% SIMIODE-TeacherStudent-Template.tex and simiode.cls for LaTeX2e
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\documentclass{simiode}
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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
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\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
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% the document. See this file for examples. This has been done for you in the document below.   
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\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
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% NOTE regarding macros %%
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% Any macros defined for your paper should be contained in 
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\begin{document}

\newpage
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% See above for instructions on toggling from Teacher Version to Student Version.
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION   \else STUDENT VERSION \fi\\Water Exit Bottle}





\markboth{Water Exit Bottle}{Water Exit Bottle}

\author{Brian Winkel,
Director, SIMIODE\\ 
Chardon OH 44024USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We offer an experiment in which data is collected to ascertain a parameter in the differential equation formulation of Torricelli's Law for water flow out of a cylindrical container.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

We consider  the rate at which a fluid drains out of a hole in a container. Our container is a cylinder  made from a 2 liter plastic soda bottle.  

\vspace*{.3 true cm}
\noindent {\bf Problem:} A small hole is drilled in the side of a cylindrical container and the height of the water level (above the hole) goes from 10 cm down to 3 cm in 68 seconds. Estimate the height at intermediate times using the following two models.
\vspace*{.3 true cm}


\begin{itemize}
\item [1)] The {\it linear model\/} where $h = h(t)$ is the height of the water in cm at time $t$ in s:
\begin{equation}\label{eq:lin}
\frac{dh}{dt} = k\, , \quad h(0) = 0\,  , \quad h(68)= 3\, .
\end{equation} 
Solve for $h(t)$. 
\item [2)] {\it Torricelli's model\/} says that the time rate of change of volume $V$  of water in the draining container is proportional to the square root of the water's depth $h$. This can be shown to yield

\begin{equation}\label{eq:Torr}
\frac{dh}{dt} = k \sqrt{h}\, , \quad h(0) = 10\, , \quad h(68) = 3\, .
\end{equation}
Solve for $h(t)$. 
\end{itemize}


The table below gives the values of $h$ predicted by each of these models for various  times. 

You will fill in the column of observed values when we perform the experiment.

\begin{center}
\begin{tabular}{|c|r|r|r|}
\multicolumn{1}{c}{Time}&	\multicolumn{1}{c}{Linear}	&	\multicolumn{1}{c}{Torricelli}&		\multicolumn{1}{c}{Observed}\\
\multicolumn{1}{c}{$t$}&		\multicolumn{1}{c}{$h$}&		\multicolumn{1}{c}{$h$}&		\multicolumn{1}{c}{$h$}\\ \hline
0	&10.0	&10.0&\\ \hline
10	&9.0	&8.7&\\ \hline
20	&7.9	&7.5&\\ \hline
30	&6.9	&6.4&\\ \hline
40	&5.9	&5.4&\\ \hline
50	&4.8	&4.4&\\ \hline
60	&3.8	&3.6&\\ \hline
68	&3.0	&3.0&\\ \hline
\end{tabular}
\end{center}

\begin{itemize}
\item [3)] Verify the values in the table from your solutions in (1) and (2), respectively.
\end{itemize}



\begin{itemize}
\item [4)] Prepare a two-liter clear plastic soft drink bottle, whose midsection is essentially cylindrical, by drilling a clean 4 millimeter hole near the bottom of the cylindrical part. Attach to the bottle a strip of masking tape with centimeters marked on it and zero corresponding to the top of the hole. Perform the experiment, being sure to have a collection container for the water as it exits the cylinder. Fill in the last column in the table and compare with the two offered models, linear and Torricelli's.
\end{itemize}




\teacher{
\section*{NOTES FOR TEACHERS}

We point out to our fellow teachers that this type of article is uncommon in the traditional literature for mathematics teachers and we felt we owed it to the authors to bring it ``back to life.''  We have paraphrased and quoted liberally from the original source article.\cite{FarmerGass1992}


\begin{thebibliography}{98}




\bibitem{FarmerGass1992} Farmer, T and F. Gass. 1992.  Physical demonstration in the calculus classroom. {\it The College Mathematics Journal\/}.  23(2): 146-148.


\end{thebibliography}

}





\end{document}
