% Math 112, Fall 2001, wile E. coyote / project 2
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{\large\bf Project Based Math 112, Fall 2001\\
Project \#2 --- While Visions of Roadrunners Danced in His Head\\
Due Wednesday, November 7}
\underline{\hspace{6in}}\\[.5cm]
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\begin{flushright}
Wile E. Coyote \\
Bleached Bones, AZ 59055\\
October 17, 2001\\
\end{flushright}

\noindent Math 112 Students\\ 
Cornell University\\
Ithaca, NY 14853\\

\noindent Dear Calculus Students,\\

\noindent HELP ME!! For the last two months, I've had a series of 
recurring nightmares that are about to drive me out of my mind!
I talked to your enterprising and resourceful instructor, who 
referred me to you.\\

\noindent The scenario is nearly always the same. I'm standing at the end
of a road that is $1000$ meters long (the road has those little green
kilometer markers on it), and there at the other end is that blasted
Roadrunner, just standing there, sticking his tongue out!  I start to go
after him, but I can only run in slow motion, about $1$ meter per second.
After one second I'm $999$ meters away from that pesky fowl, but then
\textit{the road stretches uniformly and instantaneously by 1000 meters}
so I'm $1998$ meters away!! I cannot speed up, so I'm still moving in
slow motion, at $1$ meter per second. After another second, the road
stretches again by $1000$ meters! And this just keeps on happening. Over,
and over. And over. And over. Well, you get the idea. Then I wake up,
hungry and frustrated.\\

\noindent I've gotta know: Do I ever get the silly bird, and if so, how
long does it take? Should take a snack to eat along the way!\\

%\noindent Actually, most of the dreams are the same set-up, but not the
%same details. Usually, I don't know how long the road is to begin with,
%or how fast I'm moving. All I know is that I'm always moving at some
%constant slow rate, and the road stretches uniformly and instantaneously
% by its original amount after each second.\\

%\noindent You gotta help me. Do I get the Roadrunner? If so, how long
%does it take? First describe to me what happens in the dream with the
%numbers I gave, then tell me what happens in general so I know what to
%expect the next time I have this dream with different specifics.\\

\noindent I can't take this much longer.  I need an answer by November
7th, or I'll lose my brilliant, crafty mind!\\

\hspace{8cm} Hungry as ever,\\[.7cm]

\hspace{8cm} Wile E. Coyote\\[.5cm]

\noindent P.S.  What happens if I have this dream and the road is a
different length?  Or I'm traveling at a different (yet
constant) rate?  Or the road stretches by some other (fixed) amount?  

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\Large{A Few Guidelines From Your Enterprising and Resourceful
Instructor}
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\noindent After reading Wile E. Coyote's sad tale, I have a couple of
suggestions to help you get started.  If you can solve the questions
below, you'll have an answer for Wile E. 

\begin{enumerate}
\item Suppose Wile E. is standing on road $d$ meters long, with
$a$ meters ahead of him and $b$ meters behind him. If the entire road
stretches uniformly by $c$ meters, how much of the road is now behind
him?

\item After $n$ seconds, Wile E. has taken $n$ steps, and the
road has just stretched instantly by $1000$ meters yet again.  How long is
the road now?

\item Let $d_n$ be the distance Wile E. is from his original
position, after $n$ steps and subsequent stretches. 

\begin{enumerate}
\item Find a formula for $d_n$ in terms of $d_{n-1}$.
\item Find a formula for $d_n$ in terms of $n$, then prove it using
mathematical induction.
\end{enumerate}

\item Now use what you know to determine whether Wile E. travels far
enough to reach the Roadrunner after some finite number of seconds $n$.
If so, for what $n$ does he get there? (Hint: you won't be able to guess
and check to find an $n$ value that works.) 
\end{enumerate}

\noindent Extra Credit:  Answer Wile E.'s postscript plea.\\[.5cm]

\textit{As before, refer to the handouts ``Instructions for Projects'',
and ``A Guide to Writing in Mathematics Classes'' for general guidelines
on completing your project.}


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