\documentclass[11pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[nomarginpar]{geometry}
\usepackage{color}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{fancyhdr}

\setcounter{MaxMatrixCols}{10}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.00.0.2570}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{Created=Wednesday, July 12, 2006 00:27:03}
%TCIDATA{LastRevised=Wednesday, December 03, 2014 16:59:16}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Optimization 1">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Scientific Notebook\Booklet #1 - with Instructions">}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{PageSetup=72,72,72,72,1}
%TCIDATA{Counters=arabic,1}
%TCIDATA{<META NAME="PrintViewPercent" CONTENT="100">}
%TCIDATA{ComputeDefs=
%$f\left( t\right) =20\left( 55+5\sin t\right) \left( \dfrac{1}{\left(
%55+5\sin t\right) -42}-\dfrac{1}{\left( 55+5\sin t\right) -40}\right) $
%}

%TCIDATA{AllPages=
%H=36
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \hfill }
%}


\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\geometry{left=0.5in,right=0.5in,top=0.5in,bottom=0.4in}
\pagestyle{fancy}
\lhead{\color{blue} \large Lecture Notes}
\chead{\color{black} \Large Calculus with Trigonometry}
\rhead{\large page   \ \thepage}
\cfoot{}
\lfoot{\small   \copyright $\;$   Hidegkuti,  2014}
\rfoot{\small   Last revised: December 3, 2014}
\textwidth 7.56in 
\textheight 9.5in 
\setlength{\headheight}{28pt}
\setlength{\parindent}{0in}

\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{%
dtbpF}{5.1413in}{2.7354in}{0pt}{}{}{car1.bmp}{\special{language "Scientific
Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 5.1413in;height
2.7354in;depth 0pt;original-width 0.134in;original-height 0.0856in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'car1.bmp';file-properties "XNPEU";}}

\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%
}{1.8671in}{0pt}{}{}{insert4.bmp}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
2.2122in;height 1.8671in;depth 0pt;original-width 1.7262in;original-height
1.4529in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'insert4.bmp';file-properties "XNPEU";}}

\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, \ $L=\dfrac{a}{\sin
\left( \tan ^{-1}\left( \sqrt[3]{\dfrac{a}{b}}\right) \right) }+\dfrac{b}{%
\cos \left( \tan ^{-1}\left( \sqrt[3]{\dfrac{a}{b}}\right) \right) }$\FRAME{%
dtbpF}{2.6299in}{1.9623in}{0pt}{}{}{insert4b.bmp}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "F";width 2.6299in;height 1.9623in;depth
0pt;original-width 2.1534in;original-height 1.5999in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'insert4b.bmp';file-properties
"XNPEU";}}

\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 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$

Concave up where $\cos x<0$ -\ that is,\ \ $\dfrac{\pi }{2}+2k\pi <x<\dfrac{%
3\pi }{2}+2k\pi $ \ \ \ \ \ where $k\in 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$\FRAME{dtbpFX}{2.917in}{1.9441in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.917in;height 1.9441in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "-6.28";xmax "6.28";xviewmin "-6.28";xviewmax "6.28";yviewmin
"-1.3";yviewmax "1.3";viewset"XY";rangeset"X";plottype 4;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\sin
^{3}x$};linecolor "black";linestyle 1;pointstyle "point";linethickness
2;lineAttributes "Solid";var1range "-6.28,6.28";num-x-gridlines
100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file
"T";tempfilename 'NG10QN0F.wmf';tempfile-properties "XPR";}}

\pagebreak 
\end{enumerate}

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

\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?

$\dfrac{1}{100}\left( \dfrac{100000}{3600}\right) \left( \dfrac{100}{141}%
\right) ^{2}=\allowbreak 0.139\,72\left( \dfrac{180}{\pi }\right)
=\allowbreak 8.\,\allowbreak 005\,4$%
\begin{eqnarray*}
\tan \alpha &=&\dfrac{x}{100} \\
\sec ^{2}\alpha \cdot \alpha ^{\prime } &=&\dfrac{1}{100}\cdot x^{\prime } \\
\alpha ^{\prime } &=&\dfrac{1}{100}\cdot x^{\prime }\cdot \cos ^{2}\alpha =%
\dfrac{1}{100}\left( \dfrac{100000}{3600}\right) \left( \dfrac{100}{141}%
\right) ^{2}\approx 0.139\,72\dfrac{\text{rad}}{\text{s}}\approx 8\dfrac{%
\deg }{\text{s}}
\end{eqnarray*}

\item At what position on the road is the angle $\theta $ maximized? \ \ \ \ 
$\sqrt{3500}\approx \allowbreak 59.\,\allowbreak 161$ \FRAME{dtbpF}{5.1413in%
}{2.7354in}{0pt}{}{}{car1.bmp}{\special{language "Scientific Word";type
"GRAPHIC";display "USEDEF";valid_file "F";width 5.1413in;height
2.7354in;depth 0pt;original-width 0.134in;original-height 0.0856in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'car1.bmp';file-properties "XNPEU";}}

\item How long is the longest straight rod that can be carried through the
corner shown on the picture below? \ (Assume that $a=10$ and $b=6$) \ \ \ \
\ \ \ $\theta =\tan ^{-1}\left( \sqrt[3]{\dfrac{10}{6}}\right) \approx
\allowbreak 49.\,\allowbreak 855$ \ \ \ $L=\dfrac{10}{\sin \left( \tan
^{-1}\left( \sqrt[3]{\dfrac{10}{6}}\right) \right) }+\dfrac{6}{\cos \left(
\tan ^{-1}\left( \sqrt[3]{\dfrac{10}{6}}\right) \right) }\approx \allowbreak
22.\,\allowbreak 388$\FRAME{dtbpF}{1.7642in}{1.4883in}{0in}{}{}{insert4.bmp}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 1.7642in;height 1.4883in;depth
0in;original-width 1.7262in;original-height 1.4529in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'insert4.bmp';file-properties
"XNPEU";}}

\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 $B$. \ \ \ \ \ \ \ $%
6$

\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 $B$? \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ right at the start, both at
the station

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

Concave up where $\cos x>0$ \ \ \ \ $-\dfrac{\pi }{2}+2k\pi <x<\dfrac{\pi }{2%
}+2k\pi $ \ \ \ where $k\in 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$\FRAME{dtbpFX}{4.5in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific
Word";type "MAPLEPLOT";width 4.5in;height 3in;depth 0pt;display
"USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin
"-5";xmax "5";xviewmin "-5.01";xviewmax "5.01";yviewmin
"-1.00196029564358";yviewmax "1.00196029564358";plottype 4;numpoints
100;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis
\TEXUX{y};var1name \TEXUX{$x$};var2name \TEXUX{$y$};function \TEXUX{$\sin
^{3}x$};linecolor "black";linestyle 1;pointstyle "point";linethickness
1;lineAttributes "Solid";var1range "-5,5";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename
'NG13UG0G.wmf';tempfile-properties "XPR";}}

\item Cut

Consider two posts as shown on the picture. \ The light atop post A moves
vetically up and down the post according to $h\left( t\right) =55+5\sin t$ \
\ \ ($t$ in seconds, height in meters). \ How fast is the length of the
shadow of the two-meter statue changing at $t=20$ seconds?\FRAME{dtbpF}{%
4.7478in}{2.124in}{0pt}{}{}{pic1.jpg}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 4.7478in;height 2.124in;depth 0pt;original-width
4.0629in;original-height 1.8023in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'pic1.JPG';file-properties "XNPEU";}}$\dfrac{h}{%
x_{1}}=\dfrac{40}{x_{1}-20}\ \ \ \ $so$\ \ \ \ x_{1}=\dfrac{20h}{h-40}$

$\dfrac{h}{x_{2}}=\dfrac{42}{x_{2}-20}\allowbreak $ \ \ \ \ \ \ so \ \ $%
x_{2}=\dfrac{20h}{h-42}$

$x_{2}-x_{1}=\dfrac{20h}{h-42}-\dfrac{20h}{h-40}=20h\left( \dfrac{1}{h-42}-%
\dfrac{1}{h-40}\right) $

We also need: \ $h\left( 20\right) =5\sin 20+55=59.\,\allowbreak 565$ \ \
and \ \ $h^{\prime }\left( 20\right) =5\cos 20=\allowbreak 2.\,\allowbreak
040\,4$%
\begin{eqnarray*}
\dfrac{d\left( x_{2}-x_{1}\right) }{dt} &=&20\dfrac{d}{dx}\left[ h\left( 
\dfrac{1}{h-42}-\dfrac{1}{h-40}\right) \right] \\
&=&20\left[ h^{\prime }\left( \dfrac{1}{h-42}-\dfrac{1}{h-40}\right)
+h\left( \ln \left\vert h-42\right\vert h^{\prime }-\ln \left\vert
h-40\right\vert h^{\prime }\right) \right] \\
&=&20h^{\prime }\left[ \dfrac{1}{h-42}-\dfrac{1}{h-40}+h\ln \left( \dfrac{%
h-42}{h-40}\right) \right] \\
&=&20\left( 2.\,\allowbreak 040\,4\right) \left( \dfrac{1}{59.\,\allowbreak
565-42}-\dfrac{1}{59.\,\allowbreak 565-40}+59.\,\allowbreak 565\left( \ln
\left( \dfrac{59.\,\allowbreak 565-42}{59.\,\allowbreak 565-40}\right)
\right) \right)
\end{eqnarray*}

$=20\left( \allowbreak 2.\,\allowbreak 040\,4\right) \left( \dfrac{1}{%
59.\,\allowbreak 565-42}-\dfrac{1}{59.\,\allowbreak 565-40}+59.\,\allowbreak
565\left( \ln \left( \dfrac{59.\,\allowbreak 565-42}{59.\,\allowbreak 565-40}%
\right) \right) \right) =$

$h\left( t\right) =55+5\sin t$

$f\left( a\right) =\dfrac{20a}{a-42}-\dfrac{20a}{a-40}$

$f\left( h\left( 20\right) \right) =\allowbreak =\allowbreak 6.\,\allowbreak
933\,2$

$f^{\prime }\left( h\left( 20\right) \right) =\allowbreak -0.632\,70$

$f\left( t\right) =20\left( 55+5\sin t\right) \left( \dfrac{1}{\left(
55+5\sin t\right) -42}-\dfrac{1}{\left( 55+5\sin t\right) -40}\right) $

$f^{\prime }\left( 20\right) =\allowbreak -1.\,\allowbreak 291\,0$
\end{enumerate}

\end{document}