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\begin{document}

\vfil \noindent

{\LARGE{
\hfil Math 120 Autumn 2018 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil December 8, 2018 \hfil \pp
}}

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

\medskip

\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 &       \\ \hline
 3 & 10 &      \\ \hline
4 &  10 &    \\ \hline
5 &  10 &    \\ \hline
6 &  10 &    \\ \hline
7 &  10 &    \\ \hline
Total & 70 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all seven questions. 
\item Show all work for full credit. 
\item The only calculator you may use during this exam is
a TI-30XIIs.
All other electronic devices are not allowed, and should be
turned off and put away for the duration of the exam.
\item If you use a trial-and-error or guess-and-check method
when an algebraic method is available, you will not receive full credit.
\item You may use one, two-sided, hand-written 8.5 by 11 inch page of notes.  Write
your name on your notesheet and turn it in with your exam.
\item You have 170 minutes to complete this exam. 
\end{itemize}
\vfil
.

\newpage

\begin{enumerate}

%\item
%
%A biologist is studying a colony of mole rats on the island of Socotra.  She is only able to visit the island once a year.  On her first visit she counts 19 animals in the colony.  When she returns one year later, she counts 51 animals.  Two years into the study there are 75 animals in the colony.
%
%The biologist decides to model the size of the colony as a quadratic function of time, in years.  According to this model, at what time(s) will the population of the colony be 84 animals?
%\vfill
%\newp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
Fred and Ted are running around a circular track.  The track has a radius of 60 meters.

Fred starts from the westernmost point of the track, and runs clockwise.

Ted starts at the same time from the southernmost point of the track, and runs counterclockwise.

Fred runs at a constant speed of 9 meters per second, and passes Ted for the first time after 15 seconds.

\begin{enumerate}
\item What is Ted's speed in meters per second?
\vfil \vfil
\item After running for $200$ seconds, who is farther North, Fred or Ted? Show all work.
\vfil \vfil \vfil
\end{enumerate}
\newp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
At the beginning of $2001$, Clovis invested \$ $10,000$ in an account. 
In $2013$ his investment was worth \$ $14,764$. 

Isobel made an investment at the same time as Clovis.  
Her investment doubles every $7$ years.  
In $2014$, Clovis had $8$ times as much in his account as Isobel had in hers.

Assume the values of both investments are exponential functions of time.

Take $t=0$ in $2001$.
 
\begin{enumerate}
\item  Give an exponential function relating the value of Clovis's investment $y$ to the year~$t$. 
\vfil \vfil

\item What is the annual growth rate of Clovis' investment?
\vfil

\item What was the value of Isobel's investment at the beginning of 2001?

\vfil \vfil

\item How many years after $2001$ will Clovis have five times as much money as Isobel?
\vfil 

\end{enumerate}
\newp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
Keoki and Nalani are moving in the $xy$-plane along straight lines at constant speeds. They both start at the same time.

Keoki starts from the point $(-5,9)$ and heads directly toward the point $(11,1)$, 
reaching it in 8 seconds.

Nalani starts from the point $(8,9)$ and moves toward the $y$-axis along the line $y=\frac34 x+3$.

Nalani takes twice as long to reach the $y$-axis as it takes Keoki to reach the $y$-axis.

\begin{enumerate}
\item Find the parametric equations of motion for Keoki.
\vfill
\item Find the parametric equations of motion for Nalani.
\vfill
\item How long has Keoki been moving when the distance between
Keoki and Nalani is as small as it ever gets?
\vfill 
\end{enumerate}

\newp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item
%Yolanda stands 11 miles west and 3 miles north of the {\bf northernmost} point of a circular forest with radius 26 miles. At the {\bf southernmost} point of the forest is a house made out of pork chops.\\[.1 in]
%Yolanda walks in a straight line to the house at a speed of 3 miles per hour.\\[.1 in]
%For how much time does she walk {\bf inside the forest}?
%\newp


\item Here's the graph of $f(x)$.

\vspace{-.3 in}
\begin{center}
\begin{tikzpicture}
\graphgrid{-3}{4}{-2}{3}{$x$}{$f(x)$}
\draw [very thick] (-2,1) arc (180:90:2) (0,-2) -- (3,1);
\foreach \x/\y/\col in {-2/1/black,0/-2/black,0/3/white,3/1/white} \draw [fill=\col,thick] (\x,\y) circle (.1);
\end{tikzpicture}
\end{center}
\vspace{-.3 in}
Use the graph to answer the following questions.
\begin{enumerate}
\item Compute $f(f(f(2)))$.
\vskip 1.5 in
\item Is $f$ one-to-one? Why or why not?
\vskip 1.5 in
\item Let $g(x) = f(2-x)+1$.\\[.1 in]
Sketch a graph of $g(x)$.
\vspace{-1 in}
\begin{flushright}
\begin{tikzpicture}[scale=.8]
\graphgrid{-5}{5}{-3}{4}{$x$}{$g(x)$}
\end{tikzpicture}
\end{flushright}
\end{enumerate}

\newpage
% Linear-to-linear
\item Midge is practicing her comedy act. The number of laughs she gets is a linear-to-linear rational function of how long she practices.\\[.1 in]
If she doesn't practice at all, she'll get $4$ laughs.\\[.1 in]
If she practices for an hour, she'll get $24$ laughs.\\[.1 in]
If she practices for four hours, she'll get $52$ laughs.
\begin{enumerate}
\item Write a function $f(x)$ for the number of laughs Midge gets if she practices for $x$ hours.
\vskip 5 in
\item Suppose the domain of $f(x)$ is $[0,\infty)$. What is the range?
\end{enumerate}
\newpage

% Triangle trig
\item You are standing somewhere between a mountain and a fountain, which are 100 km apart from each other.\\[.1 in]
You know that the mountain is $500$ times as tall as the fountain.\\[.1 in]
From where you stand, the mountain is at an angle of elevation of $3^\circ$, and the fountain is at an angle of elevation of $40^\circ$.\\[.1 in]
How tall is the mountain?

\newpage
% Sinusoidal
\item The temperature in Skiattle is a sinusoidal function of time.\\[.1 in]
$120$ days ago, the temperature was at its maximum value of $55^\circ$ F. The temperature has been falling since then, and $20$ days from today it will reach its minimum value of $10^\circ$ F.
\begin{enumerate}
\item Write a function $f(t)$ for the temperature in Skiattle, in Fahrenheit, $t$ days from today.
\vskip 2.5 in
\item The residents of Skiattle can only ski when the temperature is below $28^\circ$ F.\\[.1 in]Over the next $500$ days (starting from today), for how much time will it be cold enough to ski?
\end{enumerate}


\end{enumerate}


\end{document}