%% Math 131, Fall, 2003
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\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Quiz \#2 Solutions}
Professor Hwang \\
\end{center}

\be
\item The sales at Borders bookstores went from \$78~million in~1991
  to \$412~million in~1994. Find an exponential model of the form
  \[
  S = f(t) = S_0\, e^{kt}
  \]
  for Borders' sales, with time measured in years since~1991. Hint:
  In~1994, $t=$?

  \vfil

  \soln Let~$t$ be time in years starting from~1991, so that $t=3$ in
  1994. Since the sales were \$78~million in~1991 (when $t=0$), we
  have $S_0=78$. To find~$k$, set $t=3$:
  \[
  412 = 78 e^{3k},\qquad\text{or}\quad
  3k = \textstyle\ln\left(\frac{412}{78}\right)\simeq1.664314,
  \]
  so that $k=\frac{1}{3}\ln\left(\frac{412}{78}\right)\simeq0.55477$.

  \MyAnswer{$S_0=$}{\quad 78\quad}\MyAnswer{$k=$}{0.55477}

  \begin{center}
    \hrule
  \end{center}
  
\item Referring to the previous question, assume Borders' sales
  continued to grow at the same rate through the 1990s. At what
  time~$t$ did their sales reach \$1~billion?

  \vfil

  \soln Solve for~$t$ in the equation $1000=78e^{kt}$ with~$k$ as above:
  \[
  \textstyle\ln\left(\frac{1000}{78}\right)=kt,\qquad
  \text{or}\quad
  t=\displaystyle\frac{1}{k}\textstyle\ln\left(\frac{1000}{78}\right)
  = \displaystyle\frac{3}{\ln\left(\frac{412}{78}\right)}
                            \textstyle\ln\left(\frac{1000}{78}\right).
  \]

  \MyAnswer{$t\simeq$}{$4.6$~\text{years}}

\ee
\eject

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