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\begin{center}
\large
Math 120 - Summer 2006 \pp
Midterm 1 \pp
July 13, 2006 \pp
\end{center}
\normalsize
\vfill
\medskip
\hfill Name: \hrulefill \hrulefill \hrulefill
\hspace{0.5in}
%Section: \hrulefill
Section: C
\vfill
\begin{center}
{\large
\begin{tabular}{||c|c|r||} \hline 
    1     & 10 &\hspace{10mm} \hfill\\ \hline 
    2     & 10 & \\ \hline
    3     & 10 & \\ \hline
    4     & 10 & \\ \hline
    5     & 10 & \\ \hline
    Total & 50 & \\ \hline 
\end{tabular}
}
\end{center}
\vfill
\begin{itemize}
    \item You may use a calculator during this examination.  
          Other electronic devices are not allowed, and should be
          turned off for the duration of the exam.
    \item If you use a trial-and-error or guess-and-check method, or read a
          numerical solution from a graph on your calculator when an algebraic
          method is available, you will not receive full credit.
    \item You may use one hand-written 8.5 by 11 inch page of notes.
    \item Show all work for full credit.  
    \item You have 60 minutes to complete the exam. 
    \item Good luck!
\end{itemize}

\newpage

\begin{enumerate}

%\item Let 
%\begin{align*}
%f(x) &= 3x + 4\\
%g(x) &= 2cx + c\\
%h(x) &= 3x - 5
%\end{align*}
%Find $c$ so that $f(g(x)) = g(h(x))$ for all $x$.

%\newpage

\item
\begin{enumerate}
\item Let $f(x) = \sqrt{x^2 + 3x + 5},$ $\displaystyle g(x) = \frac{1}{x - 1}$
and $h(x) = 2x$.

Compute the composition $f(g(h(x)))$. \textbf{Do not simplify your answer.}
\vfill

%\item let $y = \frac{1}{2}x^2 - 2x + \frac{3}{2}$.  Find the vertex of the parabola, the $x$ intercepts, and sketch the graph.
%\vfill

\item Let $r(x) = 2x^2 + 7$.  Simplify the expression 
$\displaystyle \frac{r(x+h) - r(x)}{h}$ so that $h$ is no longer a factor  in the denominator.
\vfill

\end{enumerate}

\newpage

\item
According to the US Bureau of the Census, the population of the US at
the end of the year 2005 (at December 31, midnight, the instant 
before January 1st) was 297,534,317.  After births, deaths, and
immigration is taken into account, there is a net gain of one person every 10
seconds.

\begin{enumerate}
\item
Find a linear function $P(t)$ that gives the US population as a function
of days since the new year. (So that $P(0) =$ 297,534,317 and 
$P(1)$ = the population at the end of the day January 1st, etc.)
\vfill

\item What is the population at the end of today?  (Hint: today is the 194th day of the year).
% 299210477
\vfill

\item What day will the population reach exactly 300 million people? 

%\newpage

%\emph{Extra credit (2 points) if you have time: what date does this correspond %to (E.g. ``March 3rd'' rather than ``61'').  The table below shows the number %of days in each month to help out.}

%\begin{tabular}{|l|c||l|c|}
%\hline
%Month & \# Days & Month & \# Days \\
%\hline
%January & 31  & July & 31 \\
%February & 28 & August & 31 \\
%March & 31  & September & 30 \\
%April & 30 & October & 31 \\
%May & 31 & November & 30 \\
%June & 30 & December & 31 \\
%\hline
%\end{tabular}
\end{enumerate}

\vfill

\newpage

\item Sylvia started a microbrewery and is trying different prices to sell
her beer. During the first month she charges \$10 per six-pack and sells
100. During the second month she charges \$12 per six-pack and sells 70. 

\begin{enumerate}
\item Assume the number of six-packs sold each month is a linear function of the price.  Find this function.

\vfill

\item 
Assume that it costs Sylvia to \$5 to make each six-pack.
Then the total revenue, the total cost, and the profit that Sylvia makes
are given by:
\begin{align*}
\text{total revenue} &= \text{price} \times \text{quantity} \\
\text{total cost} &= 5 \times \text{quantity} \\
\text{profit} &= \text{total revenue} - \text{total cost}.
\end{align*}
What price should Sylvia charge to maximize her profit?  

\vfill
\end{enumerate}

\newpage

\item 
The graphs of $f(x)$ and $g(x)$ are shown below:\

\includegraphics{smoothgraph-1.pdf}
\hfill
\includegraphics{smoothgraph-2.pdf}

$g(x)$ is a scaled and translated version of $f(x)$, so $g(x) = a f(b(x-c)) + d,$ for some constants $a, b, c,$ and $d$.  Find $a, b, c,$ and $d$.

\newpage

\item 
A truck driver is driving at 50 miles per hour along a (straight) highway from
the town of Adillup to the city Beduga.  Beduga is located 50 miles east and
100 miles north of Adillup.  There is a circular thunderstorm 60 miles north of
Adillup with radius 40 miles, covering part of the road.  

\begin{enumerate}
\item Where does the driver first enter the thunderstorm?  Where does he exit the storm?

\vfill

\item How long does he spend driving through the thunderstorm?

\vfill
\end{enumerate}


\end{enumerate}

\end{document}

