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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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% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
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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 \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.
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% This option was created  by Eric Sullivan, Carroll College, Helena MT.
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\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}

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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\SALT COMPARTMENTS}





\markboth{Salt Compartments}{Insert Name of Scenario for Right Page running Heading}






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

\makeStitlePDFLaTex




\begin{abstract}
Model a phenomena in which salt mixtures from two tanks are mixed using several strategies.
\end{abstract}


\section*{SCENARIO DESCRIPTION}



Suppose we have two tanks of liquid, Tank 1, holding 100 liters of pure water, and Tank 2, holding 200 liters  of pure water - reasonable size fish tanks you might have in your home/office. We dump 100 g of salt (NaCl) into Tank 2, thoroughly dissolve and mix so the salt is uniformly distributed in the tank. Next we start a pumping action in which we pump 8 l/min of thoroughly mixed saline solution from Tank 1 to Tank 2 and at the same time we pump 8 l/min of thoroughly mixed saline solution from Tank 2 to Tank 1, thus keeping  volumes of water in the respective tanks the same throughout our experiment. Let us assume that a stirrer in each tank keeps the solution throughly mixed so that at the very instant saline solution enters a tank its salt is thoroughly mixed in that tank.

Let $x(t)$ be the amount of salt in  g in Tank 1 at time $t$ in min and $y(t)$ be the amount of salt in  g in Tank 2 at time $t$ in min.

\begin{itemize}
\item [a)] Set up a system of linear differential equations with initial conditions to describe the flow of salt in each tank at time $t$ min.
\item [b1)] Solve the system using an eigenvalue/eigenvector approach and  plot both $x(t)$ and $y(t)$ over a 60 minute time interval.
\item [b2)] Solve the system by eliminating one of the variables $x(t)$ or $y(t$) and solving for the remaining variable. Then use that solution to solve for the other variable. Plot both $x(t)$ and $y(t)$ over a 60 minute time interval.
\item [c)] From your plot describe what is happening with the salt in solution and tell why you could have predicted this at the start without any differential equations!
\item [d)] How can the eigenvalues and eigenvectors tell you the observed phenomena in (c) would happen?
\item [e)] Use {\it Mathematica\/}'s {\tt DSolve} to solve these equations and pick out the eigen information in your solutions. Also plot these solutions showing that your results are identical to those in (b1) and (b2).
\item [f)] Compute the total amount of salt in your solution from your solutions at all time. Why is this to be expected?
\end{itemize}




\teacher{
\section*{NOTES FOR TEACHERS}

We offer a complete analysis in a Mathematica notebook 5-025-T-Mma-SaltCompartments-TeacherVersion and offer this file as a pdf for those who do not read Mathematica files.   The approach is NOT dependent upon Mathematica, as we offer commands with mathematical and process comments of a general nature.

We ask students to approach this problem in several ways (1) eigenvalue/vectors, (2) replacement, and (3) technology. The first two are appropriate for hand work as the numbers re ``friendly," while plotting with technology is comforting.
}



\end{document}
