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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 - Winter 2016 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil March 12, 2016 \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 & 13 &\hspace{10mm} \hfil\\ \hline 
 2 & 13 &       \\ \hline
 3 & 13 &      \\ \hline
4 & 13 &    \\ \hline
5 & 12 &    \\ \hline
6 & 12 &    \\ \hline
7 & 12 &    \\ \hline
8 & 12 &    \\ \hline
Total & 100 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item This exam consists of EIGHT problems on NINE pages, including this cover sheet.
\item Show all work for full credit. 
\item You may use a scientific calculator during this
exam.  Graphing calculators are not permitted.
Other electronic devices are not allowed, and should be
turned off and put away for the duration of the exam.
\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 If you write on the back of the page, please indicate that you have done so!
\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

\newpage

\begin{enumerate}

\item For parts (a) through (c), consider the function $f(x) = 2\cdot3^{6-2x}$.
\begin{enumerate}
\item {\bf [5 points]} Write a formula for the inverse function, $f^{-1}(x)$.

\vskip 3 in

\item {\bf [4 points]} Suppose $f(a^2+1) = 54$. Find all possible values of $a$.

\vskip 2.5 in

\item {\bf [4 points]} Write $f(x)$ in standard exponential form.\\[.1 in] {\small (That is, write it as $f(x) = A_0 b^x$ for some constants $A_0$ and $b$.)}

\end{enumerate}

\newpage

\item {\bf [13 points]} You have 8 feet of wire which you would like to use to make two shapes: a {\em square}, and a {\em right triangle}, where one leg of the right triangle is three-fourths the length of the other leg. (You have to use all the wire.)\\[.1 in]
For example, they might look like this:\\[.1 in]
What is the {\bf minimum possible total area} of the two shapes?
\vspace{-.9 in}
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\foreach \y in {1,2} \draw (-.1,\y) -- (.1,\y);
\foreach \x in {1,2,3} \draw (\x,-.1) -- (\x,.1);

\end{tikzpicture}
\end{flushright}

\newpage

\item {\bf [13 points]} Sinead decides to track the noise level where she's standing at Sasquatch, and finds that the volume (measured in decibels) is a sinusoidal function of time.\\[.1 in]
Right now, the volume is at its maximum: 80 decibels.\\[.1 in]
The volume will decrease for the next 14 minutes until it hits its minimum: 44 decibels.\\[.1 in]
Over the next hour (starting now), for how long will the volume be above 70 decibels?

\newpage

\item Godzilla and King Kong are running around the Earth, up the prime meridian and down its antimeridian, as shown below. The radius of Earth is 4000 miles.

\begin{enumerate}
\item {\bf [3 points]} Godzilla begins at the north pole. It takes him 5 hours to run one full circle around the world.\\[.1 in]
What is Godzilla's speed, in miles per hour?
\vspace{-.6 in}
\begin{flushright}
\begin{tikzpicture}[scale=1.5]
\draw [very thick] (1,0) arc (0:360:1);

\draw (1,0) -- (0,0) -- (40:1);
\draw (0:.3) arc (0:40:.3);
\node at (20:.5) {$\phi$};

\draw [thick,->] (90:1.1) arc (90:120:1.1);
\draw [thick,->] (40:1.1) arc (40:70:1.1);

\node [circle,draw,fill=gray,label=above:{Godzilla},inner sep = 2] at (0,1) {};
\node [circle,draw,fill=gray,label=right:{King Kong},inner sep = 2] at (40:1) {};

\draw [very thick, <->] (1.7,-.5) node[below] {South} -- (1.7,-.1) node[above] {North};
\end{tikzpicture}
\end{flushright}

\vskip 1 in
\item {\bf [5 points]} King Kong starts at an angle $\phi$ north of the equator and runs in the same direction as Godzilla at a speed of 6000 miles per hour.\\[.1 in]
He passes Godzilla after 3 hours. What's $\phi$? Give your answer in radians.

\vskip 2.6 in

\item {\bf [5 points]} After 14 hours, who is farther north: Godzilla or King Kong?
\end{enumerate}

\newpage

\item {\bf [12 points]} Nessie stands 30 miles east and 10 miles north of the center of a circular lake with radius 26 miles.\\[.1 in]
She walks due west in a straight line until she hits the edge of the lake. Then, she swims in a straight line towards the westernmost point of the lake.\\[.1 in]
When she is closest to the center of the lake, how far is she from her starting position?

\newpage

\item {\bf [4 points per part]} 
\begin{enumerate}

\item A goblin starts at $(-4,2)$ at time $t=0$ and walks in a straight line at a constant speed towards the point $(5,-3)$, reaching it in $5$ seconds.\\[.1 in]
Find parametric equations for the goblin's coordinates after $t$ seconds.

\vskip 2.5 in

\item Meanwhile, a ghoul starts at $(7,3.2)$ at time $t=0$ and walks in a straight line towards the point $(-5,1)$ at a constant speed of $4$ units per second.\\[.1 in]
Find parametric equations for the ghoul's coordinates after $t$ seconds.

\vskip 3.2 in

\item A zombie stands at the origin. After $1.5$ seconds, who is closer to the zombie: the goblin or the ghoul?

\end{enumerate}

\newpage

\item {\bf [12 points]} Frank is engaged in an election campaign. The percentage of the vote he earns in the New Hampshire primary is a {\em linear-to-linear rational function} of the number of advertising dollars he spends.\\[.1 in]
If he spends 1 million dollars, he'll earn 34\% of the vote.\\[.1 in]
If he spends 4 million dollars, he'll earn 49\% of the vote.\\[.1 in]
As his advertising spending increases, his vote share will approach (but not reach) 65\%.\\[.1 in]
How much does he need to spend to earn 55\% of the vote?

\newpage

\item {\bf [12 points]} Two buildings, creatively named Tower A and Tower B, stand across from each other along a street. Tower A is 100 feet taller than Tower B.\\[.1 in]
The buildings are 40 feet apart. You are standing somewhere in the street between them (but not necessarily halfway), carefully dodging traffic while you solve a math problem.\\[.1 in]
From where you stand, the top of Tower A is at an angle of $87^\circ$ above the horizontal, and the top Tower B is at an angle of $83^\circ$ above the horizontal.\\[.1 in]
How tall is Tower A?\\[.1 in]
{\small (Ignore your own height in this problem.)}

\end{enumerate}

\end{document}