Maximum Volume of a Box under a ParaboloidChris Standley & Alexandru Raducanu253 Honors Spring 2009restart;with(Maplets[Tools]): with(Maplets[Elements]): with(plots):
StartEngine();randomize():lightorange:="#FFB300":rand15:=rand(1..5):assume(a<0,b<0,c<0);problem:=proc()
global vol, vol1, dx, dy, xsol, ysol, zsol, dxx, dyy, dxy, vol2, top;
local a, b, c, eq, xysol, testd, xmax, ymax, gr;
a:=rand15();
b:=rand15();
c:=rand15();
eq:= z = c - a*x^2 - b*y^2;
vol:=2*x*2*y*z;
vol1:=2*x*2*y*(c - a*x^2 - b*y^2);
dx:=expand(diff(vol1,x));
dy:=expand(diff(vol1,y));
xysol:=solve({ dx=0, dy=0, x > 0, y > 0}, {x,y});
xsol:=subs(xysol,x);
ysol:=subs(xysol,y);
zsol:=subs(x=xsol,y=ysol,rhs(eq));
vol2:=simplify(4*xsol*ysol*zsol);
xmax:=ceil(sqrt(c/a));
ymax:=ceil(sqrt(c/b));
gr:=plot3d(rhs(eq),x=-xmax..xmax,y=-ymax..ymax, view=0..c, shading=zhue, transparency=0.8, scaling=constrained, axes=normal);
top:=plot3d(zsol,-xsol..xsol,-ysol..ysol,labels=["x axis", "y axis", "z axis"], filled=true);
Set(MMLeq=eq);
Set(Graph=display(gr,top));
Set(TFvol="");
Set(TFdx="");
Set(TFdy="");
Set(TFxy="");
Set(TFvol2="");
Set(reply="");
end proc:hint1:=proc()
Set(reply="The length and width of the box are not just x and y. Remember the other halves."):
end proc:check1:=proc()
global vol1;
local user_vol;
if Get(TFvol)="" then
Set(reply="No answer given.");
return;
end if;
user_vol:=Get(TFvol::anything):
if simplify(vol1 - user_vol)=0 then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:show1:=proc()
global vol1;
Set(TFvol=vol1):
end proc:hintdx:=proc()
Set(reply="Remember to treat the y's as constants!"):
end proc:checkdx:=proc()
global dx;
local user_dx;
if Get(TFdx)="" then
Set(reply="No answer given.");
return;
end if;
user_dx:=Get(TFdx::anything):
if simplify(dx - user_dx)=0 then
Set(reply="Good Job!"):
else
Set(reply="Not quite..."):
end if;
end proc:showdx:=proc()
global dx;
Set(TFdx=dx):
end proc:hintdy:=proc()
Set(reply="Remember to treat the x's as constants!"):
end proc:checkdy:=proc()
global dy;
local user_dy;
if Get(TFdy)="" then
Set(reply="No answer given.");
return;
end if;
user_dy:=Get(TFdy::anything):
if simplify(dy - user_dy)=0 then
Set(reply="I oughta give you a cookie...but no milk yet! Get to solving!"):
else
Set(reply="Not quite..."):
end if;
end proc:showdy:=proc()
global dy;
Set(TFdy=dy):
end proc:hintxy:=proc()
Set(reply="Set each derivative equal to zero, and remember to get positive answers only."):
end proc:checkxy:=proc()
global xsol, ysol;
local user_xy, user_x, user_y;
if Get(TFxy)=""then
Set(reply="No answer given.");
return;
end if;
user_xy:=Get(TFxy::list, corrections=true, update=true):
user_x,user_y:=op(user_xy);
if simplify([xsol, ysol] - user_xy=[0,0]) then
Set(reply="Congratulations! Keep going!"):
elif xsol-user_x =0 then
Set(reply="Check your y value again."):
elif ysol-user_y =0 then
Set(reply="Check your x value again."):
else
Set(reply="So close, yet so far. Try again."):
return;
end if;
end proc:showxy:=proc()
global xsol, ysol;
Set(TFxy=[xsol, ysol]):
end proc:hint5:=proc()
Set(reply="Remember it is not necessary to compute the value of z, you can find the volume with onle x and y."):
end proc:check5:=proc()
global vol2;
local user_vol2;
if Get(TFvol2)="" then
Set(reply="No answer given.");
return;
end if;
user_vol2:=Get(TFvol2::anything):
if simplify(vol2 - user_vol2=0) then
Set(reply="Yes sir you are right."):
else
Set(reply="Nope. But close...I think."):
end if;
end proc:show5:=proc()
global vol2;
Set(TFvol2=vol2):
end proc:JSFHMaxVolBox:=Maplet(onstartup=RunWindow(MAIN),
Font[helv10]("helvetica",10),
Window[MAIN](title="Inscribed Box in a Paraboloid",
[ halign=none,
[ Button("New Problem", Evaluate(function=problem), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ border=true,
"Find the volume of the largest rectangular prism that can be inscribed in the paraboloid ",
MathMLViewer[MMLeq](width=150, height=45, background=white)
],
[ inset=0,
[ border=true,
caption="Plot",
Plotter[Graph](height=300,width=300)
],
[ inset=0,
[ border=true,
caption="Enter the Volume of the Inscribed Prism as a function of x and y only:",
"V =",
TextField[TFvol](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="Find the derivative of the volume with respect to x.",
"Vx =",
TextField[TFdx](width=17, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hintdx"), background=lightorange),
Button("Check", Evaluate(function="checkdx"), background=green),
Button("Show", Evaluate(function="showdx"), background=lightorange)
],
[ border=true,
caption="Find the derivative of the volume with respect to y.",
"Vy =",
TextField[TFdy](width=17, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hintdy"), background=lightorange),
Button("Check", Evaluate(function="checkdy"), background=green),
Button("Show", Evaluate(function="showdy"), background=lightorange)
],
[ border=true,
caption="Solve for the critical point.",
"[x,y] =",
TextField[TFxy](width=17, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hintxy"), background=lightorange),
Button("Check", Evaluate(function="checkxy"), background=green),
Button("Show", Evaluate(function="showxy"), background=lightorange)
],
[ border=true,
caption="Compute the maximum volume.",
"V =",
TextField[TFvol2](width=17, background=turquoise),
HorizontalGlue(),
Button("Hint", Evaluate(function="hint5"), background=lightorange),
Button("Check", Evaluate(function="check5"), background=green),
Button("Show", Evaluate(function="show5"), background=lightorange)
]
] ],
[ halign=none,
TextField[reply](width=65, editable=false, background=white)
],
[ background="#DDFFFF", inset=0, spacing=0,
Label("Programmers: Chris Standley & Alexandru Raducanu", font=helv10),
HorizontalGlue(),
Label("Copyright P. Yasskin 2009", font=helv10)
]
]
)
):Maplets[Display]( MaxVolBox );