3))]},Lighting->"Neutral",Epilog->Inset[Graphics[{},Axes->True,PlotRange->1.336,ImagePadding->0],{.5,.5},{0,0},1],
Boxed->False,ImageSize->250,PlotRange->1.36,SphericalRegion->True,ViewAngle->.298,PlotLabel->Style["volume","Label"]]},Center,2]}}],
{{tt,5,"time"},0,10}]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Relation of Radius, Surface Area, and Volume of a Sphere" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/RelationOfRadiusSurfaceAreaAndVolumeOfASphere/

Contributed by: Joe Bolte

Example 6: Related Rates

Person Walking Away from Spotlight

Manipulate[
Grid[{{
Text@Style[Column[{"Conditions:","  - Spotlight is 12 meters away ","     from the wall.","  - The person is walking at 1.6","     meters per second.","  - The person is 2 meters tall.","","Distances and Rate:",Row[{"meters away from wall = ",If[a≠12,12-a,"SPLAT!!"]}],Row[{"shadow height = ",24/a}],Row[{"shadow decrease rate = ",-24/(a^2)*1.6}]}],"Label"],
Deploy[Show[{
Plot[2*x/a,{x,0,0.5},Axes->False,PlotRange->{{0,13},{-1,20}}],
Graphics[{
{Blend[{Blue,Black},0.2],Polygon[{{0,.8},{0,20},{12,24},{.5,1}}]},
{Yellow,Opacity[0.8],Polygon[{{0,0},{12,24},{12,24/a}}]},
{Yellow,Opacity[0.8],Polygon[{{0,0},{a,2},{a+0.2,0}}]},
{Blend[{White,Gray},0.5],Polygon[{{12,0},{12,20},{13,20},{13,0}}]},
{Opacity[0.4],Blend[{Yellow,Black},0.5],Polygon[{{a+0.2,0},{a,2},{12,24/a},{12,0}}]},
{Gray,Polygon[{{0,0},{0.5,0},{.5,1},{0,.8}}]},
{White,Thickness[0.01],Line[{{.5,0},{.5,1}}]},
{Red,Thickness[0.015],Line[{{a,1},{a,2}}]},
{Red,PointSize[0.03],Point[{a+0.05,2}]},
{Red,Thickness[0.01],Line[{{a,1},{a-0.25*Cos[2*a],0}}]},
{Red,Thickness[0.01],Line[{{a,1},{a+0.25*Cos[2*a],0}}]},
{Red,Thickness[0.007],Line[{{a,1.7},{a-0.2*Cos[2*a],1}}]},
{Red,Thickness[0.007],Line[{{a,1.7},{a+0.2*Cos[2*a],1}}]},
Line[{{12,0},{12,20}}],
{Thickness[0.005],Line[{{a+0.2,-0.1},{12,-0.1}}]},
{Thickness[0.009],Line[{{12,0},{12,23/a}}]},
{Thickness[.001],Line[{{0,0},{12,0}}]},
{White,Disk[{3,17},2]},
{Blend[{Blue,Black},0.2],Disk[{4,17},2.5]},
{White,Disk[{5,15},0.1]},
{White,Disk[{6,18},0.1]},
{White,Disk[{3,17},0.1]},
{White,Disk[{2,9},0.1]},
{White,Disk[{1,5},0.1]},
{White,Disk[{4,10},0.1]},
{White,Disk[{3,7},0.1]},
{White,Disk[{2,11},0.1]},
{White,Disk[{1,13},0.1]},
{White,Disk[{4,13},0.1]},
{White,Disk[{7,16},0.1]}
}]
},ImageSize->{350,400},AspectRatio->Automatic]]
}},ItemSize->{{16,24},Automatic},Alignment->{Left}],
{{a,4,"distance from spotlight"},1,12}]

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Person Walking Away from Spotlight" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/PersonWalkingAwayFromSpotlight/

Contributed by: James Che, Brian Alford, and Mani Ramachandran (Torrey Pines High School)

Based on a problem in: J. Stewart, Calculus: Early Transcendentals, 4th ed., Pacific Grove: Brooks/Cole, 1999, p. 257, #10.