Will Meador & Matt HopperRelated Rates: Filling Conesrestart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):r25:=rand(2..5):
r13:=rand(1..3):ask:=proc()
global V0, rtermsh, V0h, dV0hdt, solution;
local R0, d0, h0, dV0, plotlist, t, coneplot, waterconeplot, watertopplot;
R0:=r25();
d0:=r25();
h0:=r13()+d0;
dV0:=r25();
V0:=(1/3)*Pi*r^2*h;
rtermsh:=R0/h0*h;
V0h:=1/3*R0^2/h0^2*Pi*h^3;
dV0hdt:=R0^2/h0^2*Pi*h^2*(dh/dt);
solution:=dV0/R0^2/Pi*h0^2/d0^2;
Set(question= sprintf(
"Water is being poured into a cone with a radius of %a meters and a height of %a meters and at the rate of %a cubic meters per minute. When the depth is %a meters what is the rate of increase of the depth?",
R0, h0, dV0, d0));
Set(ave="");
Set(rth="");
Set(newequation="");
Set(dvdt="");
Set(finalanswer="");
plotlist:=NULL;
for t from 1 to 15 do
coneplot:=plot3d([r*cos(theta),r*sin(theta),h0*r/R0], r=0..R0, theta=0..2*Pi, transparency=.5, color=red, grid=[5,25]):
waterconeplot:=plot3d([r*cos(theta),r*sin(theta),h0*r/R0], r=0..t*R0/15, theta=0..2*Pi, color=blue, grid=[2,25]):
watertopplot:=plot3d([r*cos(theta),r*sin(theta),t*h0/15], r=0..t*R0/15, theta=0..2*Pi, color=blue, grid=[2,25]):
plotlist:=plotlist,display(coneplot,waterconeplot,watertopplot, scaling=constrained, axes=normal, tickmarks=[[-R0,R0],[-R0,R0],[h0]], orientation=[30,65]);
end do:
Set(ConePlot=display([plotlist], insequence=true));
end proc:check1:=proc()
global V0;
local userV0;
userV0:=Get(ave::anything):
if userV0=V0
then Set(reply="Correct! Try another question.",
reply(background)=green, reply(foreground)=black):
else Set(reply="Incorrect. Try again.",
reply(background)=red, reply(foreground)=black):
end if;
end proc:show1:=proc()
global V0;
Set(ave=V0);
Set(reply="Try it on your own next time!",
reply(background)=yellow, reply(foreground)=black):
end proc:check2:=proc()
global rtermsh;
local userrth;
userrth:=Get(rth::anything):
if userrth=rtermsh
then Set(reply="You did it! Move on to the next step!",
reply(background)=green, reply(foreground)=black):
else Set(reply="A valiant effort, but try again!",
reply(background)=red, reply(foreground)=black):
end if;
end proc:show2:=proc()
global h0, R0, h, rtermsh;
Set(rth=rtermsh);
Set(reply="Try it on your own next time!",
reply(background)=yellow, reply(foreground)=black):
end proc:check3:=proc()
global V0h;
local userV0h;
userV0h:=Get(newequation::anything):
if userV0h=V0h
then Set(reply="You've got the right equation! Go to the next step!",
reply(background)=green, reply(foreground)=black):
else Set(reply="You don't quite have it!",
reply(background)=red, reply(foreground)=black):
end if;
end proc:show3:=proc()
global V0h;
Set(newequation=V0h);
Set(reply="Try it on your own next time!",
reply(background)=yellow, reply(foreground)=black):
end proc:check4:=proc()
global dV0hdt;
local userdV0hdt;
userdV0hdt:=Get(dvdt::anything):
if userdV0hdt=dV0hdt
then Set(reply="You've got the right derivative! Only one step left!",
reply(background)=green, reply(foreground)=black):
else Set(reply="Sorry, try again!",
reply(background)=red, reply(foreground)=black):
end if;
end proc:show4:=proc()
global dV0hdt;
Set(dvdt=dV0hdt);
Set(reply="Try it on your own next time!",
reply(background)=yellow, reply(foreground)=black):
end proc:check5:=proc()
global solution;
local usersolution;
usersolution:=Get(finalanswer::anything):
if usersolution=solution
then Set(reply="Goodness Gracious, Outer Spacious! That's correct! Are you brave enough to try again?",
reply(background)=green, reply(foreground)=black):
else Set(reply="Check your calculations, they might have gone askew.",
reply(background)=red, reply(foreground)=black):
end if;
end proc:show5:=proc()
global solution;
Set(finalanswer=solution);
Set(reply="Try it on your own next time!",
reply(background)=yellow, reply(foreground)=black):
end proc:FillingCones:=Maplet(onstartup = RunWindow(MAIN),
Window[MAIN](title="Related Rates: Filling Cones",
[ halign=none,
[ Button("New Problem", Evaluate(function = "ask"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ TextBox[question](width=80, height=3, editable=false, background=wheat)
],
[ inset=0,
[ inset=0,
[ halign=none, border=true,
caption="Step 1: Enter the formula for the Volume of a Cone. ",
"V =",
TextField[ave](width=25, background=wheat),
HorizontalGlue(),
Button("Check", Evaluate(function = "check1"), background=green),
Button("Show", Evaluate(function = "show1"), background=yellow)
],
[ halign=none, border=true,
caption="Step 2: Set up a proportion for r in terms of h.",
"r =",
TextField[rth](width=25, background=wheat),
HorizontalGlue(),
Button("Check", Evaluate(function = "check2"), background=green),
Button("Show", Evaluate(function = "show2"), background=yellow)
],
[ halign=none, border=true,
caption="Step 3: Express the Volume as a function of h.",
"V =",
TextField[newequation](width=25, background=wheat),
HorizontalGlue(),
Button("Check", Evaluate(function = "check3"), background=green),
Button("Show", Evaluate(function = "show3"), background=yellow)
],
[ halign=none, border=true,
caption="Step 4: Take the derivative of the Volume. Type dh/dt for the derivative of h.",
"dV/dt =",
TextField[dvdt](width=25, background=wheat),
HorizontalGlue(),
Button("Check", Evaluate(function = "check4"), background=green),
Button("Show", Evaluate(function = "show4"), background=yellow)
],
[ halign=none, border=true,
caption="Step 5: Plug in the known values and solve for dh/dt.",
"dh/dt =",
TextField[finalanswer](width=25, background=wheat),
HorizontalGlue(),
Button("Check", Evaluate(function = "check5"), background=green),
Button("Show", Evaluate(function = "show5"), background=yellow)
]
],
[ halign=none, border=true,
caption="Animation",
[ Button[BPlay]("Play", onclick=SetOption(ConePlot(play)=true), background=tan),
Button[BPause]("Pause", onclick=SetOption(ConePlot(pause)=true), background=tan),
Button[BForward]("Forward", onclick=Action(
SetOption(ConePlot(pause)=true),
SetOption(ConePlot(frame_forward)=true)), background=tan),
Button[BReverse]("Reverse", onclick=Action(
SetOption(ConePlot(pause)=true),
SetOption(ConePlot(frame_backwards)=true)), background=tan)
],
Plotter[ConePlot](height=250, width=250)
]
],
[ TextField[reply](width=60, editable=false)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Will Meador & Matt Hopper", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
)):Maplets[Display](FillingCones);