Maximum Rectangle Inscribed in an EllipseChris Standley & Michael Stricklandrestart;with(Maplets[Elements]):with(Maplets[Tools]):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):rand15:=rand(1..5):assume(a>0,b>0);problem:=proc()
global arear, subarear, derivarear, xmax, ymax, areamax;
local a,b,eq,yval;
a:=rand15();
b:=rand15();
eq:= x^2/a^2 + y^2/b^2 = 1;
yval:=b*sqrt(1-x^2/a^2);
arear:=4*x*y;
subarear:= 4*x*yval;
derivarear:= diff(subarear,x);
xmax := a/sqrt(2);
ymax := b/sqrt(2);
areamax:=2*a*b;
Set(MMLeq=eq);
Set(TFarear="");
Set(TFsubarear="");
Set(TFderivarear="");
Set(TFxmax="");
Set(TFymax="");
Set(TFareamax="");
Set(reply="");
end proc:hint1:=proc()
Set(reply="The length of the rectangle is not just x. Remember the other half."):
end proc:check1:=proc()
global arear;
local user_arear;
if Get(TFarear)="" then
Set(reply="No answer given.");
return;
end if;
user_arear:=Get(TFarear::anything):
if simplify(arear - user_arear)=0 then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:show1:=proc()
global arear;
Set(TFarear=arear):
end proc:hint2:=proc()
Set(reply="Solve the ellipse equation for y."):
end proc:check2:=proc()
global subarear;
local user_subarear;
if Get(TFsubarear)="" then
Set(reply="No answer given.");
return;
end if;
user_subarear:=Get(TFsubarear::anything):
if simplify(subarear - user_subarear)=0 then
Set(reply="Good Job!"):
else
Set(reply="Not quite..."):
end if;
end proc:show2:=proc()
global subarear;
Set(TFsubarear=subarear):
end proc:hint3:=proc()
Set(reply="Differentiate the area formula using the product rule and the chain rule."):
end proc:check3:=proc()
global derivarear;
local user_derivarear;
if Get(TFderivarear)=""then
Set(reply="No answer given.");
return;
end if;
user_derivarear:=Get(TFderivarear::anything):
if simplify(derivarear - user_derivarear=0) then
Set(reply="Congratulations! Keep going!"):
else
Set(reply="So close, yet so far. Try again."):
end if;
end proc:show3:=proc()
global derivarear;
Set(TFderivarear=derivarear):
end proc:hint4:=proc()
Set(reply="Set the derivative equal to zero and solve for x."):
end proc:check4:=proc()
global xmax;
local user_xmax;
if Get(TFxmax)="" then
Set(reply="No answer given.");
return;
end if;
user_xmax:=Get(TFxmax::anything):
if simplify(xmax - user_xmax=0) then
Set(reply="You rock! Finish it out!"):
else
Set(reply="Epic fail. One more try."):
end if;
end proc:show4:=proc()
global xmax;
Set(TFxmax=xmax):
end proc:hint5:=proc()
Set(reply="Plug the x value back into the equation for y."):
end proc:check5:=proc()
global ymax;
local user_ymax;
if Get(TFymax)="" then
Set(reply="No answer given.");
return;
end if;
user_ymax:=Get(TFymax::anything):
if simplify(ymax - user_ymax=0) then
Set(reply="Yes sir you are right."):
else
Set(reply="Nope. But close...I think."):
end if;
end proc:show5:=proc()
global ymax;
Set(TFymax=ymax):
end proc:JSFHhint6:=proc()
Set(reply="Plug the x and y values into area equation."):
end proc:check6:=proc()
global areamax;
local user_areamax;
if Get(TFareamax)="" then
Set (reply="No answer given.");
return;
end if;
user_areamax:=Get(TFareamax::anything):
if simplify(areamax - user_areamax=0) then
Set (reply="You did it! Try this problem again with different numbers."):
else
Set (reply="You were doing so good...one more step. Try again,"):
end if;
end proc:show6:=proc()
global areamax;
Set(TFareamax=areamax):
end proc:MaxRectinEll:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN](title="Maximum Rectangle Inscribed in an Ellipse",
[ halign=none,
[ Button("New Problem", Evaluate(function=problem), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ border=true,
"Find the area of the largest rectangle that can be inscribed in the ellipse ",
MathMLViewer[MMLeq](width=150, height=45, background=white)
],
[ border=true,
caption="Enter the Formula for the Area of the Inscribed Rectangle:",
"A = ",
TextField[TFarear](width=17, 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 Area of the Rectangle in terms of only x:",
"A = ",
TextField[TFsubarear](width=17, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint2"), background=lightorange),
Button("Check", Evaluate(function="check2"), background=green),
Button("Show", Evaluate(function="show2"), background=lightorange)
],
[ border=true,
caption="Differentiate the Area Formula:",
"A' =",
TextField[TFderivarear](width=30, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint3"), background=lightorange),
Button("Check", Evaluate(function="check3"), background=green),
Button("Show", Evaluate(function="show3"), background=lightorange)
],
[ border=true,
caption="Enter the value of x:",
"x = ",
TextField[TFxmax](width=12, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint4"), background=lightorange),
Button("Check", Evaluate(function="check4"), background=green),
Button("Show", Evaluate(function="show4"), background=lightorange)
],
[ border=true,
caption="Enter the value of y:",
"y = ",
TextField[TFymax](width=12, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint5"), background=lightorange),
Button("Check", Evaluate(function="check5"), background=green),
Button("Show", Evaluate(function="show5"), background=lightorange)
],
[ border=true,
caption="Solve for the Maximum Area of the Rectangle:",
"Max A =",
TextField[TFareamax](width=10, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint6"), background=lightorange),
Button("Check", Evaluate(function="check6"), background=green),
Button("Show", Evaluate(function="show6"), background=lightorange)
],
[ halign=none,
TextField[reply](width=65, editable=false, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Chris Standley & Michael Strickland", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
)
):Maplets[Display]( MaxRectinEll );