% m211oh77.tex - (prev.: oh72.*) The Cost of Breeding Rabbits
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\input{m211oh.tex}
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\begin{titlepage}
\vspace*{-30pt}
\begin{center}\tcr{\rmH The Cost of Breeding Rabbits} \end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  

\item[\tcg{Situation:}] \rule{0pt}{0pt}

\tcr{1)} The females of a certain rabbit species produce \tcb{$6$ females/year}.

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

\tcr{3)} \tcb{None} die.

\tcr{4)} We start with: \mc{a_{\scr0}} adult rabbit, \mc{b_{\scr0}} young.

\item[\tcg{Questions:}] \rule{0pt}{0pt}

\tcr{1)} \tcr{How many} female rabbits are there after \mc{n} years?

\tcr{2)} If \tcb{{\$}$c_{\scr k-1}$} is the feeding cost of a mature female in the \mc{k^{th}} year, what is the \tcr{total cost} \mc{T(n)} to feed the rabbits for \mc{n} years?   

\item[\tcg{Analysis:}] \rule{0pt}{0pt}

Let \begin{tabular}[t]{rcl}
\mc{a_{\scr n}} & = & \# adult females (at the end of) year \mc{n} \\
\mc{b_{\scr n}} & = & \# females born in year \mc{n}.
\end{tabular} 

Then: %\vspace{-10pt}
$\tcr{\underbrace{\tcb{\left(\begin{array}{c} a_{\scr n+1} \\ b_{\scr n+1}
\end{array}\right)}}_{\vec{v}_{\scr n+1}}} \tcb{\;\;=\;\;}
\tcr{\underbrace{\tcb{\left(\begin{array}{cc}
1 & 1 \\ 6 & 0 \end{array}\right)}}_{A}
\underbrace{\tcb{\left(\begin{array}{c} a_{\scr n} \\ 
b_{\scr n} \end{array}\right)}}_{\vec{v}_n}}\tcb{,}\quad
\begin{minipage}{5.0cm} \tcr{(recursion\\\hspace*{1pt} relation)} \end{minipage}$\sps\\
\hspace*{70pt}$\tcb{T(n) = (1,0)f(A)\vec{v}_{\scr0}}$,\sps\\
\hspace*{70pt}where $\tcb{f(t) = c_0 + c_1t + \ldots + c_{\scr n-1}t^{n-1}.}$  
\end{description}
\end{titlepage}

\end{document}                          