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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\\POTATO COOLING}



\markboth{Potato Cooling}{Potato Cooling}






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

% \makeStitle
\makeStitlePDFLaTex



\begin{abstract}
We model the cooling of a baked potato and compare it to student-collected data.
\end{abstract}

\section*{SCENARIO DESCRIPTION}
 

Several years ago a group of students wanted to determine the time of death and wrote, ``An experiment in estimating the time of death was performed where a potato was used instead of a human body. The time of death for the potato was considered to be the time it was taken out of the oven.The temperature of the potato at the time of `death' was 194 $^{\circ}$F. Temperature readings of the potato were taken every fifteen minutes for three hours and this data is offered in Table~\ref{table:data}. The ambient temperature was 75.2 $^{\circ}$F.''

Model this cooling process and using your model determine the time at which the potato reaches body temperature which is 98.6 $^{\circ}$F. Comment on whether or not this is a reasonable length of time for this to happen.

\begin{table}
\begin{center}
\begin{tabular}{|r|r|} \hline
Time  (min) & Temp ($^{\circ}$F)\\ \hline
0 & 194.0\\ \hline
15 & 183.6\\ \hline
30 & 162.5\\ \hline
45 & 144.3\\ \hline
60 & 132.8\\ \hline
75 & 122.2\\ \hline
90 & 115.9\\ \hline
105 & 109.2\\ \hline
120 & 103.6\\ \hline
135 & 99.0\\ \hline
150 & 97.7\\ \hline
165 & 94.3\\ \hline
180 & 90.5\\ \hline
\end{tabular}

\caption{\label{table:data} Data on temperature of a potato after removal from oven.} 
\end{center}
\end{table}

\teacher{
\section*{NOTES FOR TEACHERS}

This is a direct application  of Newton's Law of Cooling. If students have not seen this law then they can derive the model with hints about the way the temperature changes when the potato is hot and a great difference occurs between the temperature of the potato and the temperature of the room, as opposed to the rate of colling when the potato is cooler, and its temperature is closer to that of the room temperature.  Often, students can wave their hand a what they believe the temperature of the potato is as time proceeds. Certainly, the data, when plotted will show them this temperature-time relationship.


We have posted a fully annotated  Mathematica notebook 1-61-T-Mma-PotatoCooling-Teacher- Version.nb and an accompanying pdf file for those who do not have access to Mathematica in which we offer a step-by-step analysis of solving the differential equation for Newton's Law of Cooling with an unknown cooling parameter $k$.  Further, we determine the best estimate of this parameter in terms of minimizing the sum of squared errors between the model's prediction and the data at each observation's time. The model is given by (\ref{eq:model}) while the best fitting parameter $k$ is found to be 0.0116304. The temperature model is given by (\ref{eq:solve}). 

\vspace*{-.3 true cm}
\begin{equation}\label{eq:model}
T'(t)=-k (T(t)-75.2)\, .
\end{equation}

\vspace*{-.5 true cm}
\begin{equation}\label{eq:solve}
T(t) = 75.2\, +118.8\, e^{-0.0116304 t}\, .
\end{equation}

The time it takes until the potato reaches body temperature of 98.6 $^{\circ}$F is then 139.694 minutes, over two hours. Frankly, this seems quite long for the potato to stay above body temperature and hence appear warm to the taste, but we have not done the experiment ourselves and so we cannot say for certain what is a reasonable time for this drop in temperature from 194 $^{\circ}$F to 98.6 $^{\circ}$F.
\begin{thebibliography}{1}

\bibitem{TopBitS2010}  Woodson, T., C.~Tyndall, and C.~Stepheson.1996.  Estimating Time of Death. \href{http://people.uncw.edu/lugo/MCP/DIFF\_EQ/deproj/death/death.htm}{http://people.uncw.edu/lugo/MCP/DIFF\_EQ/deproj/death/death.htm}.

\end{thebibliography}

}
\end{document}




