\documentclass[12pt]{report}
\usepackage{palatino}
\usepackage{epsfig}
\usepackage[margin=0.8in]{geometry}
%\usepackage{draftwatermark}
%\SetWatermarkText{DRAFT}
%\SetWatermarkScale{1}
\usepackage{amsmath}
\usepackage{wrapfig}
\usepackage{tikz}
\usetikzlibrary{patterns,arrows,snakes}
\tikzstyle{printersafe}=[snake=snake,segment amplitude=0 pt]
\pagestyle{empty} %%% this results in no pagenumbers (footer is empty}
\setlength{\parindent}{0in}
\setlength{\parskip}{0.05in}
\baselineskip=20pt
\newcommand{\textbox}[1]{
\raisebox{-.06 in}{\begin{tikzpicture}
\node [draw] at (0,0) {#1};
\end{tikzpicture}}\!
}
\newcommand{\dsps}{\displaystyle}
\newcommand{\pp}{\par \noindent}
\newcommand{\newp}{\vfil \eject}
\newcommand{\Points}[1]{(#1 points)}
\tikzstyle{printersafe}=[snake=snake,segment amplitude=0 pt]
\newcommand{\graphgrid}[6]{
\foreach \x in {#1,...,#2}{
\draw (\x,0) node [above left,scale=.8] {\x};
\draw [line width=1pt,densely dotted,printersafe] (\x,#3-.2) -- (\x,#4+.2);}
\foreach \y in {#3,...,#4}{
\draw (0,\y) node [above left,scale=.8] {\y};
\draw [line width=1pt,densely dotted,printersafe] (#1-.2,\y) -- (#2+.2,\y);}
\draw[<->, line width=1 pt] (#1-.6,0) -- (#2+.4,0) node[right] {#5};
\draw[<->, line width=1 pt] (0,#3-.4) -- (0,#4+.6) node[above] {#6};
}
\begin{document}
\vfil \noindent
{\LARGE{
\hfil Math 120 Spring 2022 \hfil \pp
\hfil Final Exam\hfil \pp
\hfil June 4, 2022 \hfil \pp
}}
\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill
\medskip
\hfil Signature: \hrulefill \hrulefill \hrulefill \hspace{0.5in} Section: \hrulefill
\vfil
\begin{center}
{\Large
\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}
}\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 SEVEN} problems on {\bf FIVE} double-sided pages. The backs of the first and last page are 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 an algebraic 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 170 minutes to complete this 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}
% Cake
\item Mrs. White is in the dining room using a knife to cut this cake:
\vspace{-.4 in}
\begin{flushright}
\begin{tikzpicture}
\draw [very thick] (0,0) -- (6,4) -- (9,4) -- (9,0) -- cycle;
\draw [very thick,densely dotted,<->,printersafe,shift={(0,-.3)}] (0,0) -- node[below] {$9$} (9,0);
\draw [very thick,densely dotted,<->,printersafe,shift={(.3,0)}] (9,0) -- node[right] {$4$} (9,4);
\draw [very thick,densely dotted,<->,printersafe,shift={(0,.3)}] (6,4) -- node[above] {$3$} (9,4);
\end{tikzpicture}
\end{flushright}
\vspace{-.3 in}
\begin{enumerate}
\item {\bf [6 points]} Suppose she makes a vertical cut $x$ units from the left end of the cake. Write a multipart function for the area to the left of the cut.
\vfill
\item {\bf [4 points]} Mrs. White wants to make two vertical cuts to divide the area of the cake into three pieces of equal area. How far in should she make those two cuts?
\vfill
\end{enumerate}
\newpage
% Linear???
\item {\bf [10 points]} Colonel Mustard is in the billiard room, and has tied two billiard balls together with an $80$-inch rope.\\[.1 in]
At time $t=0$, he knocks the first ball north at a constant speed of $4$ inches per second.\\[.1 in]
Two seconds later, he knocks the second ball from the same starting position as the first ball. It travels west at a constant speed of $5$ inches per second.\\[.1 in]
When does the rope become tight?
\newpage
% Exponential
\item Professor Plum is investigating a leaky lead pipe in the conservatory. It seems to be causing the flowers to grow exponentially!
\begin{enumerate}
\item {\bf [3 points]} The number of forsythias doubles every $25$ days.\\[.1 in]
Initially, there were $20$ forsythias.\\[.1 in]
Write a function $f(t)$ for the number of forsythias after $t$ days.
\vfill
\item {\bf [4 points]} The number of geraniums is also growing exponentially.\\[.1 in]
After $16$ days, there were $10$ geraniums, and after $22$ days, there were $13$ geraniums.\\[.1 in]
Write a function $g(t)$ for the number of geraniums after $t$ days.
\vfill
\item {\bf [3 points]} When will the number of forsythias equal the number of geraniums?\\[.1 in]
Round your answer to the nearest day.
\vfill
\end{enumerate}
\newpage
% Trigonometry
\item Mrs. Peacock is standing in the study, where a candlestick is positioned on the floor.\\[.1 in]
Mrs. Peacock is $165$ centimeters tall. The candlestick is $20$ centimeters tall, and it's holding a candle which is $10$ centimeters tall.\\[.1 in]
Let $\theta$ be the angle of elevation of Mrs. Peacock's head relative to the top of the candle, as shown in the picture below.
\begin{flushright}
\begin{tikzpicture}
\draw [very thick] (-1,0) -- (6,0);
\draw [double,cap=round] (0,1) -- node[right,scale=.7] {candle} (0,1.5);
\draw [line width=2pt] (0,0) -- node [right,scale=.7] {candlestick} (0,1);
\draw [line width=2pt] (5,0) -- node [left,scale=.7] {Mrs. Peacock} (5,5);
\draw [dash pattern = on 3pt off 1pt,printersafe] (5,1.5) -- (0,1.5) -- (6,5.7);
\draw (.6,1.5) arc (0:35:.6);
\draw (0,1.5) ++ (17.5:.8) node {$\theta$} ;
\draw [<->,very thick,densely dotted,printersafe,shift={(-.2,0)}] (0,0) -- node [left] {$20$ cm} (0,1);
\draw [<->,very thick,densely dotted,printersafe,shift={(-.2,0)}] (0,1) -- node [left] {$10$ cm} (0,1.5);
\draw [<->,very thick,densely dotted,printersafe,shift={(.2,0)}] (5,0) -- node [right] {$165$ cm} (5,5);
\draw [<->,very thick,densely dotted,printersafe,shift={(0,-.2)}] (0,0) -- node [below] {$x$ cm} (5,0);
\end{tikzpicture}
\end{flushright}
\begin{enumerate}
\vspace{-3 in}
\item {\bf [5 points]} Mrs. Peacock measures $\theta$ to be $50^\circ$.\\[.1 in]
How far away from the candlestick is she?\\[.1 in]
{\em (In other words, what's $x$?)}
\vfill
\item {\bf [5 points]} The candle burns at a constant speed. After $1$ minute, the angle $\theta$ is $51^\circ$.\\[.1 in]
When will the candle burn all the way down?
\vfill
\end{enumerate}
\newpage
% Circular motion
\item Miss Scarlett is in the ballroom, dancing to a Beatles album.\\[.1 in]
Her dance proceeds clockwise in a circle of radius $20$ feet at a constant speed. It takes her $17$ seconds to make one complete lap, and she reaches the northernmost point $5$ seconds after she starts.
\begin{enumerate}
\item {\bf [3 points]} Find Miss Scarlett's linear speed.
\vskip 2 in
\item {\bf [4 points]} Impose a coordinate system with the center of the circle at the origin.\\[.1 in]
Write parametric equations for Miss Scarlett's coordinates after $t$ seconds.
\vfill
\item {\bf [3 points]} After $35$ {\bf minutes}, how far {\bf east} is Miss Scarlett from her starting point?
\vfill
\end{enumerate}
\newpage
% Sinusoidal
\item {\bf [5 points per part]} Mr. Green is in the kitchen, using a wrench to adjust the water pressure under the sink. The pressure is a sinusoidal function of time.\\[.1 in]
The pressure first reaches its maximum of $100$ psi $13$ minutes after the start.\\[.1 in]
It then decreases, reaching a minimum of $50$ psi $35$ minutes after the start.
\begin{enumerate}
\item Find a function $f(x)$ for the water pressure (in psi) $x$ minutes after the start.
\vfill
\item The maximum recommended water pressure in a home is $80$ psi. In the first hour, for how much time (total) is the pressure above this level?
\vfill
\end{enumerate}
\newpage
% Graph
\item Mr. Boddy is in the library with a linear-to-linear rational function:\[f(x) = \frac{3x+2}{x+4}\]
\begin{enumerate}
\item {\bf [4 points]} Find the following data about this function:\\[.1 in]
\begin{itemize}
\item Horizontal asymptote:\\[.2 in]
\item Vertical asymptote:\\[.2 in]
\item $x$-intercept:\\[.2 in]
\item $y$-intercept:\\[.2 in]
\end{itemize}
\item {\bf [3 points]} Compute $f(f(6))$.
\vfill
\item {\bf [3 points]} Write a formula for $f^{-1}(x)$.
\vfill
\end{enumerate}
\end{enumerate}
\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{document}