\documentclass[12pt]{report} 

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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 C - Autumn 2014 \hfil \pp
\hfil Midterm Exam Number One\hfil  \pp
\hfil October 16, 2014 \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 & 15 &\hspace{10mm} \hfil\\ \hline 
 2 & 15 &       \\ \hline
 3 & 15 &      \\ \hline
4 & 15 &    \\ \hline
Total & 60 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item This exam consists of FOUR problems on FIVE 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.
Also, 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 50 minutes to complete the exam.
\end{itemize}

\vfil
{\small --}

\newpage

\begin{enumerate}

\item {\bf[15 points]} Cooper's spaceship is hovering over the equator of a distant planet with a radius of 4100 kilometers. His ship is 1025 kilometers from the ground. At the north pole of the planet is a research station.

Cooper sends a radio signal to the research station. It travels in a straight line: first through the atmosphere at a speed of 200 kilometers per millisecond, and then through the planet at a speed of 100 kilometers per millisecond. {\bf How long, in milliseconds}, does the radio signal take to arrive at the station?
\begin{flushright}
\begin{tikzpicture}[auto,scale=.05, line width=1.5pt, inner sep=2 pt]

\draw (41,0) arc (0:360:41);

\draw (0,41) node[circle,draw,fill=black,label=above:Research Station] (R) {};
\draw (-51.25,0) node[circle,draw,fill=black,label=left:Cooper] (C) {};

\draw [dashed,dash pattern = on 4 pt off 2 pt,-stealth'] (C) -- node[below,sloped] {signal} (R);

\draw (0,0) node {Planet};
\draw (-60,-30) node (L) {1025 km};
\draw [dotted] (C) -- (-41,0);

%\draw [smooth,->] (L)++(0,5) arc (295:360:26);
\draw [smooth,->] (L)++(0,5) arc (318:340:70);

\end{tikzpicture}
\end{flushright}

\newpage

\item Wing and a rabbit are walking around the $xy$-plane for some reason.

\begin{enumerate}
\item {\bf[5 points]} Wing begins at $(-2,4)$ and walks at a constant speed in a straight line toward $(5,6)$, reaching it in 10 seconds.\\[.1 in]
Give {\bf parametric equations} for Wing's coordinates after $t$ seconds.
\vskip 2 in
\item {\bf[6 points]} The rabbit begins at $(3.6,-5)$ and hops toward $(-0.9,-2.2)$ with a constant speed of $0.5$ units per second.\\[.1 in]
Give {\bf parametric equations} for the rabbit's coordinates after $t$ seconds.
\vskip 3 in
\item {\bf[4 points]} When is Wing {\bf directly north} of the rabbit?
\end{enumerate}

\newpage

\item Consider the following multipart function:

$$f(x) = \begin{cases}
2x + 4 & \mbox{if }-2 < x \leq 0\\
2 & \mbox{if }0 < x < 4\\
3 - \sqrt{16 - (x-4)^2} & \mbox{if } 4 < x \leq 8
\end{cases}$$

\begin{enumerate}
\item {\bf[9 points]} Plot $f(x)$ in the grid below. It's okay if you're not a great artist, but please label your graph clearly.

\begin{center}
\begin{tikzpicture}[auto,scale=1, line width=2pt]

\foreach \x in {-3,-2,...,9} \draw [line width=1pt,densely dotted] (\x,-2.2)--(\x,8.2);
\foreach \x in {-3,-2,...,9} \draw (\x+.25,.25) node {\small \x};
\foreach \y in {-2,-1,...,8} \draw [line width=1pt,densely dotted] (-3.2,\y)--(9.2,\y);
\foreach \y in {-2,-1,...,8} \draw (.25,\y+.25) node {\small \y};

\draw [<->] (-3.6,0) -- (9.5,0);
\draw [<->] (0,-2.6) -- (0,8.6);

\draw (9.7,0) node {$x$};
\draw (0,8.9) node {$f(x)$};

%\draw (-2,0) -- (0,4);
%\draw (0,2) -- (4,2);
%\draw (4,-1) arc (270:360:4);

\end{tikzpicture}
\end{center}

\item {\bf[6 points]} Find the point on the graph of $f(x)$ which is {\bf closest} to $(-2,3)$.\\
{\small (Give both coordinates.)}

\end{enumerate}
\newpage

\item {\bf[15 points]} Francie is bored at home when she decides to walk around the neighborhood for a little while and make a math problem.

First, Francie walks north for $20$ minutes at a constant speed of $80$ meters per minute.

Then she stands in place for $5$ minutes, checking her phone or whatever.

Then she walks west for $10$ minutes at a constant speed of $120$ meters per minute.

Write a {\bf multipart function} $f(t)$ for the distance from Francie to her house (in meters), $t$ minutes after she starts walking.

\vfill
*Here's a compass rose, in case you get lost.
\begin{tikzpicture}[scale=.55, line width=1.5pt]
\draw (14,8) node (CN) {North};
\draw (14,4) node (CS) {South};
\draw (11.5,6) node (CW) {West};
\draw (16.5,6) node (CE) {East};
\draw (14,6) node (Compass) {};

\draw [-latex'] (Compass) -- (CN);
\draw [-latex'] (Compass) -- (CS);
\draw [-latex'] (Compass) -- (CE);
\draw [-latex'] (Compass) -- (CW);
\end{tikzpicture}

\end{enumerate}

\end{document}