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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A - Winter 2018 \hfil \pp
\hfil Midterm Exam Number Two\hfil  \pp
\hfil February 22nd, 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 & 12 &\hspace{10mm} \hfil\\ \hline 
 2 & 12 &       \\ \hline
 3 & 12 &       \\ \hline
 4 & 12 &      \\ \hline
 5 & 12 &      \\ \hline
Total & 60 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item This exam consists of FIVE problems on THREE double-sided pages.
\item Show all work for full credit. 
\item You may use a TI-30X IIS calculator during this
exam. Other calculators and electronic devices are not permitted.
\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 \textbox{Draw a box} around your final answer to each problem.
\item {\bf Do not write within 1 centimeter of the edge!} Your exam will be scanned for grading.
\item If you run out of room, write on the back of the last page and indicate that you have done so. If you still need more room, ask your TA for an extra page to staple to your exam.
\item You may use one hand-written double-sided 8.5" by 11" page of notes.
\item You have 50 minutes to complete the exam.
\end{itemize}

\vfil
{\small --}

\newpage

\begin{enumerate}

%%%%%%%%%%%%%%%%% Version A

% Full exponential thing.
\item {\bf [12 points]} I have placed some plums in an icebox. Their temperature (in Celsius) is an exponential function of time.\\[.1 in]
$10$ hours from now, the plums will be at a temperature of $11^\circ$ Celsius.\\[.1 in]
$22$ hours from now, they will be at a temperature of $5^\circ$ Celsius.\begin{enumerate}
\item Write a function $p(t)$ for the temperature of the plums (in degrees Celsius) $t$ hours from now.
\vskip 4 in
\item I will eat the plums when their temperature reaches $4^\circ$ Celsius. How many hours from now is this?
\end{enumerate}

\newpage

% Full optimization thing.
\item {\bf [12 points]} I have \$$3500$ with which I'd like to build a fence in the shape of a sector, with a partition in the shape of an arc running between the midpoints of the straight sides, as shown below.\\[.1 in]
The outside fence costs \$$5$ per foot, and the partition costs \$$2$ per foot.\\[.1 in]
What is the {\bf maximum possible total area} inside the fence?
\vspace{-.8 in}
\begin{flushright}
\begin{tikzpicture}[rotate=80]
\draw [line width=2pt] (0,0) -- (0:4) arc (0:84:4) -- cycle;
\draw [thick, dash pattern = on 5 pt off 2 pt,printersafe] (0:2) arc (0:84:2);
\end{tikzpicture}
\end{flushright}

\newpage

% Full linear-to-linear thing, with inverse.
\item {\bf [12 points]} Shuri is developing a shock-absorbing material. The amount of energy it can absorb is a linear-to-linear rational function of how long Shuri conducts her research.\\[.1 in]
Right now, the material can absorb $2$ megajoules of energy.\\[.1 in]
If she researches for $2$ more days, it will be able to absorb $2.8$ megajoules.\\[.1 in]
As Shuri's research continues, the amount of energy the material is able to absorb will approach (but not reach) $5$ megajoules.
\begin{enumerate}
\item Write a linear-to-linear rational function $f(x)$ for the amount of energy the material can absorb, in megajoules, if Shuri researches for $x$ days.
\vskip 4 in
\item Find the inverse of the function you found in part (a).
\end{enumerate}

\newpage

% Wheels, mostly straightforward
\item {\bf [12 points]} Check out these wheels:\\[.1 in]
Wheel A has a radius of $3$ inches and makes $2$ revolutions per minute.\\[.1 in]
Wheel B is connected by a belt to wheel A, and by an axle to wheel C.\\[.1 in]
Wheel C has a radius of $5$ inches and rotates at a speed of $10$ inches per minute.\\[.1 in]
What's the radius of wheel B?

\vskip 3.3 in

% Small transformation thing.
\item {\bf [12 points]} On the left is the graph of $f(x)$. On the right, please graph $g(x) = \frac{1}{2}f(x-1)-1$.

\begin{center}
\begin{tikzpicture}[auto,scale=.75, line width=2pt,inner sep=1.5]

\foreach \x in {-4,-3,...,4} \draw [printersafe, line width=1pt,densely dotted] (\x,-4.2)--(\x,4.2);
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\foreach \y in {-4,-3,...,4} \draw (0,\y) node[anchor=south west] {\scriptsize \y};

\draw [line width = 1.5 pt, <->] (-4.6,0) -- (4.7,0) node[right] {$x$};
\draw [line width = 1.5 pt, <->] (0,-4.6) -- (0,4.7) node[above] {$f(x)$};

\draw (-3,-2) -- (-1,4)
(-1,-2) -- (1,-2) -- (3,0);
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(-1,4) circle (.1)
(3,0) circle (.1);
\draw [fill=white,line width=1 pt] (-1,-2) circle (.13);
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\qquad
\begin{tikzpicture}[auto,scale=.75, line width=2pt,inner sep=1.5]

\foreach \x in {-4,-3,...,4} \draw [printersafe, line width=1pt,densely dotted] (\x,-4.2)--(\x,4.2);
\foreach \x in {-4,-3,...,4} \draw (\x,0) node[anchor=south west] {\scriptsize \x};
\foreach \y in {-4,-3,...,4} \draw [printersafe, line width=1pt,densely dotted] (-4.2,\y)--(4.2,\y);
\foreach \y in {-4,-3,...,4} \draw (0,\y) node[anchor=south west] {\scriptsize \y};

\draw [line width = 1.5 pt, <->] (-4.6,0) -- (4.7,0) node[right] {$x$};
\draw [line width = 1.5 pt, <->] (0,-4.6) -- (0,4.7) node[above] {$g(x)$};

\end{tikzpicture}
\end{center}


\end{enumerate}

\end{document}