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\lhead{\color{blue} \large Lecture Notes}
\chead{\color{black} \Large Calculus with Trigonometry}
\rhead{\large page   \ \thepage}
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\lfoot{\small   \copyright $\;$   Hidegkuti,  2014}
\rfoot{\small   Last revised: December 3, 2014}
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\begin{document}


\begin{enumerate}
\item A searchlight $100$ meters from a road is tracking a car moving at $%
100 $ kilometers per hour. \ At what rate (in degrees per second) is the
searchlight turning when the car is $141$ meters away?

\item At what position on the road is the angle $\theta $ maximized? \FRAME{%
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\item How long is the longest straight rod that can be carried through the
corner shown on the picture below? \ 

a) \ \ Assume that $a=10$ and $b=6$

b) \ Solve the problem in general, using $a$ and $\ b.$\FRAME{dtbpF}{2.2122in%
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\item Particle $A$ is moving in the plane according to $x=3\sin 3t$ and $%
y=3\cos 3t$ and particle $B$ is moving according to $x=3\cos 2t$ and $%
y=3\sin 2t$. \ Find the maximum distance between $A$ and $B$.

\item Two trains, each $50$ meters long, are moving away from the
intersection point of perpendicular tracks at the same speed. \ Where are
the trains when train $A$ subtends the largest angle as seen from the front
of train $B$?

\item Where is the function $f\left( x\right) =\sin ^{3}x$ concave up? \ \
Concave down?

\pagebreak
\end{enumerate}

\begin{center}
{\LARGE Answers}\bigskip
\end{center}

\begin{enumerate}
\item $\dfrac{1}{100}\left( \dfrac{100000}{3600}\right) \left( \dfrac{100}{%
141}\right) ^{2}\approx 0.139\,72\left( \dfrac{180}{\pi }\right) $rad $=%
\dfrac{8.\,\allowbreak 005\,4^{\circ }}{\text{s}}$

\item $\sqrt{3500}\approx \allowbreak 59.\,\allowbreak 161$

\item a) \ \ $\theta =\tan ^{-1}\left( \sqrt[3]{\dfrac{10}{6}}\right)
\approx \allowbreak 49.\,\allowbreak 855^{\circ }$ \ \ \ $L=\dfrac{10}{\sin
\left( \allowbreak 49.\,\allowbreak 855^{\circ }\right) }+\dfrac{6}{\cos
\left( \allowbreak 49.\,\allowbreak 855^{\circ }\right) }\approx \allowbreak
22.\,\allowbreak 388$

b) \ $\theta =\tan ^{-1}\left( \sqrt[3]{\dfrac{a}{b}}\right) $ \ \ \ \ $L=%
\dfrac{a}{\sin \theta }+\dfrac{b}{\cos \theta }$ \ So, \ $L=\dfrac{a}{\sin
\left( \tan ^{-1}\left( \sqrt[3]{\dfrac{a}{b}}\right) \right) }+\dfrac{b}{%
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\item $6$

\item right at the start, both at the station

\item Concave up where $\cos x>0$ -\ that is,\ \ \ $-\dfrac{\pi }{2}+2k\pi
<x<\dfrac{\pi }{2}+2k\pi $ \ \ \ where $k\in 
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Concave up where $\cos x<0$ -\ that is,\ \ $\dfrac{\pi }{2}+2k\pi <x<\dfrac{%
3\pi }{2}+2k\pi $ \ \ \ \ \ where $k\in 
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\end{enumerate}

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