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\noindent Fall 2007  \hfill Math 235 \vspace{.1in}

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{\bf \large \em Column of Books}\footnote{adapted from {\em
Calculus: Early Transcendentals}, 4th ed. Stewart, pg 780}
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\vspace{.1in}

Consider the stack of identical books pictured in Figure \ref{stack}.

\begin{figure}[hb]
\center
\includegraphics[width=3in]{stack}
\caption{A precarious stack of books.}
\label{stack}
\end{figure}

Notice the top book on the stack extends completely beyond the
footprint of the bottom book.  Show that given enough identical
books one can create a stack such that the top book extends as far
to the right of the bottom book as you want.  Let the sequence
\begin{equation}
\left\{\frac{1}{2}, \frac{1}{4}, \frac{1}{4}, \frac{1}{8}\right\}
\label{Seq1}
\end{equation}
be known as the {\bf overhang sequence}.  Here are some hints to
get you started:

\begin{enumerate}

 \item Begin by computing the center of mass of the stack of books in
 Figure \ref{stack}.  What does the center of
 mass tell about the stability of the stack?

 \item \label{arb} Assuming all of the books are of uniform size, density
 and have length 1, find the overhang sequence for $n$ books (you might say
 that the first book, centered on the origin, is zero, and count
 from the top book- Therefore, Equation \ref{Seq1} gives the
 sequence for 3 books).

 \item Compute how far the top book extends beyond the footprint of the
 bottom book for the arbitrary length overhang sequence in Part \ref{arb}.
Can it continue like that forever?  Be specific in your reasons.
\end{enumerate}

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