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\vfil \noindent

{\LARGE{
\hfil Math 120 - Winter 2015 \hfil \pp
\hfil Final Exam\hfil  \pp
\hfil March 14, 2015 \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 & 12 &\hspace{10mm} \hfil\\ \hline 
 2 & 13 &       \\ \hline
 3 & 13 &      \\ \hline
4 & 9 &    \\ \hline
5 & 15 &    \\ \hline
6 & 13 &    \\ \hline
7 & 12 &    \\ \hline
8 & 13 &    \\ \hline
Total & 100 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item This exam consists of EIGHT problems on NINE 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.
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 170 minutes to complete the exam.
\end{itemize}

\vfil
{\small $\circ$}

\newpage

\begin{enumerate}

\item {\bf [12 points]} In the following figure (not drawn to scale), find $x$.

\begin{tikzpicture}
\draw [very thick] (0,0) -- (9,0) -- (0,4) -- cycle
(0,1.5) -- (5,0)
(0,4) -- (5,0);
\draw (3.5,0) arc (180:164:1.5);
\draw (0,3.5) arc (270:320:.5);
\draw (8.2,0) arc (180:155:.8);
\draw node at (3.1,.25) {$40^\circ$};
\draw node at (.4,3.2) {$33^\circ$};
\draw node at (7.7,.25) {$50^\circ$};
\draw [very thick,densely dotted,<->] (5,-.15) -- node[below] {$4$} (9,-.15);
\draw [very thick,densely dotted,<->] (-.15,0) -- node[left] {$x$} (-.15,1.5);
\end{tikzpicture} 

\newpage

\item The number of trees in Treeattle grows exponentially.\\[.1 in]
Treeattle had 600 trees in the year 2008, and 1100 trees in the year 2015.
\begin{enumerate}
\item {\bf [4 points]} Write a function $f(x)$ for the number of trees in Treeattle, $x$ years after the year 2000.
\vskip 3 in
\item {\bf [6 points]} Compute $f^{-1}(x)$, the inverse of the function you found in part (a).
\vskip 2.5 in
\item {\bf [3 points]} When will there be 4000 trees in Treeattle? Round your answer to the nearest year.
\end{enumerate}

\newpage

\item \begin{enumerate}
\item {\bf [3 points]} Write a function $f(x)$ for an upper semicircle of radius $4$ centered at $(6,2)$, defined over the interval $2 \leq x \leq 10$.
\vskip 1.5 in
\item {\bf [3 points]} Write a function $g(x)$ for the curve obtained by taking $f(x)$ from part (a), moving it $2$ units to the left, and {\em then} scaling it horizontally by a factor of $1/2$.
\vskip 2.5 in
\item {\bf [4 points]} Find the domain and range of $g(x)$.
\vskip 2.5 in
\item {\bf [3 points]} Is $g(x)$ one-to-one? Explain, briefly.
\end{enumerate}

\newpage

\item {\bf [9 points]} In the following configuration, wheels A and B are connected by a belt, as are wheels C and D. Wheels B and C are connected by an axle.
\begin{center}
\begin{tikzpicture}[scale = .5]
\draw [line width=2 pt] (3,0) arc (0:360:3)
(11,0) arc (0:360:2)
(13,0) arc (0:360:4)
(18,0) arc (0:360:1.5);
\draw [line width=1 pt] (.3,3) -- (9,2)
(.3,-3) -- (9,-2)
(9.9,3.9) -- (17.1,1.4)
(9.9,-3.9) -- (17.1,-1.4);
\draw node at (9,0) [circle,draw,very thick,fill=gray,inner sep=1.5] {};
\draw node at (0,0) {Wheel A}
node at (20,0) {Wheel D}
node at (7,5) (B) {Wheel B}
node at (11,-5) (C) {Wheel C};
\draw [thick,-stealth'] (C.west) arc (260:200:1.5);
\draw [thick,-stealth'] (B.east) arc (60:-40:1.9);
\draw node at (9,.6) {\small axle};
\end{tikzpicture}
\end{center}
Wheel A has a radius of 7 feet and rotates at a speed of 6 revolutions per minute.\\[.1 in]
Wheel B has a radius of 4 feet, Wheel C has a radius of 8 feet, and Wheel D has a radius of 3 feet.\\[.1 in]
How many seconds does it take Wheel D to make a complete rotation?
\newpage

\item Tori and Harry are both running {\bf counter-clockwise} around a circular track of radius 10 meters. Tori begins at the northernmost point and Harry begins at the easternmost point. Harry runs faster.
\begin{enumerate}
\item {\bf [4 points]} Tori first reaches the southernmost point after 8 seconds.\\[.1 in]
What is Tori's speed, in meters per second?
\vspace{-.4 in}
\begin{flushright}
\begin{tikzpicture}[scale=1.5]
\draw [line width=1 pt] (1,0) arc (0:360:1);
\draw node at (0,1) [very thick,draw,circle,fill=gray,label=above:{Tori},inner sep=2.5]{};
\draw node at (1,0) [very thick,draw,circle,fill=gray,label=right:{Harry},inner sep=2.5]{};
\draw [line width=2.5 pt,->] (95:1) arc (95:130:1);
\draw [line width=2.5 pt,->] (5:1) arc (5:40:1);
\end{tikzpicture}
\end{flushright}
\vskip 1.3 in
\item {\bf [6 points]} Harry begins running at the same time as Tori, and catches up to her in 11 seconds.\\[.1 in]
What is Harry's speed, in meters per second?
\vskip 2.5 in
\item {\bf [5 points]} Impose a coordinate system with units in meters and the origin at the center of the circle. After 80 seconds, what are Harry's coordinates?
\end{enumerate}

\newpage

\item Consider the following multipart function:

$$f(x) = \begin{cases}
x^2+6x+8 & \mbox{if }-4 \leq x < -1\\[.1 in]
3\sin\left(\dfrac{2\pi}{5}(x+1)\right)+4 & \mbox{if } -1 \leq x < 9\\
\end{cases}
$$

\begin{enumerate}
\item {\bf [6 points]} Sketch a graph of $f(x)$. Label your graph clearly.
\begin{center}
\begin{tikzpicture}[auto,scale=.8, line width=2pt]

\foreach \x in {-5,-4,...,10} \draw [line width=1pt,densely dotted] (\x,-2.2)--(\x,8.2);
\foreach \x in {-5,-4,...,10} \draw (\x+.2,.25) node {\scriptsize \x};
\foreach \y in {-2,-1,...,8} \draw [line width=1pt,densely dotted] (-5.2,\y)--(10.2,\y);
\foreach \y in {-2,-1,1,2,...,8} \draw (.2,\y+.25) node {\scriptsize \y};

\draw [<->] (-5.6,0) -- (10.7,0) node [right] {$x$};
\draw [<->] (0,-2.6) -- (0,8.7) node [above] {$f(x)$};

\end{tikzpicture}
\end{center}
\item {\bf [7 points]} Find {\em all solutions} to the equation $f(x) = 2$. 
\end{enumerate}

\newpage

\item Chlo\"e and and Jo\"e are walking around the co\"ordinate plane. They both begin walking at the same time, in straight lines at constant speeds.
\begin{enumerate}
\item {\bf [3 points]} Chlo\"e starts at $(-2,-3)$ and walks east at a speed of $4$ units per second.\\[.1 in]
Give parametric equations for Chlo\"e's co\"ordinates after $t$ seconds.
\vskip 2 in
\item {\bf [4 points]} Jo\"e begins at the point $(6,3)$ and walks towards the point $(14,-5)$, reaching it in $4$ seconds.\\[.1 in]
Give parametric equations for Jo\"e's co\"ordinates after $t$ seconds.
\vskip 2 in
\item {\bf [5 points]} When are Chlo\"e and Jo\"e closest together?
\end{enumerate}

\newpage

\item Let $f(x)$ be the linear-to-linear rational function with an $x$-intercept of $5$ and a $y$-intercept of $-4$, passing through the point $(35,-6)$.

\begin{enumerate}
\item {\bf [7 points]} Write a formula for $f(x)$.
\vskip 4 in
\item {\bf [2 points]} Write the domain and range of $f(x)$.
\vskip 1.5 in
\item {\bf [4 points]} Solve the equation $f(f(x)) = 2$.
\end{enumerate}

\end{enumerate}

\end{document}