\documentclass[12pt]{report} 

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

\usepackage{amsmath}
\usepackage{tikz}


%\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}

\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A, B Winter 2017 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil March 11, 2017 \hfil \pp
}}

\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill

\vspace{0.2in}

\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 You should keep at least four decimal digits throughout all
calculations. Bases in exponential expressions should be kept
as precise as possible.
\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 2 hours and 50 minutes to complete the exam. 
\end{itemize}
\vfil
.

\newp

\begin{enumerate}


\item Tom and Jerry are standing 2,000 feet apart on a straight, horizontal road. They 
observe a hot-air balloon between them directly above the road. The angle of elevation 
from Tom is $62^\circ$ and from Jerry is $48^\circ$. Find the height of the balloon to the nearest foot. \\


\includegraphics[width=0.3\linewidth]{alex/17FinalBalloon}
\vskip 1.7in

\newp

\item You are selling tickets to a concert, and you need to decide on the ticket price.

From past experience, you know that the number of tickets you
sell is a linear function of the ticket price.

If you set the ticket price to be $\$10$, then you will sell $292$ tickets.

If you set the ticket price to be $\$15$, then you will sell $240$ tickets.

The \textit{revenue} is the amount of money you make from the sale of the tickets.

\begin{enumerate}
\item What ticket prices would result in a revenue of $\$2700$ ?
\vfil

\vfil

\vfil
\vfil

\item What should the ticket price be to maximize the revenue?
\vfil

\end{enumerate}
%\item You have 100 feet of fence to fully enclose a wedge-shaped garden. Compute the radius $R$ and the angle $\theta$ which result in the largest possible enclosed area.\\ 
%\includegraphics[width=0.25\linewidth]{alex/17FinalWedge}



\newp


\item You are tasked to build a tunnel through the Sine Hill, so-called because it has a sinusoidal cross section (see figure). The valley to the left is filled with water to a maximum depth of 50 m, and the top of the hill is 250 m above the water level. 
You set up a coordinate system with the $x$-axis at water level and the $y$-axis 200 m to the right of the deepest part of the water, as shown. The top of the hill has $x$-coordinate $x=400$ m.

\vspace{-0.2in}

\begin{center}\includegraphics[width=0.6\linewidth]{alex/17FinalMountain2}\end{center}

\vspace{-0.3in}

 a) Write a sinusoidal function expressing $y$ in terms of $x$ for the points on the surface of the hill, in the above coordinate system. 
\vfil \vfil

b) If your tunnel is to be dug horizontally through the hill, at a height of 150 m above the water level, how long will it be?  Round your answer to the nearest meter.

\vfil \vfil
\newp

%
%\item Rocinante is a horse and he is searching for his master on his ranch. The master is not moving, just sitting under a tree and whistling. His whistle can be heard from up to 1 mile away. Initially, Rocinante is located at a point 8/10 of a mile north and 8/10 of a mile east of his master. 
%
%
%\begin{center}\includegraphics[width=0.3\linewidth]{alex/17FinalDQ}\end{center}
%
%Rocinante starts to gallop at a constant speed of 20 {miles/hr} in a straight line towards his barn, which is located 1 mile due west of his master. After a while, he hears his master's whistle and, as soon as he hears it, he turns and continues galloping at the same speed straight towards his master. How long (in \textbf{minutes}) will it take Rocinante to reach his master?
%
%
%\newp

\item Let $A$ and $B$ be two constants, and consider the functions:
 $$f(x)=\sqrt{16-x^2}$$ $$g(x)=-2f(Ax+B)+3$$

\subitem (a)  On the graph below, draw the graph of $f(x)$, and determine its domain and range.

\begin{center}\includegraphics[width=0.5\linewidth]{alex/17grid}\end{center}

Domain of $f$ \hrulefill \hspace{0.5in} Range of $f$ \hrulefill

\medskip

\subitem (b)  What is the range of $g(x)$?\vskip 1in
\vfil

\subitem (c)  For what values of $A$ and $B$ will the domain of $g(x)$ be $-1\leq x\leq 3$? 
\vfil \vfil
\newp


%\item  Si begins walking from a point 9 km west and 6 km north of the center of
%the Circular Forest.  The Circular Forest has a circular shape and a radius of 8 km.
%
%Si walks directly toward Point $A$.
%Point $A$ is $3$ km directly east of the center of the forest.
%
%When Si reaches Point $A$, Si turns and walks due south until they have left the forest.
%
%\begin{enumerate}
%\item For what total distance does Si walk in the forest?
%\vfil \vfil
%\item On Si's walk, where is Si when they are closest to the center of the forest? 
%Give the location relative to the center of the forest.
%\end{enumerate}
%
%\newp

\item You have a pizza shaped as shown in the figure below (dimensions
are in centimeters).
You are going to make a vertical cut through the pizza.
The cut will be located a distance of $x$
centimeters from the leftmost corner of the pizza.

Express the area of the pizza to the left of the cut 
as a multipart function of $x$.

\begin{tikzpicture}[scale=0.5]
\tikzstyle{arrow}=[<->,>=stealth,very thick]
\draw[line width=3pt] (0,0) -- (10,0) -- (10,8) -- (3,4) -- (0,0);
\draw (11.25,4.5) node {$8$};
\draw (5,-1.25) node {$10$};
\draw[arrow] (3.5,4) -- (3.5,0);
\draw (4,2.5) node {$4$};
\draw (3,4) -- (4,4); 
\draw (3,4) -- (3,5);
\draw (0,0) -- (0,5);
\draw (10.25,8) -- (11.25,8);
\draw (10.25,0) -- (11.25,0);
\draw[arrow] (3,4.5) -- (0,4.5);
\draw (1.5,5.) node {$3$};
\draw[arrow] (10.75,0) -- (10.75,8);
\draw (0,-0.25) -- (0,-1.25);
\draw (10,-0.25) -- (10,-1.25);
\draw[arrow] (0,-0.75) -- (10,-0.75);
\end{tikzpicture}

\newp



\item City X and City Y are growing exponentially.  

City X had a population of $4000$ in the year 1960 and a population of $7000$ in the year 2000.

The population of City Y doubles every $24$ years.

The populations of City X and City Y will be equal in the year 2020.

When will the population of City Y be twice that of City X?

Express your answer in years after the year 2000.

\newp

\item  Paula and Jing are running around a circular track.  They start running at the same time.

Paula starts from the northernmost point and runs clockwise at $2$ radians per minute.

Jing starts from the westernmost point of the track and runs counterclockwise.

Jing and Paula pass each other after they have been running for $1.4$ minutes.

\begin{enumerate}
\item What is Jing's angular speed?

\vfil

\item Suppose the radius of the track is $140$ meters. What is the straight-line distance between
Jing and Paula after they have been running for $11.3$ minutes?
\vfil

\end{enumerate}

\end{enumerate}

\end{document}


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