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\newcommand{\pp}{\par \noindent}
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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A - Spring 2022 \hfil \pp
\hfil Midterm Exam Number Two\hfil  \pp
\hfil May 19th, 2022 \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}
\qquad \qquad {\Huge
\begin{tabular}{||c|c|r||} \hline 
 1 & 15 &\hspace{10mm} \hfil\\ \hline 
 2 & 15 &       \\ \hline
 3 & 15 &       \\ \hline
 4 & 15 &       \\ \hline
Total & 60 & \\ \hline 
\end{tabular}
}\begin{tikzpicture}
\node [scale=.7] at (0,0) {\em This grid is purely decorative.};
\node [scale=.7] at (0,-.5) {\em The exam is graded online.};
\end{tikzpicture}
\end{center}

\vfil
\begin{itemize}
\item This exam consists of {\bf FOUR} problems on {\bf FOUR} double-sided pages. The fourth page is left blank for scratch work.
\item Show all work for full credit.
\item You may use a TI-30X IIS (or equivalent) 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 one of the scratch work pages {\bf and indicate that you have done so}. If you still need more room, raise your hand and ask for an extra page.
\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

\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see first page'' below a problem.

\newpage

\begin{enumerate}

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

% Linear-to-linear
\item {\bf [15 points]} Gomba is learning to climb a new set of stairs. The time it takes him to climb them on his $n^{\text{th}}$ attempt is a linear-to-linear rational function of $n$.\\[.1 in]
On his {\bf 1$^{\text{st}}$} attempt, it took him $31$ seconds to climb the stairs.\\[.1 in]
On his {\bf 4$^{\text{th}}$} attempt, it took him $17$ seconds to climb the stairs.\\[.1 in]
As Gomba continues to practice, the time it takes will approach (but not reach) $3$ seconds.\\[.1 in]
How long does his {\bf 13$^{\text{th}}$} attempt take?  

\newpage

% Wheels
\item {\bf[15 points]} Three wheels are connected as shown in the diagram below: Wheels A and B are connected by an axle, and Wheels B and C are connected by a belt.

\begin{center}
\begin{tikzpicture}[auto,scale=.23, line width=1.5pt, inner sep=2 pt]

\draw (22,0) arc (0:360:5);
\draw (19,0) arc (0:360:2);
\draw (45,0) arc (0:360:9);

\draw [->] (11.1,0) arc (180:140:5.9);
\draw [->] (14.1,0) arc (180:120:2.9);
\draw [->] (36,9.9) arc (90:60:9.9);

\draw (17,0) node[circle,draw,fill=gray] (C) {};

\draw [thin] (17,2.1) -- (33,8.5);
\draw [thin] (17,-2.1) -- (33,-8.5);

\draw (19,9.4) node {Wheel A};
\draw [-stealth',dashed,line width=1pt,printersafe] (19,8) -- (18,5); 
\draw (14,-8) node {Wheel B};
\draw [-stealth',dashed,line width=1pt,printersafe] (15,-7) arc (210:150:6); %(16.5,-2);
\draw (36,0) node {Wheel C};
\end{tikzpicture}
\end{center}

Wheel A has a radius of $5$ meters, and rotates at a linear speed of $10$ meters per second.\\[.1 in]
Wheel C has a radius of $8$ meters, and takes $9$ seconds to make one complete rotation.\\[.1 in]
What is the radius of wheel B?

\newpage

% Exponential, compositions, inverses
\item {\bf [5 points per part]} For each part of this question, let $f(x) = 3\log_2(x)+2$.
\begin{enumerate}
\item Find a formula for $f^{-1}(x)$. {\bf Write your answer in standard exponential form.}
\vskip 2.5 in
\item Suppose $f(f(x)) = 11$. What's $x$?
\vskip 2.5 in
\item Let $g(x) = \log_2(x)$. What transformations (shifting, scaling, reflecting) will lead you from the graph of $y=g(x)$ to the graph of $y=f(x)$?
\vfill
Fill in the blanks:\\[.1 in]

\begin{itemize}
\item First, you \underline{\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad}.\\[.3 in]
\item Then, you \underline{\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad}.
\end{itemize}

\end{enumerate} 

\newpage

% Short optimization
\item {\bf [15 points]} Steve is no longer invited to parties, because he keeps trying to entertain people with optimization problems. Here's his most notorious trick:\\[.1 in]
He takes a 16 cm piece of wire, breaks it into two pieces, and uses those pieces to construct two figures: a square, and a sector with angle $0.8$ radians.\\[.1 in]
What is the {\bf minimum possible total area} of these two shapes?
\vspace{-.3 in}
\begin{flushright}
\begin{tikzpicture}
\draw [very thick,shift={(-2.5,0)}] (0,0) rectangle (1.5,1.5) (.75,0) node[below] {$x$} (0,.75) node [left] {$x$};
\draw [very thick] (0,0) -- node[below] {$r$} (20:2) arc (20:65.83:2) -- node[left] {$r$} cycle;
\draw (20:.6) arc (20:65.83:.6);
\node [scale=.6] at (40:1) {$0.8$ rad};
\end{tikzpicture}
\end{flushright}

\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see back page'' below a problem.

\newpage

You may use this page for scratch-work.\\[.1 in]
{\bf All work on this page will be ignored} unless you write \& circle ``see back page'' below a problem.

\end{enumerate}

\end{document}
