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\markright{Chapter 1: Worksheet 3}
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\Bf{\Star\ \Star\ Chapter 1: Worksheet 3  \hfill Jack K. Cohen \hfill Colorado School of Mines}

\vspace{1.5ex}
\Cb{Mathematical Modeling: Words into Mathematics}
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\noindent \Bf{Suggested Problems}
Section 1.1: 60, 62, 63;
Section 1.3: 5, 7, 9

\vspace{1.5ex}
\noindent \Bf{Credit:}  Some of the material on this worksheet comes from a forthcoming text by Paul Zorn and Arnold M. Ostebee at St. Olaf College.

\begin{enumerate}
\item Gardener Alpha plans a rectangular lettuce patch; 100 meters of rabbit-proof fencing is available.  What are the best dimensions?
        \begin{enumerate}
        \item  What does `best' mean in the context of this problem?
        \item Without any analysis, what is the answer (based on your intuition or symmetry considerations)?
        \item Draw a good-sized figure illustrating the conditions of this problem and clearly label it with appropriate dimensions.
        \item From your figure and from the optimization criteria you decided on in part (a), explain why the equation $A(x) = x(50 - x)$ is an adequate translation of the problem into mathematical symbols.  What does $x$ represent?  What does $A$ represent?  What values of $x$ are valid?
        \item Your optimization criteria should imply that we seek the $x$-value in the closed interval $[0, 50]$ that maximize $A(x)$ as given by this formula.  Of course you know the answer without calculation in this case, but let's pretend to be naive in order to set the stage for harder problems.  One way to find this optimal $x$-value is to tabulate $A$.   Here's how to use \Mma\ to do it painlessly:
        
        \Tt{Table[\{x, x(50 - x)\}, \{x, 0, 50, 5\}] //TableForm}

\noindent This statement produces the output:
        \begin{verbatim}        
        0    0
        5    225
        10   400
        15   525
        20   600
        25   625
        30   600
        35   525
        40   400
        45   225
        50   0
        \end{verbatim}

\vspace{1 ex}

It looks like the optimal value is $x= 25$ (we knew that!) with corresponding area 625 m$^2$.  What geometrical object corresponds to this $x$-value?  Still acting in a naive mode, zoom in on the critical region with the successive \Mma\ commands:

        \Tt{Table[\{x, x(50 - x)\}, \{x, 20, 30, 1\}] //TableForm
        
        Table[\{x, x(50 - x)\}, \{x, 24, 26, 0.1\}] //TableForm}
        
\item Graphics provides another approach to solving the optimization problem.  The \Mma\ command 
        
        \Tt{Plot[x(50 - x), \{x, 0, 50\}] }
        
produces the plot shown in Figure 1.   Again, zoom in on the critical region, but this time with appropriate \Mma\ plots.  Hand in a plot showing the region $24 < x < 26$.

                \begin{figure}[htb]
                \epsfysize 100pt
                \centerline{\epsffile{ws3p1.eps}}
                \caption{Graph of $x(50 - x)$ in the physical region of Problem 1.} 
                \end{figure}

\item Describe in geometric terms the location of the optimal $x$-value with respect to the zeros of the function $A(x)$.  Is there any way to justify your intuition about this?

\item We have assumed that ``best'' means maximum area.  In a real life analog to this problem, would this always be true?
                
\item Discuss the generalization of this problem to the case where gardener Alpha has $P$ meters---instead of---100 meters of fencing.  The down side of using a \Em{parameter} like $P$ instead of an explicit value such as 100 is that we can not directly use tabulation or plot techniques.  The up side is that if we can somehow solve the problem \Em{with} the parameter, we have solved a whole family of problems---this is important since we won't encounter perimeters of exactly 100 meters all that often as we proceed through our scientific careers. 

\noindent In this case, it is easy to state the optimization result in terms of the parameter $P$.  Do so!  How would you justify your answer to an obtuse, but practical, person?  Can you see anyway to prove  that your answer is correct to an acute, but pedantic, calculus professor?

 Note: ``Discuss'' means to write coherent sentences describing the situation---please do not just write down equations without giving some context.

        \end{enumerate}

\item Beta also has 100 meters of  fence.  Beta's barn will form one side of the lettuce patch.  (The barn is long enough---over 100 m, say---so that its length isn't a constraint.)
What dimensions are best now?  Can you also state the answer if the 100 is replaced by a general parameter $P$?
        
\item Gardener Gamma plans to fence exactly 1000 m$^2$ of land.  \Em{Any} rectangular shape will do.  For convenience in watering, one of side will lie along a (straight) river.  How much fencing will Gamma need for the other three sides?

\noindent \Bf{Comments}:  Draw and label a figure.  Decide on the unknown quantity.  Derive an equation for the unknown.  Decide if this is a maximum or a minimum problem.  Use Plot to get a rough solution and Table to get a numerically accurate solution.

\item You land in your space ship on a spherical asteroid. Your partner walks 1000 feet away along the smooth surface carrying a 10 feet rod, and thereby vanishes over the horizon.  When she places one end on the ground and holds  the rod straight up and down, you (lying on your stomach to rest from the arduous journey from Earth) can just barely see the tip of the rod.  The ultimate quest is to determine the radius $r$ of the asteroid, but for now:
	\begin{enumerate}
	\item Draw a careful diagram and label it with the quantities you think will be useful in getting an equation involving only $r$ and known quantities.
	\item Derive this equation from your diagram.
	\end{enumerate}
        

\end{enumerate}
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