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\begin{document}

\newpage

\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Measuring the Quality of Insulated Water Bottles}


\markboth{Insulated Water Bottles}{Newton's Law of Cooling}

\author{Brody Johnson, Elodie Pozzi \\
Saint Louis University\\
Saint Louis MO 63103 USA\\
brody.johnson@slu.edu, elodie.pozzi@slu.edu
}

\makeStitlePDFLaTex

\begin{abstract}
This project involves the application of Newton's law of cooling to the study of insulated water bottles.  Students have the option to conduct experiments with their own bottles outside of class or use data included in the student version.  The modeling scenario leads the students through an analysis of the phase line, the solution of the separable differential equation, and the implementation of the model on the data.  Finally, the notion of half-life for a water bottle is introduced, which provides a concrete point of comparison for the quality of two insulated water bottles.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

The goal of this project is to compare the effectiveness of two insulated bottles to keep a drink cold over a long period of time.  If the two bottles are filled with water of equal temperature and kept in the same environment, this comparison can be accomplished by measuring the temperatures of the contents at any later time.  The bottle with the colder contents must be better.  But what if the contents do not start at the same temperature or are not kept in the same environment?  In order to make a fair comparison when these variables are not controlled, a mathematical model is needed.  The basis for this mathematical model comes from work carried out over three hundred years ago by the famous scholar Isaac Newton.  Newton studied the evolution of the temperature of a small body over time and his experiments showed that the temperature of the body changed at a rate proportional to the temperature difference between the body and the surrounding environment.  More precisely, Newton discovered that
\begin{equation} \label{Newton}
\frac{dT}{dt}=k\left(T_E-T\right), 
\end{equation}

\noindent
where $T(t)$ represents the temperature of the small body at time $t$, $T_{E}$ is the temperature of the surrounding environment, and $k>0$ is a constant of proportionality. The first-order differential equation \eqref{Newton} is now known as \emph{Newton's Law of Cooling}, although this law applies to warming equally well.

\medskip
The basic outline for this modeling activity is as follows.  Temperature data will be collected for two water bottles over a period of at least 4 hours in an environment of more or less constant temperature.  Newton's Law of Cooling will be used to predict the future temperature of the contents from the starting temperature for any values of the constants $k$ and $T_{E}$.  The constants will then be estimated through a trial-and-error procedure in which the predicted values are made as close as possible to the observed temperatures.  The empirical estimates of $k$ and $T_{E}$ for each experiment will then be used to determine which bottle provides the better protection for a cold drink.  This general framework allows for the examination of a wide variety of interesting questions. 
\begin{itemize}
\item Are modern vacuum-insulated steel bottles better than the glass-walled thermoses that were popular 30--40 years ago?  (See \cite{Rick} for an excellent treatment.)

\item Does a large bottle offer any advantage over a small bottle of similar construction?

\item Is a brand-name insulated water bottle worth the cost?  Or, could a cheaper knock-off bottle offer comparable performance?

\item How does a typical plastic water bottle from a vending machine stack up against a vacuum-insulated water bottle?
\end{itemize}

\section*{MATERIALS}

The following materials are needed in order to conduct the experiment:
\begin{itemize}
\item two insulated water bottles;
\item a pitcher or other large container;
\item a waterproof food thermometer;
\item a clock;
\item cold water. 
\end{itemize}

\noindent
It is recommended that bottles of similar capacity be used unless the intent of the experiment is to specifically examine the effect of bottle size and, in this case, the two bottles should be of the same construction/brand.  Also note that the probe on the thermometer must be small enough to be inserted into the mouth of the water bottle and long enough to reach past the neck of the bottle.

\section{Data Collection}

Data collection will take a minimum of 4 hours, so it is recommended that this be done outside of class.  It may make sense for each student to collect data for one bottle and share their findings (as well as a description of their bottle) with the rest of the class/group.  This will likely facilitate numerous interesting comparisons.

\medskip \noindent
{\bf Student Task:} Carefully follow the procedure below, noting any deviations.

\begin{itemize}
\item {\bf Cool a large pitcher of water overnight.}

\smallskip \noindent
The actual temperature of the cool water is not terribly important as long as the water remains a liquid and will be cooler than the ambient temperature where the experiment is to be conducted.

\item {\bf Devise a means to insert the thermometer probe into the water bottle.}

\smallskip \noindent
Many insulated water bottles have a drinking spout that will allow for insertion of the probe for the food thermometer.  Make sure that each temperature reading can be made relatively quickly ($\approx 15$ seconds), so the cap is open for as little time as possible.  

\item {\bf Fill the bottle(s) with cool water to within one inch of the top.}

\noindent
It is important to make sure the temperature probe will easily reach the water.  Also, less air in the bottle will mean less opportunity for heat exchange when the cap is opened and closed.

\item {\bf Place the bottle(s) in an environment where the surrounding temperature will be relatively constant.}

\noindent
The mathematical model \eqref{Newton} assumes a constant environmental temperature, so large changes in the temperature of the surroundings will be problematic.  Good locations will be indoors and away from windows and heat/cooling vents.  If it is possible to measure/estimate the environmental temperature, do so.

\item {\bf Record the water temperature at regular intervals for at least 4 hours.}

\noindent
A high quality insulated water bottle can keep water cool for several hours.  It would be a good idea to keep the bottle(s) close by as you work on homework or binge watch a new streaming series.  Ideally, the temperature of the water should be recorded every 30 minutes for 3 or 4 hours.  Record the temperature every 15 minutes if the insulating properties of the bottle are not expected to be good, e.g., for a clear plastic water bottle or single-walled steel bottle.
\end{itemize}

For those who are unable to perform their own experiment, data sets for three different water bottles (of comparable size) are provided in Table \ref{datasets}.

\begin{table}[htp]
\caption{Temperature data for three water bottles.} \label{datasets}
\centering
\begin{small}
\begin{tabular}{|c|c|c|c|}
 \hline
        &  Bottle 1     & Bottle 2     & Bottle 3 \\ 
  Time  &  Temperature  & Temperature  & Temperature \\
 (minutes) & ($^{\circ}$F) & ($^{\circ}$F) & ($^{\circ}$F) \\ 
 \hline
0 & 39.0 & 47.5 & 39.2 \\
\hline
30 & 42.8 & 48.2 & 49.5 \\
\hline
60 & 44.9 & 49.5 & 55.2 \\
\hline
90 &  46.2 & 50.2 & 59.9 \\
\hline
120 & 48.3 & 51.1 & 63.3 \\
\hline
150 &  49.9 & 52.2 & 65.5 \\
\hline
180 &  51.1 & 52.9 & 67.5 \\
\hline
210 &  52.7 & 54.0 & 69.3 \\
\hline
240 & 54.7 & 54.5 & 70.7 \\
\hline
270 & 56.3 & 55.2 & 71.6 \\
\hline
300 & 57.6 & 55.9 & 72.5 \\
\hline
330 & 58.1 & 56.5 & 73.0 \\
\hline
360 & 59.2 & 57.0 & 73.6 \\
\hline
390 & 60.8 & 57.4 & 73.9 \\
\hline
420 & 62.1 & 58.1 & 74.3 \\
\hline
\end{tabular}
\end{small}
\end{table}

\section{Qualitative Analysis}

Qualitative analysis leads to the understanding of certain characteristics of the solutions of a differential equation (what is going to happen) without explicitly solving the differential equation (no formula is used).  This is especially important when the differential equation cannot be solved using available techniques.  Notice that the differential equation \eqref{Newton} is \emph{autonomous}, which means the slope $\frac{dT}{dt}$ at any point $(t,T)$ is dependent only on $T$.  Recall that the \emph{phase line} of an autonomous differential equation $\frac{dT}{dt} = f(T)$ is essentially an annotated graph of $f(T)$ over the $T$-axis.  Positive values of $f(T)$ correspond to positive values of $\frac{dT}{dt}$, which means $T$ will increase (move to the right) as $t$ increases.  Similarly, negative values of $f(T)$ correspond to negative values of $\frac{dT}{dt}$, which means $T$ will decrease (move to the left) as $t$ increases.  These directions are noted on the $T$-axis using arrows to the right or left.  If $f(T)=0$, however, then $\frac{dT}{dt}=0$ and $T(t)$ will remain constant.  Such a point $T$ is called an \emph{equilibrium point} of the differential equation.  If nearby solutions tend towards the equilibrium point it is said to be \emph{stable}.  Otherwise, the equilibrium point is said to be \emph{unstable}.

\bigskip \noindent
{\bf Student Task:} Construct a phase line for the differential equation \eqref{Newton} in the space provided below.  Assume that both $k$ and $T_{E}$ are positive.

\begin{center}
\begin{tikzpicture}
    \begin{axis}[width=4.0in,height=1.75in,
            xlabel=$T$,ylabel=$\frac{dT}{dt}$,xmin=0,xmax=105,ymin=-3,ymax=3,
            axis y line=left,axis x line=middle,
            xtick={63},ytick=\empty,
            xticklabels={$T_{E}$},
             every axis x label/.style={at={(current axis.right of origin)},anchor=west},
             every axis y label/.style={at={(current axis.north west)},anchor=south}]
\end{axis}
\node[scale=1] at (-0.2,1.25) {$0$};
\end{tikzpicture}
\end{center}

\bigskip 
\begin{samepage}
\noindent
Questions:

\begin{enumerate}[label=\thesection.\arabic*,start=1]
\item What are the equilibrium points of this model?  Describe the stability of each equilibrium point.
\item What can you say about $T(t)$ for large values of $t$?  Does your answer depend on $T(0)$?  Explain.
\item How does the parameter $T_{E}$ affect the phase line?  Explain.
\item How does the parameter $k$ affect the phase line?  Explain.
\end{enumerate}
\end{samepage}


\section{Model Solution} \label{modsol}

In order to validate the model using real data, it will be necessary to predict the temperatures $T(t)$ for $t>0$ for any values of $k$ and $T_{E}$ from a given initial condition, $T(0)$.  In other words, a solution of \eqref{Newton} must be found.  Since this differential equation is separable, it is possible to find an explicit solution via integration.  Notice that \eqref{Newton} can be written as
$$ \frac{dT}{dt} = -k (T-T_{E}),$$

\noindent
which leads to the integral equation
\begin{equation} \label{separable}
\int \frac{1}{T-T_{E}} \; dT = \int -k \; dt.
\end{equation}

\medskip \noindent
{\bf Student Task:} Solve the integral equation \eqref{separable} and find an expression for $T(t)$, $t>0$, in terms of the initial condition, $T(0)$.  Be careful not to forget the constant of the integration!

\medskip
\begin{samepage}
\noindent
Questions:

\begin{enumerate}[label=\thesection.\arabic*,start=1]
 \item What happens to $T(t)$ as $t\rightarrow \infty$?  Does your answer agree with the findings of qualitative analysis?

 \item Assume $T(0)=39^{\circ} \text{F}$ and $T_{E} = 75^{\circ} \text{F}$.  Find the expression for $T(t)$, which will still contain the constant $k$. 
 
 \item Graph $T(t)$ for $k=0.1$ and $k=0.2$ for $0\le t\le 60$, again using $T(0)=39^{\circ} \text{F}$ and $T_{E}=75^{\circ} \text{F}$.  What effect does $k$ have on the solution curve?
 
 \item How do the constants $k$ and $T_{E}$ relate to the quality of the insulated water bottle?  How would a well-insulated bottle be distinguished from a poorly insulated bottle using these constants?
 
\end{enumerate}
\end{samepage}

\section{Model Implementation}

The final step in the project is to incorporate real data in order to estimate the constants $k$ and $T_{E}$ for each experiment.  This will be accomplished using a spreadsheet.

\medskip \noindent
{\bf Student Task:} Create a spreadsheet to compare the mathematical model with the observed temperature data.  Adjust the constants $k$ and $T_{E}$ so that the sum of the squared error (SSE) is as small as possible.  Let $T_n$ represent the temperature predicted by the mathematical model at time $t_{n}$ and let $\widehat{T}_{n}$ represent the experimental temperature observed at time $t_{n}$.  The SSE can be computed as
$$ \text{SSE} = \sum_{n=1}^{N} \left ( T_{n}-\widehat{T}_{n} \right )^{2}. $$

\noindent
Tips for the Spreadsheet: (see Figure \ref{spreadsheet})

\begin{figure}[h]
\centering
\includegraphics[width=4.0in]{spreadsheet-eg.png}
\caption{Screenshot of a sample spreadsheet.} \label{spreadsheet}
\end{figure}
\vspace*{-.4 true cm}
\begin{itemize}
\item Use the following columns: (A) Time $t_{n}$, (B) Temperature $T_{n}$ (predicted), (C) Temperature $\widehat{T}_{n}$ (observed), (D) Squared Error.
\item Create cells for $k$, $T_{E}$, $T(0)$, and $\text{SSE}$.
\item Enter the value of $T(0)$. 
\item Create a scatter plot to compare $T_{n}$ (predicted) with $\widehat{T}_{n}$ (observed).
\item Start with $k=0.1$ and $T_{E}=72$.  (If you measured the ambient temperature you may want to start with this value instead of $T_{E}=72$.)  Make small adjustments to $k$ and $T_{E}$ while watching the graph and the sum of the squared error.  Think about how the change will affect the graph of $T_{n}$ (predicted) before implementing it.  
\item {\sc Optional:} If the spreadsheet is equipped with a solver, use the solver to optimize the values of $k$ and $T_{E}$ so that the sum of the squared errors is minimized.  The value of $T_{E}$ should be constrained to a limited range, say, $68^{\circ} \text{F}$ to $80^{\circ} \text{F}$, in most cases.

\item Repeat these steps for each experiment.
\end{itemize}




\begin{samepage}
\noindent
Questions:

\begin{enumerate}[label=\thesection.\arabic*,start=1]
\item Does the model provide a good approximation of the data for each experiment?  Are there any data points that stand out?  Explain.

\item What were the optimal values of $k$ and $T_{E}$ for each experiment?  Do the values for $T_{E}$ make sense based on the location where the experiment was conducted?

\item Define the \emph{half-life}, $t_{0.5}$, of a water bottle as the time for which
$$ T(t_{0.5}) = \frac{T_{E}+T(0)}{2}.$$

\noindent
Use the general solution of \eqref{Newton} found in Section \ref{modsol} to derive a formula for $t_{0.5}$ depending only on $k$.

\item Calculate the half-life for each bottle studied.  Does one bottle have superior insulating properties?  If so, is the difference in half-life substantial?
\end{enumerate}
\end{samepage}


\begin{thebibliography}{98}
\bibitem{Rick}
Sorensen, Rick. 2021. {The Thermodynamics of Vacuum-Insulated Bottles: An Investigation.}  \url{https://www.vernier.com/2020/03/10/the-thermodynamics-of-vacuum-insulated-bottles-an-investigation/}. Accessed 17 June 2021. 
\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

\subsection*{Background Knowledge and Placement in a Course}

This modeling scenario introduces {\bf Newton's Law of Cooling} and assumes that students are familiar with the notion of a {\bf phase line} and able to solve {\bf separable differential equations}.  As such, this project could be used very early in most ODE courses.  Instructors are also encouraged to look at a related modeling scenario by Eli Goldwyn that also examines water bottles \cite{Gold}.

The intent of the project is to have students conduct experiments in which the water temperature of competing water bottles is monitored in order to determine which one does a better job keeping its contents cold.  The introduction to the student version proposes a few research questions that the class could choose to pursue, but any two water vessels can be used.

In most cases, it will not be practical to collect data during a class meeting due to the amount of time required.  If the instructor has one or two waterproof thermometers that can be loaned out, it might be a good idea to seek volunteers who can collect data a few days before the activity is assigned.  Alternatively, the instructor could collect the data or use data provided in the instructor resources.  Three of these data sets are already included in the student version.

\subsection*{Materials}

The following materials are needed in order to conduct the experiment:
\begin{itemize}
\item at least one insulated water bottle;
\item a pitcher or other large container;
\item a waterproof food thermometer;
\item a clock;
\item cold water. 
\end{itemize}

\noindent
The only unusual items on this list for students would be the large pitcher and the waterproof food thermometer.  Note that placing a well-insulated water bottle in a refrigerator overnight may not achieve the desired result, due to the excellent insulating properties of the bottle.  An open-topped pitcher will allow the water to cool much faster, but if the experimenter has access to sufficiently cold running water, the pitcher can be eliminated from the materials.  As mentioned above, it might be smart for the instructor to acquire one or two suitable thermometers that can be loaned to student volunteers who will collect data for the class.

\subsection*{Customization}

As written, this modeling scenario can be used in a variety of ways.
\begin{itemize}
\item {\bf In-Class Activity:} Students could work through the different sections of the project in groups during a single class meeting.  The instructor would have the option to plan for data collection ahead of this class or to use data provided in the instructor resources.

\item {\bf Out-of-Class Assignment:} The instructor could ask students to complete the modeling scenario over about one week and turn in written solutions.  It is still advised that data collection be planned out ahead of time so all students will have access to data as soon as the assignment is given.  The option to use included data could also be adopted.
    
\item {\bf In-Class Demonstration:} The instructor could choose to incorporate the content of this modeling scenario in one class or over a series of class meetings.  One approach might involve using a small portion of the lessons devoted to autonomous differential equations, separable equations, and modeling to work through the corresponding aspects of this project, culminating with the application of Newton's law of cooling to the experimental data.  The written questions could be used as discussion questions throughout these lessons.
\end{itemize}

\subsection*{Solutions}

\noindent
{\bf Qualitative Analysis}

\noindent
The phase line for Newton's law of cooling, \eqref{Newton}, is given below.

\begin{center}
\begin{tikzpicture}
    \begin{axis}[width=4.0in,height=1.75in,
            xlabel=$T$,ylabel=$\frac{dT}{dt}$,xmin=0,xmax=105,ymin=-3,ymax=3,
            axis y line=left,axis x line=middle,
            xtick={63},ytick=\empty,
            xticklabels={$T_{E}$},
             every axis x label/.style={at={(current axis.right of origin)},anchor=west},
             every axis y label/.style={at={(current axis.north west)},anchor=south}]
             
        \addplot[red,domain=-5:105,samples=106]{0.035*(63-x)};
        \node at (axis cs:63,0)[circle,fill,inner sep=2pt]{};     
        \node at (axis cs:8,0) {$>$}; 
        \node at (axis cs:20,0) {$>$}; 
        \node at (axis cs:32,0) {$>$}; 
        \node at (axis cs:44,0) {$>$}; 
        \node at (axis cs:56,0) {$>$}; 
        \node at (axis cs:70,0) {$<$}; 
        \node at (axis cs:82,0) {$<$}; 
        \node at (axis cs:94,0) {$<$}; 
             
\end{axis}
\node[scale=1] at (-0.2,1.25) {$0$};
\end{tikzpicture}
\end{center}

\medskip \noindent
Questions:

\begin{enumerate}[label=2.\arabic*,start=1]
\item What are the equilibrium points of this model?  Describe the stability of each equilibrium point.

\begin{proof}[Answer:]
The model has a single equilibrium point at $T=T_{E}$.  This equilibrium point is stable because nearby solutions tend towards $T=T_{E}$.
\end{proof}

\item What can you say about $T(t)$ for large values of $t$?  Does your answer depend on $T(0)$?  Explain.

\begin{proof}[Answer:]
If $T(0)>T_{E}$, then $T(t)$ will decrease towards $T_{E}$ as $t\rightarrow \infty$, while if $T(0)<T_{E}$, $T(t)$ will increase towards $T_{E}$.  If $T(0)=T_{E}$, then $T(t)$ will remain constant as $t$ increases.  The limiting value is independent of $T(0)$, although the shape of the graph changes depending on the initial value.
\end{proof}

\item How does the parameter $T_{E}$ affect the phase line?  Explain.

\begin{proof}[Answer:]
As expected, increasing $T_{E}$ will shift the equilibrium temperature to the right (warmer), while decreasing $T_{E}$ will shift the equilibrium temperature to the left (colder).
\end{proof}

\item How does the parameter $k$ affect the phase line?  Explain.

\begin{proof}[Answer:]
The phase line for any two positive values of $k$ will be identical, so the phase line alone does not provide much intuition about the role that $k$ plays in the mathematical model. 
\end{proof}
\end{enumerate}

\medskip \noindent
{\bf Model Solution}

\noindent
Notice that
$$ \int \frac{1}{T-T_{E}} \; dT = \int -k \; dt \qquad \text{which becomes} \qquad \ln{\vert T-T_{E}\vert} = -kt + C. $$

\noindent
Exponentiating on both sides leads to
$$ \vert T-T_{E} \vert = e^{-kt+C}.$$

\noindent
The expression $e^{C}$ is simply another unknown constant and can be replaced by $A$ for simplicity.  This new constant $A$ can also absorb the sign of $T-T_{E}$ as the absolute value is removed.  (This is possible because the sign of $T-T_{E}$ is independent of $t$.) It follows that $T-T_{E} = A e^{-kt}$, leading to the final form of the solution,
$$ T(t) = T_{E} + A e^{-kt}, \qquad t>0.$$

\noindent
Questions:

\begin{enumerate}[label=3.\arabic*,start=1]
\item What happens to $T(t)$ as $t\rightarrow \infty$?  Does your answer agree with the findings of qualitative analysis?

\begin{proof}[Answer:]
The constant $k$ is positive, so $e^{-kt}\rightarrow 0$ as $t\rightarrow \infty$, which means $T(t)\rightarrow T_{E}$ as $t\rightarrow \infty$.  This is exactly what the phase line indicated.
\end{proof}

\item Assume $T(0)=39^{\circ} \text{F}$ and $T_{E} = 75^{\circ} \text{F}$.  Find the expression for $T(t)$, which will still contain the constant $k$.

\begin{proof}[Answer:]
At $t=0$, $T(0) = T_{E} + A e^{k\cdot 0}$ and $e^{0}=1$, so that $A=T(0)-T_{E}$.  Using this formula, one finds that $A=39-75=-36$ and thus
$$ T(t) = 75 - 36 e^{-kt}.$$
\end{proof}
 
\item Graph $T(t)$ for $k=0.1$ and $k=0.2$ for $0\le t\le 60$, again using $T(0)=39^{\circ} \text{F}$ and $T_{E}=75^{\circ} \text{F}$.  What effect does $k$ have on the solution curve?

\begin{proof}[Answer:]
The two solution curves are shown in the figuire below..  It is evident that increasing $k$ causes the rate of change to increase, which means the temperature will increase more quickly and approach the ambient temperature in less time.


\end{proof}


\begin{center}
\includegraphics[width=3.5in]{kcomp.png}
\end{center}


\item How do the constants $k$ and $T_{E}$ relate to the quality of the insulated water bottle?  How would a well-insulated bottle be distinguished from a poorly insulated bottle using these constants?
   
\begin{proof}[Answer:]
Recall that $T_{E}$ represents the ambient temperature at the location where the bottle is placed.  As such, $T_{E}$ is not a property of the bottle.  The insulating property of the bottle is conveyed by the constant $k$, with a smaller value for $k$ indicating that the temperature of the water will change more slowly.  So, a smaller value of $k$ indicates better insulation.
\end{proof}
\end{enumerate}


\medskip \noindent
{\bf Model Implementation}

\noindent
The instructor resources include a spreadsheet with one sheet devoted to each of the provided data sets.  The instructor can copy any of these sheets and modify it to incorporate data collected for their class.  The solutions below will focus on the comparison of Bottle 1 and Bottle 2 using the data sets presented in Table \ref{datasets}.  The graph below includes both the experimental data (circular data markers) and the optimized models (solid curves) for each experiment.

\begin{center}
\includegraphics[width=3.5in]{bottlecomp.png}
\end{center}

\medskip \noindent
Questions:

\begin{enumerate}[label=4.\arabic*,start=1]
\item Does the model provide a good approximation of the data for each experiment?  Are there any data points that stand out?  Explain.

\begin{proof}[Answer:]
Yes.  There are a few data points that stray a little bit from the curve given by the model, but none are off by a significant amount.  It can take several seconds for the thermometer reading to settle on one number, so high readings may be the result of accepting a reading too quickly.  Sometimes a drop of water may cling to a part of the probe that is not immersed in the water and this may affect the temperature reading slightly as well.
\end{proof}

\item What were the optimal values of $k$ and $T_{E}$ for each experiment?  Do the values for $T_{E}$ make sense based on the location where the experiment was conducted?

\begin{proof}[Answer:]
The optimal values for Bottle 1 were $k=1.200\times 10^{-3}$ and $T_{E}=74.0$, while the experiment with Bottle 2 produced optimal values of $k=2.615\times 10^{-3}$ and $T_{E}=72.8$.  The experiments for Bottle 1 and Bottle 2 were not conducted in the same location and were performed on different days, so the difference in ambient temperature is completely plausible.
\end{proof}

\item Define the \emph{half-life}, $t_{0.5}$, of a water bottle as the time for which
$$ T(t_{0.5}) = \frac{T_{E}+T(0)}{2}.$$

\noindent
Use the general solution of \eqref{Newton} found in Section \ref{modsol} to derive a formula for $t_{0.5}$ depending only on $k$.

\begin{proof}[Answer:]
Using the solution from Section \ref{modsol} along with the fact that $A=T(0)-T_{E}$, it follows that
$$ T(t) = T_{E} + (T(0)-T_{E}) e^{-kt}.$$

\noindent
At $t=t_{0.5}$, this becomes
$$ \frac{T(0)+T_{E}}{2} = T_{E} + (T(0)-T_{E}) e^{-k t_{0.5}}.$$

\noindent
Moving $T_{E}$ to the left-hand side leads to
$$ \frac{T(0)-T_{E}}{2} = (T(0)-T_{E}) e^{-k t_{0.5}} \qquad \longrightarrow \qquad \frac{1}{2} = e^{-k t_{0.5}}.$$

\noindent
Finally, applying the natural logarithm and dividing by $-k$ produces a formula for the half-life:
$$ t_{0.5} = -\frac{1}{k} \ln{ \left ( \frac{1}{2} \right )} = \frac{\ln{2}}{k}.$$
\end{proof}

\item Calculate the half-life for each bottle studied.  Does one bottle have superior insulating properties?  If so, is the difference in half-life substantial?

\begin{proof}[Answer:]
The half-life for Bottle 1 is approximately 577.6 minutes, while the half-life for Bottle 2 is about 265.1 minutes.  The goal of an insulated water bottle is to preserve the temperature difference between the environment and its contents.  The half-life measures the amount of time required for the temperature difference to be cut in half.  The fact that Bottle 1's half-life is more than twice that of Bottle 2 is significant and implies that the temperature difference for Bottle 2 will be reduced by a factor of four before that of Bottle 1 is cut in half!
\end{proof}
\end{enumerate}

\begin{thebibliography}{99}
\bibitem{Gold} 
Goldwyn, Eli. 2019. 1-068-T-WaterBottleCooling. \url{https://www.simiode.org/resources/6255} Accessed 15 June 2021. 
\end{thebibliography}



\subsection*{Numerical Methods (Optional Section)}

This optional section may be included at the instructor's discretion or possibly used for an out-of-class project.  No solutions will be provided for this section, although the spreadsheet included in the instructor's resources will have one sheet that implements the numerical solution described below.

Outside the confines of a controlled experiment, it may be unreasonable to assume that $T_{E}$ in \eqref{Newton} remains constant.  If $T_{E}$ is allowed to depend on $t$, one could still hope to employ analytical methods to solve the differential equation, e.g., if $T_{E}(t)$ increases linearly or follows a simple periodic function.  However, as long as it is possible to accurately measure $T_{E}$ during an experiment, numerical methods provide a practical alternative to this more general problem.  The improved Euler method, summarized in Figure \ref{IEMethod}, is adopted here.  The corrective nature of the improved Euler method should make it possible to implement a numerical solution using the same $\Delta t$ that was used for the collection of temperature data, although it would probably be a good idea to use $\Delta t \le 15$ minutes.

\begin{figure}[thp]
\begin{center}
\fbox{\begin{minipage}{0.625\textwidth}
\begin{small}
\noindent
{\bf Improved Euler Method:}

\noindent
Choose $\Delta t>0$ and consider the initial value problem
$$ \frac{dT}{dt} = f(t,T) \quad \text{subject to} \quad T(t_{0})=T_{0}.$$

\noindent
Define $t_{n}=t_{0}+n\Delta t$.  The approximate value of $T(t_{n})$, denoted by $T_{n}$, can be computed from $T_{n-1}$ as follows:

\begin{enumerate}
\item Compute the initial slope estimate: $m_{1} = f(t_{n-1},T_{n-1})$.

\medskip
\item Compute a temporary estimate for $T$: $\widetilde{T} = T_{n-1} + m_{1} \Delta t$.

\medskip
\item Compute the second slope estimate: $m_{2} = f(t_{n}, \widetilde{T})$.

\medskip
\item Compute $T_{n}$: 
$$T_{n} = T_{n-1} + \left ( \frac{m_{1}+m_{2}}{2} \right ) \Delta t.$$
\end{enumerate}
\end{small}
\end{minipage}}
\end{center}
\caption{The Improved-Euler Method.} \label{IEMethod}
\end{figure}

\bigskip \noindent
{\bf Student Task:} Create a spreadsheet to implement the improved Euler Method on \eqref{Newton} for your experimental data (including measurements of $T_{E}$ over time) using $\Delta t$ equal to the time interval between your temperature readings.  Optimize the value for $k$ by minimizing the sum of the squared error.

\noindent
Tips for the Spreadsheet:
\begin{itemize}
\item Use the following columns: (A) Time $t_{n}$, (B) Temperature $T_{n}$ (predicted), (C) Slope $m_{1}$, (D) Temporary Prediction $\widetilde{T}_{n}$, (E) Slope $m_{2}$, (F) Temperature $\widehat{T}_{n}$ (observed), (G) Environmental Temperature $T_{E,n}$ (observed), (H) Squared Error.
\item Create cells for $k$, $\Delta t$, and SSE.
\item Create a scatter plot to compare $T_{n}$ (predicted) with $\widehat{T}_{n}$ (observed).
\item Start with $k=0.1$.  Make small adjustments to $k$ while watching the graph and the sum of the squared error.  Think about how the change will affect the graph of $T_{n}$ (predicted) before implementing it.
\item Repeat these steps for each experiment.
\end{itemize}

\noindent
Questions:

\begin{enumerate}[label=5.\arabic*,start=1]
\item Does the approximate solution obtained using the improved Euler method provide a good fit with the data?  Explain.

\item Discuss the optimal value of $k$ and the half-life for each experiment.  Does one bottle have superior insulating properties?
\end{enumerate}

}
\end{document}
