\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 2018 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 17, 2018 \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 Suppose that the height of a certain tree is
a linear-to-linear rational function of time.
Today the tree is $3$ meters tall.
One year from now, the tree will be $4$ meters tall.
Ten years from now, the tree will be $6$ meters tall.

\begin{enumerate}
\item Express the height of the tree as a linear-to-linear rational function
of time.
\vfil \vfil
\item When will the tree be $6.4$ meters tall?
\end{enumerate}

\newp

\item City A had a population of 10000 in the year 1990.
City A's population grows at a constant rate of 3$\%$ per year.

City B has a population that is growing exponentially.
In the year 2000, there were half as many people in B as in A.
In the year 2010, the population of A was $20\%$ more than the population of B.

When will the populations be equal?  Give your answer in years after 1990.

\newp

\item \begin{enumerate}
\item Restrict the domain of $f(x)=2x^2+6x-1$ to $x\le -\frac{3}{2}$ and find 
the function's inverse.
\vfil
\item Find the fixed points of the function $g(x)=2x-\frac{1}{x+1}$.
\end{enumerate}

\newp

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

\begin{tikzpicture}[auto,scale=0.5,line width=2,>=stealth]

\bikeChain{0}{2}{1.5}{6}{0}{4}
\bikeChain{6}{0}{2}{14}{2}{3}

\draw (0,2) node {A};
\draw (6,0) node {B};
\draw (14,2) node {D};
\draw (6,4.5) 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 $1.5$ cm and rotates at the rate of one revolution every $12$ seconds.

Pulley $B$ has a radius of $2$ cm.

Pulley $D$ has a radius of $3.4$ cm and rotates at the rate of $0.1$ radians per second.

What is the radius of pulley $C$?




\end{enumerate}

\end{document}


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