ODE Mixing ProblemsJonathan Garcia, Michael Finerestart; with(Maplets[Tools]): with(Maplets[Elements]): with(plots):StartEngine();lightorange:="#FFB300":great:=[
" Bullseye. ",
" Right on the mark. ",
" Great. ",
" That's absolutely right. ",
" Awesome. You're unstoppable. ",
" Correct. ",
" WOW, nailed that one. ",
" You got it! ",
" YES! Right on. ",
" Way to go. ",
" Don't stop now. ",
" I knew you could do it! ",
" Perfect. You're unstoppable. ",
" Radical Dude. ",
" Power house. ",
" Sweet! Keep on Rockin'. ",
" You rock! ",
" You got those mad skills. ",
" Awesome! ",
" Wowzers! That was great! ",
" Awesome man! ",
" Good job! You really know what you are doing. ",
" You're awesome kid. ",
" Tearing it up. ",
" Right on the money. ",
" High Five!!!!! ",
" Cool Beans. ",
" What a nerd! ",
" You're so money. ",
" You're a GENIUS!",
" You hit the nail on the head. ",
" Excellent! ",
" If I had a cookie, I'd give it to you. "," You're a winner. "," That's a winner. "," Whoop! Way to go. "," That's the way, ah ha ah ha, I like it. "," Oh yeah! "," You're the bomb! "," That was fantastic! "," Oh Snap! "," That's krunk! "]:randgreat:=rand(1..nops(great)):nextquest:=[" Keep it up."," Keep going."," On to the next problem."," Try another differential equation."," Solve another differential equation."]:randnext:=rand(1..nops(nextquest)):sorry:=[" Sorry, that's not right. "," Nope, but don't give up. "," Not quite, it's a little off. "," Bummer, thought you would get it this time. "," Close, but no cigar. "," Slightly off. "," If you need help, please ask a human. "," I don't know where you went wrong. "," Sorry, killer problem. "," No, but don't blow it off. "," D'oh! "," That's whack. "," Yikes! Really close. You'll get it next time. "," Pucha! Thought you would get it this time. "," Nice try, but your answer is wrong. "," Ask your neighbor for help. "," Brutal kid. "," That won't fly. "," So close and yet so far. "," So, that was just practice. "," Ooooh...close. "," A swing and a miss. "," Better luck next time. "," Oops. "," Tilt. Play again. "," I know you can do better. "," Sorry Charlie. "," Better luck next time. "," Tisk, tisk. It's a tough one. "," It takes a lot of wrongs to make a right. "," Even Einstein didn't get everything right. "," Oh man, you were so close. "," Today's not your lucky day. "," Don't quit your day job. "," You're colder than a polar bear's toenail. "," Back to the drawing board. "]:randsorry:=rand(1..nops(sorry)):tryagain:=[" Fix your answer and check it again."," Give it another try."," Try again."," Please try again."]:randtry:=rand(1..nops(tryagain)):showit:=[" OK That's just practice. But try doing it yourself."," That's fine for practice. Try doing it yourself."," Think about it and enter an answer yourself."," Next time, compute an answer yourself."," So that was practice. Try doing it yourself."]:randshow:=rand(1..nops(showit)):firsthints:=[
" The rate the solute enters the vat depends on the concentration and rate of the incoming solution.",
" The rate the solute enters the vat depends on the combined rates and concentrations of the incoming solutions. Remember that each bucket contributes separately.",
" Remember that the total volume of solution remains constant since solution is draining at the same volumetric rate as it is pouring in. Use this information to find the rate at which the solute leaves the vat.",
" Click check and move on to the next section."
]:randh1:=rand(1..(nops(firsthints)-1)):bounds:=plot([[0,0],[0,20],[20,20],[20,0],[0,0]], color=white):
vat:=plot([[1,8],[4,2],[16,2],[19,8],[1,8]], color=black):
bucket1:=plot([[1,15],[2,12],[5,12],[3.33,17],[1,15]], color=black):
bucket2:=plot([[19,15],[18,12],[15,12],[16.67,17],[19,15]], color=black):
vatwater:=display([
polygonplot([[1.5,7],[18.5,7],[16,2],[4,2],[1.5,7]], color=cyan),
plot([[1.5,7],[18.5,7]], color=navy)
]):
stream1:=display([
polygonplot([[1.33,14],[5,14],[6,13],[7,11],[7.5,9],[7.75,7],[5,7],[5,12],[2,12],[1.33,14]], color=cyan),
plot([[[1.33,14],[5,14],[6,13],[7,11],[7.5,9],[7.75,7]],[[5.5,12],[6,10.5],[6.25,8],[6.25,7]],[[5,12],[5,7]],[[5,7],[7.75,7]]], color=[blue,blue,blue,cyan])
]):
stream2:=display([
polygonplot([[18.67,14],[15,14],[14,13],[13,11],[12.5,9],[12.25,7],[15,7],[15,12],[18,12],[18.67,14]], color=cyan),
plot([[[18.67,14],[15,14],[14,13],[13,11],[12.5,9],[12.25,7]],[[14.5,12],[14,10.5],[13.75,8],[13.75,7]],[[15,12],[15,7]],[[15,7],[12.25,7]]], color=[blue,blue,blue,cyan])
]):
spicket:=display([
polygonplot([[[16,3],[17,3],[17,2],[16,2],[16,3]],[[17,3],[18,2.75],[19,2],[19.67,1],[20,0],[18,0],[17.75,1],[17.5,1.67],[17,2],[17,3]]], color=cyan),
plot([[[17,3],[18,2.75],[19,2],[19.67,1],[20,0]],[[18,2],[18.5,1],[19,0]],[[17,2],[17.5,1.67],[17.75,1],[18,0]]], color=blue)
]):nb:=rand(1..2):
gm:=rand(2):
vv:=rand(5..100) * 10:
c:=rand(0..20):
r:=rand(1..20):
tf:=rand(10..60):Procedure used to establish all variables and give problem statement.start:=proc()
global numbuckets, findtime, givemass, vatvolume, initconc, conc, rate, finalc, finalt,
initmass, ratein, rateout, diffeq, S, solution:
local situation, question:
Set(inrate="", inrate(editable)=true):
Set(outrate="", outrate(editable)=true):
Set(hinta(enabled)=true, checka(enabled)=true, showa(enabled)=true):
Set(fb1="", fb1(background)=white):
Set(TFdeq="", TFdeq(editable)=false):
Set(hintb(enabled)=false, checkb(enabled)=false, showb(enabled)=false):
Set(fb2="", fb2(background)=white):
Set(massinit="", massinit(editable)=false):
Set(hintc(enabled)=false, checkc(enabled)=false, showc(enabled)=false):
Set(fb3="", fb3(background)=white):
Set(eqn="", eqn(editable)=false):
Set(hintd(enabled)=false, checkd(enabled)=false, showd(enabled)=false):
Set(fb4="", fb4(background)=white):
Set(answer="", answer(editable)=false):
Set(hinte(enabled)=false, checke(enabled)=false, showe(enabled)=false):
Set(fb5="", fb5(background)=white):
Set(feedback="", feedback(background)=white):
numbuckets, givemass:=nb(), gm():
if numbuckets=1 then
rate[2]:=0
else
rate[2]:=r()
end if:
vatvolume, initconc, conc[1], rate[1], conc[2], finalt:=vv(), c(), c(), r(), c(), tf():
while conc[1]=initconc do conc[1]:=c(); end do:
if numbuckets=2 then
if conc[1]=0 then conc[1]:=1 end if:
if conc[2]=0 then conc[2]:=2 end if:
end if:
initmass:=initconc * vatvolume:
ratein:=conc[1] * rate[1] + conc[2] * rate[2]:
rateout:=(rate[1] + rate[2]) * S(t) / vatvolume:
diffeq:=diff(S(t), t) = ratein - rateout:
solution:=dsolve([diffeq, S(0)=initmass], S(t)):
finalc:=evalf(subs(t=finalt, rhs(solution))/vatvolume):
if numbuckets=1 then
if conc[1]=0 then
situation:=cat(
" A bucket is pouring pure water into the vat at a rate of ",
rate[1],
" L/min.")
else
situation:=cat(
" A bucket is pouring solution of concentration ",
conc[1],
" mol/L into the vat at a rate of ",
rate[1],
" L/min.")
end if:
else
situation:=cat(
" Two buckets are pouring solution into the vat, the first pouring solution of concentration ",
conc[1],
" mol/L at a rate of ",
rate[1],
" L/min and the second pouring ",
conc[2],
" mol/L at ",
rate[2],
" L/min.")
end if:
question:=cat(
" Find the concentration of the solution in the vat at time t = ",
finalt,
" min."):
if initconc=0 then
Set(problem=cat(
"A large vat is initially filled with ",
vatvolume,
" L of pure water. ",
situation,
" The solution is kept thoroughly mixed, and it drains at a rate of ",
(rate[1]+rate[2]),
" L/min. Let S(t) be the total amount of solute dissolved in the vat at time t. ",
question)):
else
Set(problem=cat(
"A large vat is initially filled with ",
vatvolume,
" L of solution with a concentration of ",
initconc,
" mol/L. ",
situation,
" The solution is kept thoroughly mixed, and it drains at a rate of ",
(rate[1]+rate[2]),
" L/min. Let S(t) be the total amount of solute dissolved in the vat at time t. ",
question)):
end if:
if numbuckets=1 then
Set(picture=display([bounds, vatwater, stream1, vat, bucket1, spicket], axes=none)):
else
Set(picture=display([bounds, vatwater, stream1, stream2, vat, bucket1, bucket2, spicket], axes=none)):
end if:
Set(Lfintime(caption)=sprintf(" at time t = %a min.", finalt));
end proc:hint1:=proc()
local userin, userout, inright, outright, index:
userin:=Get(inrate::anything):
userout:=Get(outrate::anything):
inright:=evalb(userin=ratein):
outright:=evalb(userout=rateout):
index:=randh1():
if numbuckets=1 and index=2 then
index:=1
elif numbuckets=2 and index=1 then
index:=2
end if:
if inright and (index=1 or index=2) then
index:=3:
end if:
if outright and index=3 then
index:=4:
end if:
Set(fb1="hint",
fb1(background)=lightorange, fb1(foreground)=black):
Set(feedback=firsthints[index],
feedback(background)=lightorange, feedback(foreground)=black):
end proc:check1:=proc()
global ratein, rateout;
local userin, userout, inright, outright:
userin:=Get(inrate::anything):
userout:=Get(outrate::anything):
inright:=evalb(userin=ratein):
outright:=evalb(userout=rateout):
if inright then
Set(inrate(editable)=false)
end if:
if outright then
Set(outrate(editable)=false)
end if:
if inright and outright then
Set(fb1="correct",
fb1(background)=green, fb1(foreground)=black):
Set(feedback=great[randgreat()],
feedback(background)=green, feedback(foreground)=black):
Set(TFdeq(editable)=true):
Set(hintb(enabled)=true, checkb(enabled)=true, showb(enabled)=true):
elif inright then
Set(fb1="almost",
fb1(background)=lightorange, fb1(foreground)=black):
Set(feedback=cat(" Your Rate Out is wrong. Fix it.", tryagain[randtry()]),
feedback(background)=lightorange, feedback(foreground)=black):
elif outright then
Set(fb1="almost",
fb1(background)=lightorange, fb1(foreground)=black):
Set(feedback=cat(" Your Rate In is wrong. Fix it.", tryagain[randtry()]),
feedback(background)=lightorange, feedback(foreground)=black):
else
Set(fb1="incorrect",
fb1(background)=red, fb1(foreground)=wheat):
Set(feedback=sorry[randsorry()],
feedback(background)=red, feedback(foreground)=wheat):
end if:
end proc:show1:=proc()
global ratein, rateout;
Set(inrate=ratein, inrate(editable)=false):
Set(outrate=rateout, outrate(editable)=false):
Set(fb1="shown",
fb1(background)=lightorange, fb1(foreground)=black):
Set(feedback=showit[randshow()],
feedback(background)=lightorange, feedback(foreground)=black):
Set(TFdeq(editable)=true):
Set(hintb(enabled)=true, checkb(enabled)=true, showb(enabled)=true):
end proc:hint2:=proc()
Set(fb2="hint",
fb2(background)=lightorange, fb2(foreground)=black):
Set(feedback=" The rate of change of the amount of solute is equal to the rate of inflow minus the rate of outflow.",
feedback(background)=lightorange, feedback(foreground)=black):
end proc:check2:=proc()
global diffeq;
local userdiff:
userdiff:=Get(TFdeq::anything):
if userdiff=rhs(diffeq) then
Set(fb2="correct",
fb2(background)=green, fb2(foreground)=black):
Set(TFdeq(editable)=false):
Set(feedback=great[randgreat()],
feedback(background)=green, feedback(foreground)=black):
Set(massinit(editable)=true):
Set(hintc(enabled)=true, checkc(enabled)=true, showc(enabled)=true):
else
Set(fb2="incorrect",
fb2(background)=red, fb2(foreground)=wheat):
Set(feedback=cat(sorry[randsorry()], tryagain[randtry()]),
feedback(background)=red, feedback(foreground)=wheat):
end if:
end proc:show2:=proc()
global diffeq;
Set(TFdeq=rhs(diffeq), TFdeq(editable)=false):
Set(fb2="shown",
fb2(background)=lightorange, fb2(foreground)=black):
Set(feedback=showit[randshow()],
feedback(background)=lightorange, feedback(foreground)=black):
Set(massinit(editable)=true):
Set(hintc(enabled)=true, checkc(enabled)=true, showc(enabled)=true):
end proc:hint3:=proc()
Set(fb3="hint",
fb3(background)=lightorange, fb3(foreground)=black):
Set(feedback=" Use the given values of initial concentration and volume in the vat to find a value for S(0), the amount of solute initially in the vat.",
feedback(background)=lightorange, feedback(foreground)=black):
end proc:check3:=proc()
global initmass;
local usermass:
usermass:=Get(massinit::anything):
if usermass=initmass then
Set(fb3="correct",
fb3(background)=green, fb3(foreground)=black):
Set(massinit(editable)=false):
Set(feedback=great[randgreat()],
feedback(background)=green, feedback(foreground)=black):
Set(eqn(editable)=true):
Set(hintd(enabled)=true, checkd(enabled)=true, showd(enabled)=true):
else
Set(fb3="incorrect",
fb3(background)=red, fb3(foreground)=wheat):
Set(feedback=cat(sorry[randsorry()], tryagain[randtry()]),
feedback(background)=red, feedback(foreground)=wheat):
end if:
end proc:show3:=proc()
global initmass;
Set(massinit=initmass, massinit(editable)=false):
Set(fb3="shown",
fb3(background)=lightorange, fb3(foreground)=black):
Set(feedback=showit[randshow()],
feedback(background)=lightorange, feedback(foreground)=black):
Set(eqn(editable)=true):
Set(hintd(enabled)=true, checkd(enabled)=true, showd(enabled)=true):
end proc:hint4:=proc()
Set(fb4="hint",
fb4(background)=lightorange, fb4(foreground)=black):
Set(feedback=" Use the methods you have learned for linear or separable differential equations to solve the one you found above. Remember to apply the initial condition.",
feedback(background)=lightorange, feedback(foreground)=black):
end proc:check4:=proc()
global solution;
local usereqn:
usereqn:=Get(eqn::anything):
if usereqn=rhs(solution) then
Set(fb4="correct",
fb4(background)=green, fb4(foreground)=black):
Set(eqn(editable)=false):
Set(feedback=great[randgreat()],
feedback(background)=green, feedback(foreground)=black):
Set(answer(editable)=true):
Set(hinte(enabled)=true, checke(enabled)=true, showe(enabled)=true):
else
Set(fb4="incorrect",
fb4(background)=red, fb4(foreground)=wheat):
Set(feedback=cat(sorry[randsorry()], tryagain[randtry()]),
feedback(background)=red, feedback(foreground)=wheat):
end if:
end proc:show4:=proc()
global solution;
Set(eqn=rhs(solution), eqn(editable)=false):
Set(fb4="shown",
fb4(background)=lightorange, fb4(foreground)=black):
Set(feedback=showit[randshow()],
feedback(background)=lightorange, feedback(foreground)=black):
Set(answer(editable)=true):
Set(hinte(enabled)=true, checke(enabled)=true, showe(enabled)=true):
end proc:hint5:=proc()
Set(fb5="hint",
fb5(background)=lightorange, fb5(foreground)=black):
Set(feedback=" Plug the value of t into S(t), and then divide by the vat volume to find the concentration.",
feedback(background)=lightorange, feedback(foreground)=black):
end proc:check5:=proc()
global finalc;
local userans:
userans:=evalf(Get(answer::anything)):
if abs(userans - finalc)/finalc < 0.05 then
Set(fb5="correct",
fb5(background)=green, fb5(foreground)=black):
Set(answer(editable)=false):
Set(feedback=cat(great[randgreat()], nextquest[randnext()]),
feedback(background)=green, feedback(foreground)=black):
else
Set(fb5="incorrect",
fb5(background)=red, fb5(foreground)=wheat):
Set(feedback=cat(sorry[randsorry()], tryagain[randtry()]),
feedback(background)=red, feedback(foreground)=wheat):
end if:
end proc:show5:=proc()
global finalc;
Set(answer=finalc, answer(editable)=false):
Set(fb5="shown",
fb5(background)=lightorange, fb5(foreground)=black):
Set(feedback=cat(showit[randshow()], nextquest[randnext()]),
feedback(background)=lightorange, feedback(foreground)=black):
end proc:MixingProblem:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN](title="Mixing Problem",
[ halign=none,
[ halign=none,
Button("New Problem", Evaluate(function="start"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ halign=none,
TextBox[problem](width=30, height=5,
editable=false, background=white),
Plotter[picture](width=200, height=200)
],
[ halign=none,
"Rate In =",
TextField[inrate](width=11, editable=false, background=white),
"Rate Out =",
TextField[outrate](width=11, editable=false, background=white),
HorizontalGlue(),
Button[hinta]("Hint", Evaluate(function="hint1"),
enabled=false, background=lightorange),
Button[checka]("Check", Evaluate(function="check1"),
enabled=false, background=green),
TextField[fb1](width=5, editable=false, background=white),
Button[showa]("Show", Evaluate(function="show1"),
enabled=false, background=lightorange)
],
[ halign=none,
"dS/dt =",
TextField[TFdeq](width=30, editable=false, background=white),
HorizontalGlue(),
Button[hintb]("Hint", Evaluate(function="hint2"),
enabled=false, background=lightorange),
Button[checkb]("Check", Evaluate(function="check2"),
enabled=false, background=green),
TextField[fb2](width=5, editable=false, background=white),
Button[showb]("Show", Evaluate(function="show2"),
enabled=false, background=lightorange)
],
[ halign=none,
"S(0) =",
TextField[massinit](width=10, editable=false, background=white),
" ",
HorizontalGlue(),
" ",
HorizontalGlue(),
Button[hintc]("Hint", Evaluate(function="hint3"),
enabled=false, background=lightorange),
Button[checkc]("Check", Evaluate(function="check3"),
enabled=false, background=green),
TextField[fb3](width=5, editable=false, background=white),
Button[showc]("Show", Evaluate(function="show3"),
enabled=false, background=lightorange)
],
[ halign=none,
"S(t) = ",
TextField[eqn](width=30, editable=false, background=white),
HorizontalGlue(),
Button[hintd]("Hint", Evaluate(function="hint4"),
enabled=false, background=lightorange),
Button[checkd]("Check", Evaluate(function="check4"),
enabled=false, background=green),
TextField[fb4](width=5, editable=false, background=white),
Button[showd]("Show", Evaluate(function="show4"),
enabled=false, background=lightorange)
],
[ halign=none,
"Concentration = ",
TextField[answer](width=10, editable=false, background=white),
Label[Lfintime](" at time t = 00 min."),
HorizontalGlue(),
Button[hinte]("Hint", Evaluate(function="hint5"),
enabled=false, background=lightorange),
Button[checke]("Check", Evaluate(function="check5"),
enabled=false, background=green),
TextField[fb5](width=5, editable=false, background=white),
Button[showe]("Show", Evaluate(function="show5"),
enabled=false, background=lightorange)
],
[ TextBox[feedback](width=50, height=3, editable=false)
],
[ background="#DDFFFF", inset=0, spacing=0,
"Programmers: Jonathan Garcia, Michael Fine",
HorizontalGlue(),
"@ Copyright P.B. Yasskin 2007"
]
]
)
):Maplets[Display]( MixingProblem );