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\centerline{Mathematics 132 -- Calculus for Physical and Life Science 2}
\centerline{Group Work on Differential Equation Models}
\centerline{March 19, 2008}
\vskip 10pt
\noindent
{\it The Story}
\bigskip
Homer needs an operation to clear carotid arteries blocked
by fatty plaque deposits (these are the arteries
in the neck that feed blood to the brain, so it's pretty
serious!).  When he is brought into the hospital,
he is given a sedative to help him sleep.  The doctor wants
to operate as soon as possible, but cannot safely do so
until the concentration of sedative in Homer's body is less than 
$0.03$ milligrams per liter.  You will use the following table
of data to determine how many hours the doctor will need to 
wait to start the operation.  Blood samples were taken 
every 10 minutes and the sedative concentration was determined
in milligrams per liter:
\[
\begin{array}{lcccr}
\hline
\vline & \text{time (minutes)} & \vline & \text{Concentration (mg/L)} & \vline \\ \hline
\vline & 0 & \vline & 10.0 & \vline \\
\vline & 10 & \vline & 6.07 & \vline \\
\vline & 20 & \vline & 3.68 & \vline \\
\vline & 30 & \vline & 2.23 & \vline \\
\vline & 40 & \vline & 1.35 & \vline \\
\vline & 50 & \vline & .820 & \vline \\
\vline & 60 & \vline & .498 & \vline \\
\vline & 70 & \vline & .302 & \vline \\
\vline & 80 & \vline & .183 & \vline \\
\vline & 90 & \vline & .111 & \vline \\ \hline
\end{array}
\]
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\noindent
{\it Questions}
\bigskip
\begin{enumerate}
\item Is the rate of change of the concentration of the sedative a constant?  Explain.
\item Estimate the rate of change of the concentration of the sedative at 30 minutes, 60 minutes, and 
80 minutes.
\item Show that the rate of change of the concentration of the sedative is (roughly)
proportional to the concentration at the times from question 2.  
\item Assuming that the pattern remains true for all times, write the relationship 
given in question 3 as a differential equation for 
the function $C(t) = $ concentration of the sedative at time $t$.  
\item What is numerical value of the constant of proportionality in your equation?
\item Solve the differential equation from question 4 using the constant from question 5.
\item When is the concentration of sedative half of what it was at $t = 0$?
\item If Homer received the sedative at 2:00 pm, when can the doctor start the operation?
\end{enumerate}
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\noindent
Turn in your answers at the end of the period.  
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