%m211oh63.tex (prev.: oh61) - The Rabbbit Problem 
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\input{m211oh.tex}
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\begin{titlepage}
\vspace*{-20pt}

\begin{center}\tcr{\rmH The Rabbit Problem} \end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  

\item[\tcr{Situation:}] 
\tcr{1)} The \tcg{females} of a certain rabbit species produce \mc{6} \tcg{females} per year.

\tcr{2)} These \tcg{mature} in \mc{1} year. 

\tcr{3)} \tcg{None} die.\vspace{-10pt}

\tcr{4)} We start with \mc{\left\{\begin{array}{ll} a_{\scr0} & \mbox{adults}\\
b_0 & \mbox{young} \end{array} \right.}.

\item[\tcr{Questions:}] \tcr{1)} How many female rabbits are there after \mc{n} years?

\tcr{2)} If \mc{c_{\scr k-1}} is the cost (in dollars) of feeding a  \tcg{mature} female in the \mc{k^{th}} year, what is the total cost \mc{T(n)} to feed the rabbits for \mc{n} years? \newline
\tcr{Assume:} there is \tcr{no cost} to feed the \tcg{young}.\vspace{-10pt}  
 
\item[\tcr{Analysis:}] Let 
\raisebox{1.5cm}{\begin{minipage}[t]{13.25cm} 
\tcb{\beqn
a_{\scr n} & = & \# \mbox{adult females at end of year}\, n
\\
b_{\scr n} & = & \# \mbox{females born in year}\, n
\eeqn} 
\end{minipage}}\vspace{-1.5cm} %\newline

\tcr{Then:} \vspace{-10pt}
\begin{eqnarray}
\tcb{b_{\scr n+1}} & \tcb{=} & \tcb{6a_{\scr n}}\hspace{60pt} \mbox{(by hypothesis 1))} \\
\tcb{a_{\scr n+1}} & \tcb{=} & \tcb{a_{\scr n} + b_{\scr n}}\hspace{20pt} \mbox{(by hypotheses 2), 3))}\\
\tcb{T(n)} & \tcb{=} & \underbrace{\tcb{c_{\scr0}a_{\scr0}}}_{\parbox{1.75cm}{\large \,cost \,of \\feeding \\1st year}} \tcb{+} \underbrace{\tcb{c_{\scr1}a_{\scr1}}}_{\parbox{1.85cm}{\large \,cost \,of \\feeding \\2nd year}}\tcb{ + \ldots + }\underbrace{\tcb{c_{\scr n-1}a_{\scr n-1}}}_{
\parbox{1.9cm}{\large \,cost of \\\,feeding \\nth year}}
\end{eqnarray}
\iffalse
\tcr{or}
$\left(\begin{array}{c} a_{\scr n+1} \\ b_{\scr n+1}\end{array} \right)
\left(\begin{array}{cc} 1 & 1 \\ 0 & 6\end{array} \right)
\left(\begin{array}{c} a_{\scr n} \\ b_{\scr n}\end{array} \right)$
\fi
\end{description}
\end{titlepage}

\end{document}                          