\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=0.3in} \pagestyle{fancy} \lhead{\color{blue} \Large Math 207} \chead{\color{black} \LARGE Related Rates} \rhead{\large page \ \thepage} \cfoot{} \lfoot{\small \copyright \; Hidegkuti, 2015} \rfoot{\small Last revised: November 2, 2015} \textwidth 7.6in \textheight 9.65in \setlength{\headheight}{28pt} \setlength{\parindent}{0in} \begin{document} \begin{enumerate} \item a) \ 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 $15\unit{mi}$? b) \ A city is of a circular shape. \ The radius of the city is growing at a constant rate of $0.3\dfrac{\unit{mi}}{\unit{y}}$ \ (miles per year). \ How fast is the area growing at the time when the radius of the circle is exactly $8\unit{mi}$? \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 A cube is decreasing in size so that its surface is changing at a constant rate of $-0.5\dfrac{\unit{m}^{2}}{\unit{min}}$. \ How fast is the volume of the cube changing when it is $27\unit{m}^{3}$?% %TCIMACRO{\TeXButton{2col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \item A tank, shaped like a cone shown on the picture, 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}{1.6129in}{1.8507in}{0pt}{}{}{insert21.bmp}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.6129in;height 1.8507in;depth 0pt;original-width 1.8663in;original-height 2.1465in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert21.bmp';file-properties "XNPEU";}}% %TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion %TCIMACRO{\TeXButton{2col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \item A rotating light is located $20$ feet from a wall. The light completes one rotation every $5$ seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is $15$ degrees from perpendicular to the wall.\FRAME{dtbpF}{2.0833in}{1.8118in}{0pt% }{}{}{related_rates1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.0833in;height 1.8118in;depth 0pt;original-width 2.7804in;original-height 2.4128in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '//psf/Home/Desktop/aaaaahere/related_rates1.bmp';file-properties "XNPEU";}} %TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item The altitude of a triangle is increasing at a rate of $2.2$ centimeters/minute while the area of the triangle is increasing at a rate of $1.5$ square centimeters/minute. At what rate is the base of the triangle changing when the altitude is $11$ centimeters and the area is $87$ square centimeters? \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 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 $t_{2}$. %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion \item At a distance of $12\,000$ meters from the launch site, a spectator is observing a rocket being launched vertically. \ What is the speed of the rocket at the instant when the distance of the rocket from the spectator is $% 13\,000$ meters and is increasing at the rate of $480$ meters per second?% \FRAME{dtbpF}{3.0822in}{1.4909in}{0pt}{}{}{relatedrates3.bmp}{\special% {language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.0822in;height 1.4909in;depth 0pt;original-width 0.1375in;original-height 0.0761in;cropleft "0";croptop "0.9636";cropright "1";cropbottom "0.0364";filename '../../../../Math207/math207_fa14/GH/exams/exam3/sample1/relatedrates3.bmp';file-properties "XNPEU";}% }% %TCIMACRO{\TeXButton{multicol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item A ladder $20\unit{ft}$ long leans against 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 At noon, ship A is $50$ miles due west of ship B. Ship A is sailing west at $24$ miles per hour and ship B is sailing north at $18$ miles per hour. How fast is the distance between the ships changing at $4$ PM? {\LARGE \pagebreak } \end{enumerate} \begin{center} {\LARGE Answers\bigskip \bigskip } \end{center} 1.) $\ $a) \ $\dfrac{1}{15\pi }\dfrac{\unit{mi}}{\unit{y}}$ \ \ \ \ \ \ b) \ $4.\,\allowbreak 8\pi \dfrac{\unit{mi}^{2}}{\unit{y}}$ \ \ \ \ \ \ \ 2.) $\ 12.\,\allowbreak 8\pi \dfrac{\unit{m}^{3}}{\unit{s}}\approx 40.\,\allowbreak 212\,386\dfrac{\unit{m}^{3}}{\unit{s}}$\bigskip\ \ \ \ \ \ \ 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.) \ $-0.375\dfrac{\unit{m}^{3}}{\unit{min}}$\ \ \ \ \ \ 5.) \ $\dfrac{1}{2}% \dfrac{\unit{ft}}{\unit{min}}$ \ \ \ \ \ \ 6.) \ $8\pi \sec ^{2}\left( 15^{\circ }\right) \dfrac{\unit{ft}}{\unit{s}}\approx 26.\,\allowbreak 93719% \dfrac{\unit{ft}}{\unit{s}}$ \bigskip\ \ \ \ \ \ \ \ 7.) \ $-2.89091\dfrac{% \unit{cm}}{\unit{min}}$ 8.) \ 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) $ \ \ \bigskip\ \ \ \ \ \ 9.) \ a) \ $p^{\prime }=-\dfrac{p^{2}}{% q^{2}}q^{\prime }$ \ \ \ \ \ \ b) \ $-18$ \ \ \ \ \ \ 10.) \ a) \ $-0.24\dfrac{\unit{in}}{\unit{s}}$ \ \ \ \ b) \ $\dfrac{5}{48}% \sqrt{2}\dfrac{\unit{in}}{\unit{s}}\approx \allowbreak 0.147\,314\dfrac{% \unit{in}}{\unit{s}}$\ \ \ \ \ \ \ 11.) \ $1248$ meters per second\ \ \ \ \ \ \ \ 12.) \ $1\dfrac{\unit{ft}}{\unit{s}}$ \bigskip 13.) \ $\allowbreak 29.\,\allowbreak 4861656$ miles per hour{\LARGE \bigskip \bigskip \bigskip \bigskip } \bigskip \bigskip \bigskip \bigskip \bigskip \bigskip \end{document}