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%\markboth{Dr. Matthew M. Conroy - University of Washington}{Dr. Matthew M. Conroy - University of Washington}

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\newcommand{\pp}{\par \noindent}
\newcommand{\newp}{\vfil \eject}

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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 - Autumn 2014 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil December 6, 2014 \hfil \pp
}}

\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.25in} Student ID no. : \hrulefill

\vfil

\hfil Signature: \hrulefill \hrulefill \hrulefill \hrulefill \hrulefill \hspace{0.25in} Quiz Section (e.g., AB, CA, etc.): \rule{0.4in}{\linethickness} 
 
\vfil
\begin{center}
{\Large
\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
8 &  10 &    \\ \hline
Total & 80 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all eight questions. 
\item Show all work for full credit. 
\item You may use a scientific calculator during this
examination.  Graphing calculators are not allowed.
Also, 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 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 the exam. 
\end{itemize}
\vfil
%{\small{.$_{.}$}}
$.$

\newp

\begin{enumerate}


\item You are lost in the desert when, in the distance, 
you notice a very large tower. You measure the angle between the top of the tower and the ground, and it is $6^\circ$.

Then you walk towards the tower at a constant speed of 80 meters per minute. After 4 minutes, you stop and 
measure the angle again; this time it is $7^\circ$.

If you keep walking towards the tower at a constant speed of 80 meters per minute, how much longer will it take you to get there?

\newp

\item Denise stands 28 feet east and 120 feet south of Aaron. Timothy stands 80 feet east and 78 feet north of Denise.

First, Aaron walks due south until he is exactly 100 feet from Denise. Then, he turns and walks in a straight line towards Timothy.

When Aaron is closest to Denise, how far from Denise is he?

\newp

\item \begin{enumerate}

\item Every 5 seconds, the number of cursed goblets in Bellatrix's vault doubles!


Right now, there are 70 goblets.

Write a function $g(t)$ for the number of goblets $t$ seconds from now.

\vfil

\item Oh no, this vault is also full of cursed coins! The number of coins is increasing exponentially.

10 seconds from now, there will be twice as many goblets as coins.

5 seconds ago, there were 20 more goblets than coins.

Write a function $c(t)$ for the number of coins $t$ seconds from now.

\vfil \vfil

\item Each coin weighs 2 ounces, and each goblet weighs 9 ounces.

When will the total weight of the coins equal the total weight of the goblets?

\vfil

\end{enumerate}

\newp

\item 
In the year 2000, Adam invested \$8000 and his investment increased at a constant rate 
of \$120 a year for the first 5 years. In 2005, the value of Adam's investment 
started to decrease and it reached \$8495 in 2012. Assume that the investment decreased linearly from 2005 to 2012. 

  \begin{enumerate} 

    \item Write a formula for A(t), the value of Adams's investment $t$ years after 2000, valid for $ 0 \leq t \leq 12$.

    \vfill    

\item Find all solutions to the equation $A(t)=8540$.
%     \item Sketch a graph of $A(t)$
%     \vfill 

%   \item Bob's investment, in the same time period  $0\leq t \leq12$, 
%   is given by the formula $B(t)=6000+|2400-400t|$. 
%   When was Adam's investment equal to Bob's investment? 
%   Give the answer as a decimal number.

\vfill

\end{enumerate}

\newp

%\item Below you are given the graph of $y=f(x)$.  
%\begin{figure}[h]
%\includegraphics[scale=0.25]{elena/2014final1.eps}
%\end{figure}
%Which of the following graphs is the graph of $3f(-x+1)-2 ? $ Circle the correct one.
%\begin{tabular}{ll}
%\includegraphics[scale=0.2]{elena/2014final2.eps} &
%\includegraphics[scale=0.2]{elena/2014final4.eps} \\
%\includegraphics[scale=0.2]{elena/2014final5.eps} &
%\includegraphics[scale=0.2]{elena/2014final6.eps} \\
%\end{tabular}
%%\item Assume the domain of $f$ is $-2 \leq x \leq 5$ and the range of f is $-2 \leq f(x) \leq 3.5$. Find positive constants $C$ and $D$ such that the domain and range of 
%%$y=2f(-x+C)+D$ are $ 0 \leq x \leq 7$ and $-1 \leq y \leq 10$ respectively
%%\vfill
%%\end{enumerate}
%\newp

\item Bob is running counterclockwise around a circular track with a radius of $2$ km.

Bob's angular speed is 1.6$\pi$ rad/hr . 

Bob starts at position $P$ and it takes him 0.5 hr to reach position $Q$. 
%\epsfysize=70mm
%\epsfbox{elena/2014final3.eps}
\begin{enumerate}
\begin{minipage}[t]{0.5\textwidth}
\vspace{-1.5in} %%%%%%%%%%%%%%%%%%%%%%% brute force to fix a vertical alignment problem!!!! Works!!!
\item When will Bob reach position $R$ ?  
\end{minipage}
\begin{minipage}[m]{0.47\textwidth}
\begin{tikzpicture}[scale=2.0]
\draw[line width=1.5pt] (0,0) circle (1.0);
\draw[color=gray] (-2,0) -- (2,0);
\draw[color=gray] (0,2) -- (0,-2);
\draw[style=dashed] (0,0) -- (0.866,0.5);
\draw[style=dashed] (0,0) -- (-0.866,0.5);
\draw (-1.05,0.5) node {$Q$};
\draw (1.05,0.5) node {$P$};
\draw (0.1,-1.15) node {$R$};
\draw (0.6,0.15) node {$\alpha$};
\draw (-0.6,0.15) node {$\alpha$};
\draw (0.5,0) arc (0:30:0.5);
\draw (-0.5,0) arc (180:150:0.5);
\def \circRad{0.04};
\fill[black] (0,-1) circle(\circRad);
\fill[black] (0.866,0.5) circle(\circRad);
\fill[black] (-0.866,0.5) circle(\circRad);
\end{tikzpicture}

\end{minipage}


\vfil

\item Find the coordinates of Bob at time $t$ with respect to a coordinate system with the origin at the center of the circular track.

\vfil

\item How far (in a straight line) is Bob from the point $(2,0)$ after running for $2.6$ hours?

\vfil

\end{enumerate}

\newp

%6. quadratic functions and modeling, quadratic optimization
\item 
%You want to draw rectangles inside the triangle formed by the line $y=5-\frac{1}{3}x$ and the $x$ and $y$-axes.

You plan to put two rectangles into the triangle formed by $y=5-\frac{1}{3}x$ and the $x$- and $y$-axes.

The first rectangle will have one vertex at the origin, one side extending along the positive $x$-axis,
one side extending along the positive $y$-axis,
and one vertex lying on the line \break  $y=5-\frac{1}{3}x$.


The second rectangle will sit on top of the first one and be half as wide (horizontal dimension).
The second rectangle will also have a corner on the line $y=5-\frac{1}{3}x$. 

Let $w$ be the width of the lower rectangle.

Find the value of $w$ that
maximizes the combined area of the two rectangles.


%\begin{enumerate}
%\item
%Consider a rectangle formed with one vertex at the origin, one side extending along the positive $x$-axis,
%one side extending along the positive $y$-axis,
%and one vertex lying on the line $y=5-\frac{1}{3}x$. (The figure below gives an example rectangle.) What should the dimensions of this rectangle be
%to achieve the maximum possible area?
%
%\begin{tikzpicture}[scale=0.7]
%\draw (0,5) -- (15,0);
%\draw[color=gray] (0,0) -- (18,0);
%\draw[color=gray] (0,0) -- (0,6);
%\draw[color=gray] (0,2) -- (9,2) -- (9,0);
%\end{tikzpicture}


%\vfil


\begin{tikzpicture}[scale=0.7]
\draw (0,5) -- (15,0);
\draw[color=black] (0,0) -- (18,0);
\draw[color=black] (0,0) -- (0,6);
\draw[color=black] (0,2) -- (9,2) -- (9,0);
\draw[color=black] (0,3.5) -- (4.5,3.5) -- (4.5,2);
\draw[-{Stealth[length=2.5mm]}] (4,-0.5) -- (0,-0.5);
\draw[-{Stealth[length=2.5mm]}] (5,-0.5) -- (9,-0.5);
\draw (4.5,-0.5) node {$w$};
\draw[-{Stealth[length=2.5mm]}] (1.7,1.5) -- (0,1.5);
\draw[-{Stealth[length=2.5mm]}] (2.6,1.5) -- (4.5,1.5);
\draw (2.25,1.5) node {$\frac{w}{2}$};
\draw[-{Stealth[length=2.5mm]}] (7,4.5) -- (6,3.1);
\draw (8.5,4.6) node {$y=5-\frac{1}{3}x$};
\end{tikzpicture}






\newp

%9. linear-to-linear rational functions 
\item The height of a certain cactus is a linear-to-linear rational function of time.
Today, the cactus is 10 feet high.

Over the next 30 years, the height of the cactus will increase by 25 feet.

The cactus will always increase in height, and the height will approach, but not exceed, 60 feet.

\begin{enumerate}
\item Express the height of the cactus as a function of time.

\vfil \vfil
\vfil \vfil

\item Find the inverse of the function you found in part (a).

\vfil \vfil

%\item When will the cactus be $50$ feet high? Give your answer in years from today.

\end{enumerate}

\newp

%12. sinusoidal functions and inverses
\item Marika has a fever.  This causes her temperature to be a sinusoidal function of time.
At 3 AM today, her temperature was at a maximum of 40 degrees celsius.
Her temperature dropped, reaching a minimum of 36 degrees celsius at 10 AM today.

\begin{enumerate}

\item Give the function $f(t)$ that yields Marika's temperature $t$ hours after midnight last night.
\vfil

\item For how much of today (from midnight to midnight) will Marika's temperature be above 38.2 degrees celsius?

\end{enumerate}



\end{enumerate}

\end{document}


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% angle=0 }
