\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 Mid-Term Exam Number Two\hfil  \pp
\hfil May 18, 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
Total & 40 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all four 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 50 minutes to complete the exam. 
\end{itemize}
\vfil
.

\newp

\begin{enumerate}

\item You have a system of four pulleys connected in pairs by belts.

\begin{tikzpicture}[auto,scale=0.75,line width=2,>=stealth]
\bikeChain{0}{0}{1}{5}{0}{3}
\bikeChain{5}{0}{2}{11}{0}{0.5}

\draw (0,0) node {A};
\draw (5,0) node {B};
\draw (11,0) node {D};
\draw (8.5,3.2) node {C};

\draw [decoration={markings,mark=at position 1 with
    {\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate}] (8.25,3) -- (7.3,2.1);


\end{tikzpicture}

Pulley $B$ and pulley $C$ are attached to the same axle and rotate together.

Pulley $A$ has a radius of $3$ cm and an angular speed of $13$ rpm.

Pulley $B$ has a radius of $4$ cm.

Pulley $D$ takes $2.1$ seconds to make one revolution, and has a radius of $1$ cm.

What is the radius of pulley $C$?

\newp

\item An island has a population of boars and monkeys.

In the year 1930, there were $400$ boars on the island.

The boar population increases at the rate of $2.1$ percent per year.

The monkey population triples in the time it takes the boar population to double.

In 1980, there were $5000$ monkeys on the island.

When will there be $10$ monkeys per boar on the island? (Give your answer in the form ``X years after the year Y".)

\newp

\item You are studying for an exam. You know that the more you study,
the better your score on the exam will be, but that no matter how hard you
study you will not do better than a score of $96$.
If you study for $10$ hours, your score will be $66$.
If you study for $20$ hours, your score will be $87$.

Assume that your score on the exam is a linear-to-linear rational function of the
number of hours that you study.

How many hours do you need to study to achieve a score of $92$?

\newp

\item Let $f(x)=|\frac{1}{2}x-3|$.  
\begin{enumerate}
\item Let $g(x)$ be the function whose graph is the graph of $f(x)$
shifted $2$ units to the right and $1$ unit up.  Write the multipart rule for $g(x)$.
\vfil
\item Find the fixed points of $g(x)$.

 
\end{enumerate}


\end{enumerate}

\end{document}


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