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\large\bf Project Based Math 112, Fall 2001\\
Project \#1 --- Houdini's Escape\\
Due Thursday, September 27\\[.25in]
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\noindent
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Harry Houdini was a famous escape artist. In this project we relive a
trick of his that challenged his mathematical prowess, as well as his
skill and bravery. It will challenge these qualities in you as well.

Houdini had his feet shackled to the top of a concrete block which was
placed on the bottom of a giant laboratory flask. The cross-sectional
radius of the flask, measured in feet, was given as a function of the
height $z$ from the ground by the formula

$$r(z)={10\over\sqrt{z}}$$

\noindent
with the bottom of the flask at $z=1$ foot. The flask was then filled with
water at a steady rate of $22\pi$ cubic feet per minute. Houdini's job
was to escape the shackled before he was drowned by the rising water in
the flask.

\indent
Now Houdini knew it would take him exactly ten minutes to escape the
shackles. For dramatic impact, he wanted to time his escape so it was
completed precisely at the moment the water level reached the top of his
head. Houdini was exactly six feet tall.
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In the design of the apparatus, he was allowed to specify only one thing:
the height of the concrete block he stood on.

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\begin{enumerate}
\item Your first task is to find out how high this block should be.
Express the volume of water in the flask as a function of the height of
the liquid above the ground level. What is the volume when the water level
reaches the top of Houdini's head? (Neglect Houdini's volume and the
volume of the block.) What should the height of the block be?

\item Let $h(t)$ be the height of the water above ground level at time
$t$. In order to check the progress of his escape moment by moment,
Houdini derives the equation for the rate of change $dh/dt$ as a function
of $h(t)$ itself.  Derive this equation. How fast is the water level
changing when the flask first starts to fill? How fast is it changing when
the water just reaches the top of his head? Express $h(t)$ as a function
of time.
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\item Houdini would like to be able to perform this trick with any flask.  
Help him plan his next trick by generalizing the derivation of part (b).
Consider a flask with cross sectional radius $r(z)$ (an arbitrary function
of $z$) and a constant inflow rate $dV(t)/dt = A$. Find $dh/dt$ as a
function of $h(t)$.
\end{enumerate}

\noindent
{\bf Extra Credit:} How would you modify your calculations to take into
account Houdini's volume, given Houdini's cross-sectional area as a
function of height?

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