\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}{Definition} \newtheorem{example}{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=0.4in,right=0.4in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture Notes} \chead{\color{black} \LARGE Related Rates} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright \; Hidegkuti, Powell, 2009} \rfoot{\small Last revised: December 6, 2013} \textwidth 7.6in \textheight 9.5in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} \begin{center} {\LARGE Sample Problems\bigskip } \end{center} \begin{enumerate} \item A city is of a circular shape. \ The area of the city is growing at a constant rate of $2\dfrac{\unit{mi}^{2}}{\unit{y}}$ \ (square miles per year). \ How fast is the radius growing when it is exactly $15\unit{mi}$? \item A sphere is growing in such a manner that its radius increases at $0.2% \dfrac{\unit{m}}{\unit{s}}$ (meters per second). \ How fast is its volume increasing when its radius is $4\unit{m}$ long? \item A sphere is growing in such a manner that its volume increases at $0.2% \dfrac{\unit{m}^{3}}{\unit{s}}$ (cubic meters per second). \ How fast is its radius increasing when it is $7\unit{m}$ long? \item The area of a rectangle is kept fixed at $100$ square meters while the legths of the sides vary. \ Express the rate of change of the length of the vertical side in terms of the rate of change in the length of the other side when a) \ the horizontal side is $18$ meters long \ \ \ b) \ the rectangle is a square \item A ladder $20\unit{ft}$ long leans a gainst a vertical building. If the top of the ladder slides down at a rate of $\sqrt{3}\dfrac{\unit{ft}}{\unit{s% }}$, how fast is the bottom of the ladder sliding away from the building when the top of the ladder is $10\unit{ft}$ above the ground? \item A tank, shaped like a cone shown on the picture below, is being filled up with water. \ The top of the tank is a circle with radius $5~\unit{ft}$, its height is $15~\unit{ft}$. \ Water is added to the tank at the rate of $% V^{\prime }\left( t\right) =2\pi \dfrac{\unit{ft}^{3}}{\unit{min}}$. How fast is the water level rising when the water level is $6$ $\unit{ft}$ high? \ (The volume of a cone with height $h$ and base radius $r$ is $V=\dfrac{\pi r^{2}h}{3}$.)\FRAME{dtbpF}{1.8005in}{2.066in}{0pt}{}{}{insert21.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.8005in;height 2.066in;depth 0pt;original-width 1.8663in;original-height 2.1465in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert21.bmp';file-properties "XNPEU";}} \item Two quantities $p$ and $q$ depending on $t$ are subject to the relation $\dfrac{1}{p}+\dfrac{1}{q}=1.$ a) \ Express $p^{\prime }\left( t\right) $ in terms of $q^{\prime }\left( t\right) $. \ \newline b) At a certain moment, $p\left( t_{0}\right) =\dfrac{4}{3}$ and $p^{\prime }\left( t_{0}\right) =2.$ \ Find $q\left( t_{0}\right) $ and $q^{\prime }\left( t_{0}\right) .$ \item The base radius and height of a cylinder are constantly changing but the volume of the cylinder is kept at a constant $600\pi $ $\unit{in}^{3}$. \ a) \ At a time $t_{1}$ the base radius is $r\left( t_{1}\right) =10\unit{in}$ and its rate of change is $r^{\prime }\left( t_{1}\right) =0.2\dfrac{\unit{in% }}{\unit{s}}.$ \ Compute the rate of change of the height of the cylinder $% h\left( t\right) $ at time $t_{1}$. b) \ At a time $t_{2}$ the height is $h\left( t_{2}\right) =12\unit{in}$ and its rate of change is $r^{\prime }\left( t_{2}\right) =-0.5\dfrac{\unit{in}}{% \unit{s}}.$ \ Compute the rate of change of the radius of the cylinder $% r\left( t\right) $ at time $t_{2}$. \item An object, dropped from a height of $h$ has a location of $y\left( t\right) =-16t^{2}+h$ feet after $t$ seconds. \ We dropped a small object from a height of $60$ feet.\FRAME{dtbpF}{2.4275in}{2.527in}{0pt}{}{}{% related1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.4275in;height 2.527in;depth 0pt;original-width 2.5728in;original-height 2.6801in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../Math207/math207_fa13/quizzes/sample_quiz14/related1.bmp';file-properties "XNPEU";}% } a) \ Where is the object and what is its velocity after $1.5$ seconds? b) \ Suppose there is a $30$ feet tall street light $10$ feet away from the point where the object will land. \ How far is the shadow of the object from the base of the street light at $t=1.5$? c) \ How fast is the obejct's shadow moving at $t=1.5$? \ {\LARGE \pagebreak } \end{enumerate} \begin{center} {\LARGE Sample Problems - Answers\bigskip \bigskip } \end{center} 1.) $\ \dfrac{1}{15\pi }\dfrac{\unit{mi}}{\unit{y}}$ \ \ \ \ \ \ \ \ \ 2.) $% \ 12.\,\allowbreak 8\pi \dfrac{\unit{m}^{3}}{\unit{s}}\approx 40.\,\allowbreak 212\,385\,965\,\allowbreak 949\,4\dfrac{\unit{m}^{3}}{\unit{% s}}$ \ \ \ \ \ \ \ \ 3.) \ $\dfrac{1}{980\pi }\dfrac{\unit{m}}{\unit{s}}% \approx \allowbreak 3.\,\allowbreak 248\,06\times 10^{-4}\dfrac{\unit{m}}{% \unit{s}}$\bigskip 4.) \ a) \ $v^{\prime }\left( t\right) =-\dfrac{25}{81}h^{\prime }\left( t\right) $\ \ \ \ \ \ \ \ \ \ \ \ b) \ $v^{\prime }\left( t\right) =-h^{\prime }\left( t\right) $ \ \ \ \ \ \ \ 5.) \ $1\dfrac{\unit{ft}}{\unit{% s}}$ \ \ \ \ \ \ \ 6.) \ $\dfrac{1}{2}\dfrac{\unit{ft}}{\unit{min}}$ \ \ \ \ \ \ \bigskip 7.) \ a) \ $p^{\prime }=-\dfrac{p^{2}}{q^{2}}q^{\prime }$ \ \ \ \ \ \ b) \ $% -18$ \ \ \ \ \ \ \ \ \ 8.) \ a) \ $-0.24\dfrac{\unit{in}}{\unit{s}}$ \ \ \ \ b) \ $\dfrac{5}{48}\sqrt{2}\dfrac{\unit{in}}{\unit{s}}\approx \allowbreak 0.147\,314\dfrac{\unit{in}}{\unit{s}}$ \ \ \ \ \bigskip 9.) \ a) \ \ $y\left( 1.5\right) =24\unit{ft}$ \ and $y^{\prime }\left( 1.5\right) =-48\dfrac{\unit{ft}}{\unit{s}}$ \ \ \ \ b) $50\unit{ft}$\ \ \ \ c) \ $-400\dfrac{\unit{ft}}{\unit{s}}$\ \ \ \ \end{document}