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\newcommand{\pp}{\par \noindent}
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\begin{document}
\noindent
\vfil \noindent
\large
\hfil Math 120 - Autumn 2005 \hfil \pp
\hfil Final Exam\hfil \pp
\hfil December 10, 2005 \hfil \pp
\hfil Answers \hfil \pp
\normalsize
\begin{enumerate}
\item 1998
\item 0.9760860118 hours
\item 31.70 days
\item 8.60808 hours
\item \begin{enumerate}
\item $x(t)=35-7t, y(t)=\frac{18}{5}t$
\item $\frac{22\sqrt{2}}{3}$ seconds
\item $x(t)=5+\frac{3}{\sqrt{2}}t, y(t) = 22- \frac{3}{\sqrt{2}}t$
\end{enumerate}
\item 550 mice
\item \begin{enumerate}
\item $\frac{2\pi}{92}$ radians per second
\item 58.56901906 seconds
\item $(58.56901906 \cos (\frac{2 \pi}{86}t),58.56901906 \sin (\frac{2 \pi}{86}t))$
\item 44.45 seconds
\end{enumerate}
\item \begin{enumerate}
\item $x=9.5$
\item \[
g(x) = \left \{ \begin{array}{ll}
-4x+17 & \mbox{if $x<2$} \\
-2x+13 & \mbox{if $2 \le x < 4$} \\
-5x+52 & \mbox{if $x \ge 4$}
\end{array}
\right.
\]
\end{enumerate}
\newp
\item The populations of termites and spiders in a certain house are growing exponentially.
The house contains 100 termites the day you move in. After 4 days,
the house contains 200 termites. On day 3 after moving in, there
are two times as many termites as spiders. On day 8, there were
four times as many termites as spiders.
How long (in days) does it take the population of spiders to triple?
\newp
\item As a result of an attack by aliens, in which his brain was exposed to
zeta waves, Howard has become Sometimes Smart Guy.
His IQ is a sinusoidal function of time.
His IQ is at its lowest, 67 points, 4.25 hours after the attack.
His IQ then increases to a maximum of 253 points 9.75 hours after the attack.
During the first 24 hours following the attack, for how many hours is Sometimes Smart Guy's
IQ at or above 200?
\newp
\item Fran and Zoe live in the coordinate plane.
At midnight, Fran starts out from the point (35,0).
She moves at a constant speed along a straight line
and will pass through the point (0,18) after 5 seconds.
At midnight, Zoe starts out from the point (5,22) and heads toward the fourth quadrant
along the line $y=-(x-5)+22$ at the rate of 3 units per second.
\epsfig{file=fran01.eps,width=7cm,angle=0}
\begin{enumerate}
\item Find parametric equations for Fran's location $t$ seconds after midnight.
\vfil \vfil
\item How long does it take Zoe to reach the $x$-axis?
\vfil \vfil
\item Find parametric equations for Zoe's location $t$ seconds after midnight.
\end{enumerate}
%Express the distance between Zoe and Fran as a function of $t$, the number of seconds
%past midnight. Don't worry about simplifying the expression.
\newp
\item Maude is conducting an orchestra of mice. She knows she can achieve
a loudness of 80 decibels with 100 mice.
With more mice, the loudness will approach, but never exceed, 120 decibels.
Of course, with no mice, she achieves a loudness of 0 decibels.
If the loudness of the orchestra is a linear-to-linear function of the number of
mice, how many mice would she need to achieve a loudness of 110 decibels?
\newp
\item Pasha and Olga are running around a circular track.
They start out together, but run in opposite directions.
Pasha runs at 4 meters per second, and it takes him 92 seconds to complete each lap of the track.
Olga takes 86 seconds to complete each lap.
\epsfig{file=olga01.eps,width=11cm,angle=0}
\begin{enumerate}
\item What is Pasha's angular speed in radians per second?
\vfil
\item What is the radius of the track?
\vfil
\item What are Olga's coordinates $t$ seconds after she starts running?
\vfil \vfil
\item When do Pasha and Olga pass each other for the first time?
\end{enumerate}
\newp
%\item At Linear Orchard, the number of apples each tree produces per season is a linear
%function of the number of trees in the orchard.
%If there are 100 trees in the orchard, a total of 12000 apples are produced per season in the
%whole orchard.
%If there are 200 trees in the orchard, a total of 13000 apples are produced per season in the whole
%orchard.
%
%How many trees should there be in the orchard to produce the maximum possible number of
%apples per season?
%\newp
\item Let $f(x)$ be defined as follows:
\[
f(x) = \left\{ \begin{array}{ll}
5-x & \mbox{if $x<4$} \\
18-2x & \mbox{if $x \ge 4$}
\end{array}
\right.
\]
\begin{enumerate}
\item Find all solutions to the equation $f(x) = -1$.
\vfil
\item Write the multipart rule for the function
\[
g(x) = |x-2| + 3f(x).
\]
\vfil
\end{enumerate}
\end{enumerate}
\end{document}