\documentclass[12pt]{report} 

\usepackage{palatino}
\usepackage{epsfig}
\usepackage[margin=0.8in]{geometry}

\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{math}

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


\pagestyle{empty} %%% this results in no pagenumbers (footer is empty}

\setlength{\parindent}{0in}
\setlength{\parskip}{0.05in}

\baselineskip=20pt

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



\newcommand{\bikeChain}[6]{
	%%% written by Matthew Conroy (conroy "at" math.washington.edu)
	%%% April 2015
	%%%
	%%% format is \bikeChain{x1}{y1}{r1}{x2}{y2}{r2}
	%%% where one circle is centered at (x1,y1) with radius r1,
	%%% and the other is centered at (x2,y2) with radius r2
	%%%
	%%% known bug: fails if x1=x2
	%%%


	\draw (#1,#2) circle (#3 cm);
	\draw (#4,#5) circle (#6 cm);
	\def\xa{#1}
	\def\ya{#2}
	\def\xb{#4}
	\def\yb{#5}
	\def\ra{#3}
	\def\rb{#6}
	

	\tikzmath{
		if ((\xa==\xb) && (\yb<\ya)) then {\betty=-90;} 
		else {
				if ((\xa==\xb) && (\yb>\ya)) then {\betty=90;}
				else {
					if ((\xa!=\xb)) then {\betty=atan((\yb-\ya)/(\xb-\xa));};
				};
		};
	}		
		
	
	%% distance between circle centers
	\pgfmathsetmacro{\alpha}{ sqrt((\xa-\xb)*(\xa-\xb)+(\ya-\yb)*(\ya-\yb))} 
	%% length of free chain segment
	\pgfmathsetmacro{\gamma}{ sqrt((\alpha)*(\alpha)-(\ra-\rb)*(\ra-\rb)) }  
	%% angle between radii and line through centers
	\pgfmathsetmacro{\theta}{(90-atan(abs(\ra-\rb)/\gamma))}; 
	%% if rb>ra, we have theta plays a different role
	\pgfmathsetmacro\theta{ifthenelse(\rb>\ra,180-\theta,\theta)}\pgfmathresult; 
	%% another geometry flip
	\pgfmathsetmacro\theta{ifthenelse(\xb<\xa,180-\theta,\theta)}\pgfmathresult; 
	%% angle the line through centers makes with horizontal
	%\pgfmathsetmacro\beta{betafun(\xa,\ya,\xb,\yb)}\pgfmathresult; 
	\pgfmathsetmacro\beta{\betty}\pgfmathresult;
	
	%\draw (\xa,-4) node {$\beta$};
	
	%% calculate points of contact with chain
	\pgfmathsetmacro{\xxa}{\xa+\ra*cos(\theta+\beta*1)};
	\pgfmathsetmacro{\yya}{\ya+\ra*sin(\theta+\beta*1)};
	\pgfmathsetmacro{\xxb}{\xb+\rb*cos(\theta+\beta*1)};
	\pgfmathsetmacro{\yyb}{\yb+\rb*sin(\theta+\beta*1)};
	%% draw first part of chain
	\draw (\xxa,\yya) -- (\xxb,\yyb);
	%% calculate the symmetrically located contact points
	\pgfmathsetmacro{\xxa}{\xa+\ra*cos(-\theta+\beta*1)};
	\pgfmathsetmacro{\yya}{\ya+\ra*sin(-\theta+\beta*1)};
	\pgfmathsetmacro{\xxb}{\xb+\rb*cos(-\theta+\beta*1)};
	\pgfmathsetmacro{\yyb}{\yb+\rb*sin(-\theta+\beta*1)};
	%% draw second part of chain
	\draw (\xxa,\yya) -- (\xxb,\yyb);
}	


\usetikzlibrary{arrows,decorations.markings}



\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A Spring 2017 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil June 3, 2017 \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 &       \\ \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
.

\newp

\begin{enumerate}

\item The radiation level measured on earth from a certain star is a sinusoidal
function of time.  At 2:30 AM today, the radiation was at its maximum, $22$.
The level then decreased to its minimum of $3$ at 5:30 AM today.  

\begin{enumerate}
\item Determine the sinusoidal function that gives the radiation level $t$ hours
after midnight last night.
\vfil

\item Starting from midnight last night, how long will it be until the
level has been above $16$ for a total of exactly $5$ hours?


\vfil


\end{enumerate}

\newp

\item Alex and Charlie start running around a circular track at the same time.
The track has a radius of $160$ meters.
Alex starts at the westernmost point and runs counterclockwise, taking $200$ seconds
to run one lap.  

Charlie runs clockwise at $3$ meters per second and passes Alex for the first time
after they have been running for $25$ seconds.

\begin{enumerate}
\item From when he begins running, 
how long does it take Charlie to reach the northernmost point of the track?

\vfil

\item How many meters north or south of his starting point is Charlie after
he has been running for $1200$ seconds?  Specify the number of meters, and whether
Charlie is north or south of his starting point.


\end{enumerate}

\newp

\item You are trying to determine the height of a building whose base is
on a level plane on which you stand.
You are at the base of a vertical pole that you are able to climb.
From the base of the pole, you measure the angle to the top of the building to be
$30^\circ$.  You climb a certain distance up the pole and measure again: the
angle is now $28^\circ$.  After climbing $10$ feet further up the pole, you
measure once more, and find the angle is $27^\circ$.

How tall is the building?

\newp

\item You are making a rectangular enclosure with fencing.
The fencing costs $5$ dollars per foot.
Also, you want to make two partitions, each parallel to the
same side of the rectangle (which will separate the enclosure
into three sections).  The partitions will be made with special
fencing material which costs $8$ dollars per foot.

For example, the enclosure might look like this, with the
regular fencing shown as the solid line, and the special fencing
as the dashed lines.

\begin{tikzpicture}[auto,scale=0.75,line width=2,>=stealth]

\draw (0,0) -- (10,0) -- (10,4) -- (0,4) -- (0,0);
\draw[dashed] (3,0) -- (3,4);
\draw[dashed] (5,0) -- (5,4);



\end{tikzpicture}

You have a budget of $1000$ dollars.  What should the dimensions
of the enclosure be to maximize its area?

\newp

\item Alex and Bah are walking in the $xy$-plane.
They start walking at the same time from different points, and
walk at constant speeds without ever changing direction.

Alex starts from the point $(10,6)$ and heads directly toward the point
$(-4,-3)$, reaching it in $8$ seconds.

Bah starts from the point $(-8,7)$ and passes through the point
where Alex crosses the $x$-axis $1.5$ seconds after Alex passes that point.

How long has Bah been walking when Bah cross the $y$-axis?

\newp

\item 

\begin{enumerate}

\item Let $f(x)=|x+5|$ and $g(x)=\frac{1}{2}x-3$. Write the multipart rule for $g(f(g(x)))$.
\vfil \vfil

\item Let $\dsps h(x)=7+\frac{3x+1}{2x-5}$.  Find $h^{-1}(x)$.
\vfil

\end{enumerate}



\newp

\item The Circular Forest is a region in the $xy$-plane represented by 
the circle
\[
x^2-18x+y^2+5y=108.75
\]
with units in kilometers (so, for example, the point $(2,3)$ is
$2$ kilometers east and $3$ kilometers north of the origin).

Charlie is camped $4$ kilometers due north of the center of the forest.
He then hikes from there $1$ km due east.  He then turns and walks due south
until leaving the forest.

At what point does Charlie leave the forest?


\end{enumerate}

\end{document}


% \epsfig{file=pizza01.eps, 
% width=11cm,
% angle=0 }