Michael Cook & Rachel GlennMaximum and Minimum Area Enclosed by Two Wire ShapesMath 152 - Section 201restart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):randL:=rand(2..20):randshape:=rand(1..3):shapes:=["Circle","Square","Equilaterial Triangle"]:problem:=proc()
global TArea,DArea,MaxiArea,MinArea,Maxx,Minx,sh1,sh2,a1,a2;
local areat, areac, areasq, L, areas, areat2, areac2, areasq2, areas2, xsol, TAsol, TA0, TAL, n1,n2;
L:=randL();
areat:=(3^(1/2)/4)*(x/3)^2;
areac:=(x^2)/(4*Pi);
areasq:=(x/4)^2;
areat2:=(3^(1/2)/4)*((L-x)/3)^2;
areac2:=((L-x)^2)/(4*Pi);
areasq2:=((L-x)/4)^2;
areas:=[areac,areasq,areat];
areas2:=[areac2,areasq2,areat2];
n1:=randshape();
n2:=randshape();
sh1:=shapes[n1];
sh2:=shapes[n2];
a1:=areas[n1];
a2:=areas2[n2];
Set(probstate(caption)=sprintf("One piece of length x is bent into a %s and the other piece is bent into a %s.", sh1, sh2));
TArea:=a1+a2;
DArea:=diff(TArea,x);
xsol:=solve(DArea=0,x);
TAsol:=eval(TArea,x=xsol);
TA0:=eval(TArea,x=0);
TAL:=eval(TArea,x=L);
MaxiArea:=max(TAsol,TA0,TAL);
MinArea:=min(TAsol,TA0,TAL);
if TAsol=MaxiArea then Maxx:=xsol end if;
if TA0=MaxiArea then Maxx:=0 end if;
if TAL=MaxiArea then Maxx:=L end if;
if TAsol=MinArea then Minx:=xsol end if;
if TA0=MinArea then Minx:=0 end if;
if TAL=MinArea then Minx:=L end if;
Set(TFlength=L);
Set(TFTArea="");
Set(TFDArea="");
Set(TFMaxx="");
Set(TFMaxiArea="");
Set(TFMinx="");
Set(TFMinArea="");
end proc:hint1:=proc()
Set(reply="Express each area in terms of its perimeter. Add areas of the shapes."):
end proc:check1:=proc()
global TArea;
local user_area;
if Get(TFTArea)="" then
Set(reply="No answer given.");
return;
end if;
user_area:=Get(TFTArea::anything):
if simplify(TArea - user_area)=0 then
Set(reply="Awesome job, Keep going."):
else
Set(reply="Looks like you need to go to Tech."):
end if;
end proc:show1:=proc()
global TArea;
Set(TFTArea=TArea):
end proc:hint2:=proc()
Set(reply="Find the Derivative."):
end proc:check2:=proc()
global DArea;
local user_darea;
if Get(TFDArea)="" then
Set(reply="No answer given.");
return;
end if;
user_darea:=Get(TFDArea::anything):
if simplify(DArea - user_darea)=0 then
Set(reply="Keep it up."):
else
Set(reply="How does Austin sound?."):
end if;
end proc:show2:=proc()
Set(TFDArea=DArea):
end proc:hint3x:=proc()
Set(reply="Check the endpoints of the function."):
end proc:check3x:=proc()
global Maxx;
local user_maxx;
if Get(TFMaxx)=""then
Set(reply="No answer given.");
return;
end if;
user_maxx:=Get(TFMaxx::anything):
if simplify(Maxx - user_maxx=0) then
Set(reply="Way to go."):
else
Set(reply="Sad Day."):
end if;
end proc:show3x:=proc()
global Maxx;
Set(TFMaxx=Maxx):
end proc:hint3a:=proc()
Set(reply="Check the endpoints of the function."):
end proc:check3a:=proc()
global MaxiArea;
local user_maxiarea;
if Get(TFMaxiArea)=""then
Set(reply="No answer given.");
return;
end if;
user_maxiarea:=Get(TFMaxiArea::anything):
if simplify(MaxiArea - user_maxiarea=0) then
Set(reply="Sweetness."):
else
Set(reply=":(."):
end if;
end proc:show3a:=proc()
global MaxiArea;
Set(TFMaxiArea=MaxiArea):
end proc:hint4x:=proc()
Set(reply="Check the endpoints of the function."):
end proc:check4x:=proc()
global Minx;
local user_minx;
if Get(TFMinx)="" then
Set(reply="No answer given.");
return;
end if;
user_minx:=Get(TFMinx::anything):
if simplify(Minx - user_minx=0) then
Set(reply="Amazing."):
else
Set(reply="Almost at the end, just keep trying."):
end if;
end proc:show4x:=proc()
global Minx;
Set(TFMinx=Minx):
end proc:hint4a:=proc()
Set(reply="Check the endpoints of the function."):
end proc:check4a:=proc()
global MinArea;
local user_minarea;
if Get(TFMinArea)="" then
Set(reply="No answer given.");
return;
end if;
user_minarea:=Get(TFMinArea::anything):
if simplify(MinArea - user_minarea=0) then
Set(reply="Superb, Now try another problem."):
else
Set(reply="Not there yet."):
end if;
end proc:show4a:=proc()
global MinArea;
Set(TFMinArea=MinArea):
end proc:WireShapes:=Maplet(onstartup=RunWindow(MAIN),
Font[helv10]("helvetica",10),
Window[MAIN](title="Maximum and Minimum Area Enclosed by Two Wire Shapes",
[ halign=none,
[ Button("New Problem", Evaluate(function=problem), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
BoxColumn( halign=none, border=true,
[ inset=0,
"A wire of length",
TextField[TFlength](width=6, editable=false, background=white),
"m is cut into two pieces.",
HorizontalGlue()
],
[ inset=0,
Label[probstate]("One piece of length x is bent into a ............... and the other piece is bent into a ............... ."),
HorizontalGlue()
],
[ inset=0,
"How should the wire be cut so that the total area enlcosed is a) maximum? b) minimum?",
HorizontalGlue()
]
),
[ border=true,
caption="Enter the Formula for the Total Area of the two shapes:",
"A =",
TextField[TFTArea](width=25, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint1"), background=lightorange),
Button("Check", Evaluate(function="check1"), background=green),
Button("Show", Evaluate(function="show1"), background=lightorange)
],
[ border=true,
caption="Enter the Formula for the Derivative of the Total Area:",
"A' =",
TextField[TFDArea](width=25, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint2"), background=lightorange),
Button("Check", Evaluate(function="check2"), background=green),
Button("Show", Evaluate(function="show2"), background=lightorange)
],
BoxColumn( border=true,
caption="Find the x which Maximizes the Area and the Maximum Area:",
[ inset=0,
"x =",
TextField[TFMaxx](width=4, background=turquoise),
"cm",
HorizontalGlue(),
Button("Hint", Evaluate(function="hint3x"), background=lightorange),
Button("Check", Evaluate(function="check3x"), background=green),
Button("Show", Evaluate(function="show3x"), background=lightorange)
],
[ inset=0,
"A =",
TextField[TFMaxiArea](width=15, background=turquoise),
"cm^2",
HorizontalGlue(),
Button("Hint", Evaluate(function="hint3a"), background=lightorange),
Button("Check", Evaluate(function="check3a"), background=green),
Button("Show", Evaluate(function="show3a"), background=lightorange)
]
),
BoxColumn( border=true,
caption="Find the x which Minimzes the Area and the Minimum Area:",
[ inset=0,
"x =",
TextField[TFMinx](width=20, background=turquoise),
"cm",
HorizontalGlue(),
Button("Hint", Evaluate(function="hint4x"), background=lightorange),
Button("Check", Evaluate(function="check4x"), background=green),
Button("Show", Evaluate(function="show4x"), background=lightorange)
],
[ inset=0,
"A =",
TextField[TFMinArea](width=30, background=turquoise),
"cm^2",
HorizontalGlue(),
Button("Hint", Evaluate(function="hint4a"), background=lightorange),
Button("Check", Evaluate(function="check4a"), background=green),
Button("Show", Evaluate(function="show4a"), background=lightorange)
]
),
[ TextField[reply](width=55, editable=false, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Michael Cook & Rachel Glenn", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
)):Maplets[Display]( WireShapes );