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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 - Winter 2018 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil March 10th, 2018 \hfil \pp
}}

\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill

\vfil

\hfil Signature: \hrulefill \hrulefill \hrulefill \hspace{0.5in} Section: \hrulefill 
 
\vfil
\begin{center}
{\Huge
\begin{tabular}{||c|c|r||} \hline 
 1 & 10 &\hspace{10mm} \hfil\\ \hline 
 2 & 10 &\hspace{10mm} \hfil\\ \hline 
 3 & 10 &\hspace{10mm} \hfil\\ \hline 
 4 & 10 &\hspace{10mm} \hfil\\ \hline 
 5 & 10 &\hspace{10mm} \hfil\\ \hline 
 6 & 10 &\hspace{10mm} \hfil\\ \hline 
 7 & 10 &\hspace{10mm} \hfil\\ \hline 
Total & 70 & \\ \hline 
\end{tabular}
}
\end{center}
\begin{itemize}
\item This exam consists of SEVEN problems on FIVE double-sided pages.
\item Show all work for full credit.
\item You may use a TI-30X IIS calculator during this
exam. Other calculators and electronic devices are not permitted.
\item You do not need to simplify your answers.
\item If you use a trial-and-error or guess-and-check method
when a more rigorous method is available, you will not receive full credit.
\item \textbox{Draw a box} around your final answer to each problem.
\item {\bf Do not write within 1 centimeter of the edge!} Your exam will be scanned for grading.
\item If you run out of room, write on the back of the first or last page and indicate that you have done so. If you still need more room, ask us for an extra page to staple to your exam.
\item You may use one hand-written double-sided 8.5" by 11" page of notes.
\item You have 170 minutes to complete the exam.
\end{itemize}

\vfil
{\small --}

\newpage

\begin{enumerate}

% Triangle trig
\item {\bf [10 points]} In the following figure (not drawn to scale), find $x$.
\vspace{-.3 in}
\begin{flushright}
\begin{tikzpicture}
\draw [very thick] (-1,0) -- (3,0) -- (3,5) -- cycle
(-1,0) -- (3,2)
(3,2) -- (1.2,0);
\draw (-0.2,0) arc (0:20:.8) node[right] {$40^\circ$} arc (20:27:.8)
(3,1.2) arc (270:245:.8) node[below] {$20^\circ$} arc (245:229:.8)
(3,4.2) arc (270:245:.8) node[below] {$25^\circ$} arc (245:230:.8)
(2.7,0) -- (2.7,.3) -- (3,.3);
\draw [very thick,densely dotted, printersafe,<->] (-1,-.2) -- node[below] {$5$} (1.2,-.2);
\draw [very thick,densely dotted, printersafe,<->] (3.2,2) -- node[right] {$x$} (3.2,5);
\end{tikzpicture}
\end{flushright}

\newpage

% Circular motion
\item Polya and Baxter begin running around a circular racetrack at the same time.\\[.1 in]
Polya starts at the northernmost point of the track and runs clockwise at a speed of $6$ meters per second. It takes him $50$ seconds to run one complete lap.\\[.1 in]
Baxter runs counterclockwise at a speed of $4$ meters per second.\\[.1 in]
Polya and Baxter first pass each other after $10$ seconds.
\begin{enumerate}
\item {\bf [3 points]} Find Baxter's angular speed in radians per second.
\vskip 2.5 in
\item {\bf [7 points]} Find Baxter's coordinates after two minutes.\\[.1 in]
{\small (Place the origin at the center of the track, and measure units in meters.)}
\end{enumerate}

\newpage

% Sinusoidal modeling
\item The tides in Sea-attle are a sinusoidal function of time.\\[.1 in]
The first high tide today was at 3:00 AM, with a depth of $5$ meters above low tide.\\[.1 in]
The first low tide was at 6:30 AM.
\begin{enumerate}
\item {\bf [4 points]} Write a function $y=f(t)$ for the water depth today, $t$ hours after midnight. Let $y=0$ be low tide.
\vskip 3 in
\item {\bf [6 points]} Today (from midnight to midnight), for how much time is the water at a depth greater than $3$ meters above low tide?
\end{enumerate}

\newpage

% Two exponentials
\item Sea-attle is also full of marine life.
\begin{enumerate}
\item {\bf [3 points]} The number of flounders in Sea-attle doubles every $24$ years.\\[.1 in]
In the year 2000, there were $7500$ flounders in Sea-attle.\\[.1 in]
Write a function $f(t)$ for the number of flounders in Sea-attle, $t$ years after 2000.
\vskip 2 in
\item {\bf [3 points]} The number of groupers in Sea-attle increases by $4.2\%$ per year.\\[.1 in]
In the year 2018, there were $2$ groupers per flounder.\\[.1 in]
Write a function $g(t)$ for the number of groupers in Sea-attle, $t$ years after 2000.
\vskip 2.5 in
\item {\bf [4 points]} In what year will there be 3 groupers per flounder?

\end{enumerate}

\newpage

% Quadratic thing?
\item I'm starting a business selling flounder-and-grouper sandwiches.\\[.1 in]
The number of sandwiches I'll sell per day is a linear function of the price.\\[.1 in]
If I charge $\$3$, I'll sell $99$ sandwiches per day.\\[.1 in]
If I charge $\$7$, I'll sell $39$ sandwiches per day.
\begin{enumerate}
\item {\bf [4 points]} Write a function $f(x)$ for the number of sandwiches I'll sell per day if I charge $\$x$ per sandwich.
\vskip 2 in
\item {\bf [6 points]} Suppose it costs me $\$2$ to make each sandwich.\\[.1 in]
How much should I charge to maximize my daily profit?\\[.1 in]
{\small (Remember: profit is the amount of money I earn from selling sandwiches, minus the cost of making those sandwiches.)}
\end{enumerate}

\newpage

% Linear to linear & composition
\item Let $f(x)$ be a linear-to-linear rational function whose graph passes through the points $(0,-4)$, $(3,-16)$, and $(4,12)$.
\begin{enumerate}
\item {\bf [5 points]} Write a formula for $f(x)$.
\vskip 4 in
\item {\bf [2 points]} Find the asymptotes of $f(x)$.
\vskip 1 in
\item {\bf [3 points]} Let $g(x) = f(f(x))$. Find the asymptotes of $g(x)$.
\end{enumerate}

\newpage

% Multipart function?
\item {\bf [10 points]} Eleanor and Jason are walking around the coordinate plane.\\[.1 in]
Eleanor starts at the origin. She walks west at a speed of $80$ meters per minute for $1$ hour.\\[.1 in]
Jason also starts at the origin, but his path is a little more complicated:\\[.1 in]
First, he walks north at a speed of $60$ meters per minute for $30$ minutes.\\[.1 in]
Then, he waits in place for $10$ minutes.\\[.1 in]
Finally, he walks to the west at a speed of $90$ meters per minute for $20$ minutes.\\[.1 in]
Write a multipart function for the distance between Eleanor and Jason after $t$ minutes.

\end{enumerate}

\end{document}