% SIMIODE-TeacherStudent-Template.tex and simiode.cls for LaTeX2e
% Version 22 November 2016.
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% Appreciation to Mark Frantz, IUPUI; John Thoo, Yuba College; and Eric Sullivan, Carroll College.
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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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% 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 only 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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% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review, but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
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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 \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.   
% Rename the pdf after you compile each final version.
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\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
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\usepackage{url}
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\renewcommand{\baselinestretch}{1.3}
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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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% as \newtheorem or \newenvironment should also go in the 
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% Do not \input your macros as separate files within your 
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% Top matter is above and the beginning of the document below.
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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\\Newton Watson  Time of Death}




\markboth{Newton Watson Time of Death}{Newton Watson Time of Death}

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

\makeStitlePDFLaTex


\begin{abstract}
Sherlock Holmes determines the time of death for a body found on a street in London and we need to reproduce his astute analysis.
\end{abstract}

\section*{SCENARIO DESCRIPTION}

It was hot for a London fall day -- 70$^{\circ}$F. Holmes arrived at Barker Avenue Annex to find the inspector hunched over the body. ``It is important that we determine the exact time of death, sir, for in that way we may immediately determine the motive," said the inspector. Not wishing to pursue the unpursuable non sequitur, Holmes took out his  thermometer and after a few moments of discrete(!) investigation, announced, ``I say! 94.6$^{\circ}$F. (What no metric system?) And it is presently noon." With that he departed into the London fog, to return to the body at the same spot in one hour. After performing another investigation Holmes declared, ``93.4$^{\circ}$F at 1 o'clock." And then silence \ldots  After some analysis, Holmes predicted the time of death to the minute.

Holmes shared with Watson a hint about a law of Newton at which point Watson said something like, ``Here, here, Holmes, you don't mean to say he fell out of a window.''  Holmes sensing the gravity of the situation proceeded to explain, as he always did with Watson(!),  how he did his analysis.


\begin{enumerate}
\item Using a differential equation model show exactly the analysis needed to come to this the conclusion Holmes offered and offer up Holmes's answer. Spare no details, for Holmes ALWAYS tried to make his explanations, ``Elementary, my dear Watson!'' State any and all assumptions clearly and precisely.
\item 	At the autopsy later that day it was discovered that the dead man had a bacterial infection at the time of death which, when interpreted by the medical staff, suggested that he was running a high fever of around 104$^{\circ}$F at the time of death. Use this information to revise the estimated time of death.
\end{enumerate}


\teacher{
\section*{NOTES FOR TEACHERS}

This material comes from an article written by the author in 1979\cite{Winkel1979} as a filler in the {\it Mathematics Magazine\/}. The approach to take in solution, even in the modification due to fever, is a direct application of Newton's Law of Colling using the two data points, initial temperature and hour later temperature, to determine the parameters, $k$ and $T_0$ in

\begin{equation}
T'(t)  = -k (T(t) - T_{\rm Env})\, \quad T(0) = T_0\, ,
\end{equation}
where $T(t)$ is the temperature of the body in degrees Fahrenheit at time $t$ in hours since death.




\begin{thebibliography}{99}

\bibitem{Winkel1979} Winkel, B. 1979. Elementary (My Dear Watson) Differential Equation. {\it Mathematics Magazine\/}.  52(3): 215. 

\end{thebibliography}
}





\end{document}
