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\markboth{}{MAT 1110 \hfill Project 2 \hfill }

\hspace*{-0.33in}%
\begin{tabular}{lr}
From: \parbox[t]{2in}{%
A. Richmond Birnan \newline
Springfield, USA  \newline
October 22, 2007 \newline}
&
\hspace{-0.3in}\parbox{2.7in}{
\includegraphics[width=2.6in]{NNSA} \newline
\textsc{\small National Nucular Insecurity Agency}} \\

To: \parbox[t]{4in}{%
Math 1110 Students \newline
Appalachian State University}
&
\end{tabular} \\[2ex]

\noindent%
Dear Calculus Students:

We have decided that the time is ripe to diversify our business. With the shortage of flu vaccines last fall and rising deductibles on prescriptions, the market seems perfectly situated to introduce the \emph{Cough and Sniffle} brand over-the-counter flu remedy. When we went looking for help in determining how much to produce, your enterprising and resourceful professor naturally referred us to you.

We have some approximations for the potential costs and revenues. There will be a small initial fixed cost of approximately $\$10,\!000$ to convert a portion of our facility from handling Illudium Q--36 explosive space modulator waste to producing the flu remedy.

The plan is to distribute the remedy to retailers in cases containing $1,\!000$ doses each. Upon the advice of your enterprising and resourceful professor, my top-notch R\&D department has also determined, through extensive research, the materials costs, the labor costs, and the price the market will bear for several different levels of production.

\begin{center}
\begin{tabular}{||r|| *{6}{c|} |}
\hline\hline
No of Cases Produced & $100$ & $600$ & $1100$ & $2100$ & $3100$ & $4100$ \\
\hline
Materials cost per case (\$) & $167$ & $72$ & $17$ & $34$ & $217$ & $568$ \\
\hline
Labor cost per case (\$) & $300$ & $175$ & $87$ & $7$ & $60$ & $247$ \\
\hline
Selling price per case (\$) & $69$ & $71$ & $72$ & $67$ & $62$ & $34$ \\
\hline\hline
\end{tabular}
\end{center}

There are several vital pieces of information we need to know in order to plan our production. Specifically, we would like to know 
\begin{itemize}
\item the number of cases we should manufacture to maximize total profit
\item the number to produce to maximize the profit per unit sold
\item the break even point
\end{itemize}
In addition, we would also like to know the number of cases that
\begin{itemize}
\item minimizes the average cost per case
\item minimizes the total cost.
\end{itemize}

I will need your report by Tuesday, October 30, in order to prepare for production.


\subsection*{A Few Comments From Your Enterprising and Resourceful Professor}
The information requested will certainly not occur at one of the given data points, so I would suggest that you try to find a function that models the materials cost, a function that models the labor cost, and one that models the selling price reasonably well. Plotting these points should give you an idea of the types of functions that will be likely candidates. Once you have these functions, you should be able to determine the requested information without difficulty.

Write your report to Mr Birnan --- don't assume that he knows all of calculus; explain carefully what you have done and why.

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