Shelby Lee & Kevin LammersMinimize the Time to Cross the RiverMath 152 - Section 202restart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):TEXTLprogrammers:="Programmers: D.B. Meade & P.B. Yasskin":TEXTLcopyright:="Copyright: Maplets for Calculus 2008":TEXTLversion:="M4C 2008":TEXTLasstprogrammer:="Assistant Programmers: Shelby Lee and Kevin Lammers":rand27:=rand(2..7):rand815:=rand(8..15):rand48:=rand(4..8):rand1115:=rand(11..15):problem:=proc()
global Rowdist, Rowtime, Rundist, Runtime, Dist, Totaltime, Rounddist;
local Riverwidth, Riverlength, Rowspeed, Runspeed, pl, TPA, TPC, TPD, TPB, TPx, priver, RP;
Riverwidth:=rand27();
Set(riverwidth=Riverwidth);
Riverlength:=rand815();
Set(riverlength=Riverlength);
Rowspeed:=rand48();
Set(rowspeed=Rowspeed);
Runspeed:=rand1115();
Set(runspeed=Runspeed);
Rowdist:=sqrt(x^2+Riverwidth^2);
Rowtime:=Rowdist/Rowspeed;
Rundist:=Riverlength-x;
Runtime:=Rundist/Runspeed;
Totaltime:=Rowtime+Runtime;
Dist:=fsolve(diff(Totaltime, x)=0, x=0..Riverlength);
Rounddist=((round(Dist*1000))/1000);
pl:=plot([[0,Riverlength],[Riverwidth, Riverlength-Dist],[Riverwidth,0]], thickness=4, color=blue);
TPA:=textplot([0,Riverlength,"A"], align=LEFT, font=[Helvetica,bold,16]);
TPC:=textplot([Riverwidth,Riverlength, "C"], align=RIGHT, font=[Helvetica,bold,16]);
TPD:=textplot([Riverwidth, Riverlength-Dist, "D"], align=RIGHT, font=[Helvetica,bold,16]);
TPB:=textplot( [Riverwidth,0, "B"], align=RIGHT, font=[Helvetica,bold,16]);
TPx:=textplot( [Riverwidth, Riverlength-Dist/2, "x"], align=LEFT, font=[Helvetica,bold,14]);
priver:=rectangle( [0,Riverlength],[Riverwidth,0], color=aquamarine );
RP:=display(pl,TPA, TPC, TPD, TPB, TPx, priver, axes=none, scaling=constrained);
Set(TFrowdist="");
Set(TFrowtime="");
Set(TFrundist="");
Set(TFruntime="");
Set(TFdist="");
Set(TFtotaltime="");
Set(reply="");
Set(RiverPlot=RP);
end proc:hintrowdist:=proc()
Set(reply="Draw a picture and set the distance where the man lands as x km downstream. Use the distance formula to find the distance."):
end proc:checkrowdist:=proc()
local user_rowdist;
if Get(TFrowdist)="" then
Set(reply="No answer given.");
return;
end if;
user_rowdist:=Get(TFrowdist::anything):
if simplify(Rowdist- user_rowdist)=0 then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:showrowdist:=proc()
global Rowdist;
Set(TFrowdist=Rowdist):
end proc:hintrowtime:=proc()
Set(reply="The formula for time is distance over speed."):
end proc:checkrowtime:=proc()
local user_rowtime;
if Get(TFrowtime)="" then
Set(reply="No answer given.");
return;
end if;
user_rowtime:=Get(TFrowtime::anything):
if simplify(Rowtime- user_rowtime)=0 then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:showrowtime:=proc()
global Rowtime;
Set(TFrowtime=Rowtime):
end proc:hintrundist:=proc()
Set(reply="Subtract the distance downstream where he lands from the total distance downstream he must go."):
end proc:checkrundist:=proc()
global Rundist;
local user_rundist;
if Get(TFrundist)=""then
Set(reply="No answer given.");
return;
end if;
user_rundist:=Get(TFrundist::anything):
if simplify(Rundist - user_rundist=0) then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:showrundist:=proc()
global Rundist;
Set(TFrundist=Rundist):
end proc:
intruntime:=proc()
Set(reply="The formula for time is distance over speed."):
end proc:checkruntime:=proc()
global Runtime;
local user_runtime;
if Get(TFruntime)="" then
Set(reply="No answer given.");
return;
end if;
user_runtime:=Get(TFruntime::anything):
if simplify(Runtime - user_runtime=0) then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:showruntime:=proc()
global Runtime;
Set(TFruntime=Runtime):
end proc:hinttotaltime:=proc()
Set(reply="Add together the run time and row time."):
end proc:checktotaltime:=proc()
global Totaltime;
local user_totaltime;
if Get(TFtotaltime)="" then
Set(reply="No answer given.");
return;
end if;
user_totaltime:=Get(TFtotaltime::anything):
if simplify(Totaltime - user_totaltime=0) then
Set(reply="Correct."):
else
Set(reply="Incorrect."):
end if;
end proc:showtotaltime:=proc()
global Totaltime;
Set(TFtotaltime=Totaltime):
end proc:hintdist:=proc()
Set(reply="Find the equation for total time. Then take the derivative and set it equal to zero to find the distance traveled when time is minimized."):
end proc:checkdist:=proc()
global Dist;
local user_dist;
if Get(TFdist)="" then
Set (reply="No answer given.");
return;
end if;
user_dist:=Get(TFdist::anything):
if abs(evalf(Dist - user_dist))<=1/1000 then
Set (reply="Correct."):
else
Set (reply="Incorrect."):
end if;
end proc:showdist:=proc()
global Dist;
Digits:=4;
Set(TFdist=evalf(round(Dist*1000)/1000)):
Digits:=10;
end proc:River:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN](title="Minimize the Time to Cross the River",
[ [ Button("New Problem", Evaluate(function=problem), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
BoxColumn( border=true, halign=none, spacing=0,
[
"A man is at point A on a bank of a straight river ",
TextField[riverwidth](width=6, editable=false, background=white),
" km wide, and wants to reach point B, ",
TextField[riverlength](width=6, editable=false, background=white),
" km downstream",
HorizontalGlue()
],
[
"on the opposite bank, as quickly as possible. He could row his boat directly across the river to point C and then run to point B or",
HorizontalGlue()
],
[ "he could row directly to point B or he could row to some point D a distance x from C and then run to point B.",
HorizontalGlue()
],
[
"If he can row at ",
TextField[rowspeed](width=6, editable=false, background=white),
" km/h and run at ",
TextField[runspeed](width=6, editable=false, background=white),
" km/h, where should he land to reach point B as soon as possible?",
HorizontalGlue()
]
),
[ halign=none, inset=0, spacing=0,
[ halign=none, inset=0, spacing=0,
[ border=true,
caption="What is the expression for rowing distance (in terms of x) from A to D?:",
"AD =",
TextField[TFrowdist](width=17, background=turquoise),
"km",
HorizontalGlue(),
Button("Hint", Evaluate(function="hintrowdist"), background=lightorange),
Button("Check", Evaluate(function="checkrowdist"), background=green),
Button("Show", Evaluate(function="showrowdist"), background=lightorange)
],
[ border=true,
caption="What is the expression for rowing time? (in terms of x):",
"Row Time =",
TextField[TFrowtime](width=17, background=turquoise),
"h",
HorizontalGlue(),
Button("Hint", Evaluate(function="hintrowtime"), background=lightorange),
Button("Check", Evaluate(function="checkrowtime"), background=green),
Button("Show", Evaluate(function="showrowtime"), background=lightorange)
],
[ border=true,
caption="What is the expression for running distance from D to B in terms of x?:",
"DB =",
TextField[TFrundist](width=17, background=turquoise),
"km",
HorizontalGlue(),
Button("Hint", Evaluate(function="hintrundist"), background=lightorange),
Button("Check", Evaluate(function="checkrundist"), background=green),
Button("Show", Evaluate(function="showrundist"), background=lightorange)
],
[ border=true,
caption="What is the expression for running time (in terms of x)?:",
"Run Time =",
TextField[TFruntime](width=17, background=turquoise),
"h",
HorizontalGlue(),
Button("Hint", Evaluate(function="hintruntime"), background=lightorange),
Button("Check", Evaluate(function="checkruntime"), background=green),
Button("Show", Evaluate(function="showruntime"), background=lightorange)
],
[ border=true,
caption="What is the expression for total time (in terms of x)?:",
"Total Time =",
TextField[TFtotaltime](width=25, background=turquoise),
"h",
HorizontalGlue(),
Button("Hint", Evaluate(function="hinttotaltime"), background=lightorange),
Button("Check", Evaluate(function="checktotaltime"), background=green),
Button("Show", Evaluate(function="showtotaltime"), background=lightorange)
],
BoxColumn( border=true, inset=0, spacing=0,
caption="How far downstream should he land to minimize time traveled?",
[ "x =",
TextField[TFdist](width=17, background=turquoise),
"km",
HorizontalGlue(),
Button("Hint", Evaluate(function="hintdist"), background=lightorange),
Button("Check", Evaluate(function="checkdist"), background=green),
Button("Show", Evaluate(function="showdist"), background=lightorange)
],
[ inset=0,
" (Round to the nearest thousandth.)",
HorizontalGlue()
]
)
],
[ border=true,
caption="The River",
[ Plotter[RiverPlot](width=300, height=350)
]
]
],
[ halign=none,
TextField[reply](width=65, editable=false, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Shelby Lee & Kevin Lammers", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
)):Maplets[Display]( River );