\begin{frame} \frametitle{Related (Dependent) Rates} \begin{exampleblock}{} Two cars are headed for the same road intersection: \begin{itemize} \item car $A$ is traveling west with $50$mi/h \item car $B$ is traveling north with $60$mi/h \end{itemize} At what rate are the cars approaching \\ when $A$ is $0.3$mi and $B$ is $0.4$mi from the intersection? \medskip \begin{minipage}{.39\textwidth} \begin{center} \begin{tikzpicture}[default] \draw (-0.3,0) -- node[pos=.4,above] {$\mpause[1]{x}$} (3,0); \draw (0,.3) -- node[pos=.4,left] {$\mpause[2]{y}$} (0,-2.3); \node at(0,0) [anchor=south east] {$C$}; \begin{scope}[cred] \draw[dashed] (0,-1.5) -- node[below,anchor=north west] {$\mpause[3]{z}$} node [at start,left] {$B$} node [at end,above] {$A$} (2,0); \end{scope} % \begin{scope} % \mpause[4]{ % \draw[->] (2.3,.5) -- node [above] {$\frac{d}{dt}x = -50$} ++(-.6,0); % } % \mpause[5]{ % \draw[->] (-.5,-1.8) -- node [left] {$\frac{d}{dt}y = -60$} ++(0,.6); % } % \end{scope} \end{tikzpicture} \end{center} \end{minipage} \begin{minipage}{.59\textwidth} \begin{itemize} \pause \item $x(t) =$ distance of $A$ to crossing \pause \item $y(t) =$ distance of $B$ to crossing \pause \item $z(t) =$ distance of $A$ to $B$ \end{itemize} \pause \begin{malign} \frac{d}{dt}x = -50 && \frac{d}{dt}y = -60 \end{malign} \end{minipage} \pause \begin{talign} &z^2 = x^2 + y^2 % \mpause[1]{\;\implies\; \frac{d}{dt}z^2 = \frac{d}{dt} (x^2 + y^2)}\\ \mpause[1]{\;\implies\; 2z\frac{dz}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}}\\ &\mpause[2]{\frac{dz}{dt} = \frac{x}{z}\frac{dx}{dt} + \frac{y}{z}\frac{dy}{dt}} \mpause[5]{\implies\; \frac{dz}{dt} = \frac{0.3}{0.5}(-50) + \frac{0.4}{0.5}(-60)} \mpause[6]{=-78} \end{talign} \pause\pause\pause When $x = 0.3$ \& $y = 0.4$, we get $z = \pause 0.5$. \pause\pause\pause The answer is \alert{$78$mi/h}. \end{exampleblock} \end{frame}