\documentclass [12pt] {article}

\usepackage{graphicx, xypic, amsmath}

\topmargin 0pt
\advance \topmargin by -\headheight
\advance \topmargin by -\headsep
\textheight 8.9in
\oddsidemargin 0pt
\evensidemargin \oddsidemargin
\marginparwidth 0.5in
\textwidth 6.5in

\begin{document}
\begin{center}

\Huge{
Optimization}

\normalsize{part two}
\end{center}
\normalsize

\begin{enumerate}
\item An industrial production process costs $C(q)$ million dollars to produce $q$ million units; these units then sell for $R(q)$ million dollars.  If $C(2.1)=5.1$, $R(2.1)=6.9$, $MC(2.1)=0.6$, and $MR(2.1)=0.7$, find the following:
\begin{enumerate}
\item The profit earned by producing 2.1 million units.
\vskip .3in
\item Should the company increase or decrease production to maximize profit?
\vskip .3in
\end{enumerate}

\item Production of an item has fixed costs of \$10,000 and each item costs \$2 to produce.  Assume the relationship between price ($p$) and quantity  demanded ($q$) is \emph{linear}.  Market research shows that 10,100 items are sold when the price is \$5 and 12,872 items are sold when the price is \$4.50.

\begin{enumerate}
\item Express the cost, $C$, of producing $q$ items.
\vskip .7in
\item Recall that the demand curve is \emph{linear}.  Express $p$, as a function of $q$.
\vskip 1.5in
\item Recall if you sell $q$ items for \$$p$, then you will have $p\cdot q$ dollars of revenue.  Use the work from (b) to express the revenue, $R$, from selling $q$ items as \emph{only} a function of $q$.
\vskip .7in
\item Express the profit earned as a function of $q$.  Use this to find how many items the company should produce to maximize profit.
\end{enumerate}

\newpage
\item An offshore oil well is located in the ocean at a point W, which is 5 miles from the closest shorepoint A on a straight shoreline.  The oil is to be piped to a shorepoint B that is 8 miles from A by piping it on a straight line under water from W to some shorepoint P between A and B ad then on to B via a pipe along the shoreline.  If the cost of laying pipe is \$100,000 per mile under water and \$75,000 per mile over land, where should the point P be located to minimize the cost of laying the pipe?

\vskip -2in
\hskip -1in\includegraphics[height=6in]{OptimizationProb1}	


\item You run a small furniture business.  You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by tthe customer later.  The price will be \$90 per chair up to 300 chairs, and above 300, the price will be reduced by \$0.25 per chair (on the whole order) for every additional chair over 300 ordered. 
\begin{enumerate}
\item Write down the revenue as a function of number of chairs sold $q$.  Note, you might want a piece-wise defined function here.
\vfill
\item What is the largest and smallest revenues your company can make under this deal?
\vfill
\end{enumerate}


\end{enumerate}

\end{document}
