\begin{frame}
  \frametitle{Related (Dependent) Rates}

  \vspace{-.5ex}
  \begin{exampleblock}{}
    A water tank has the shape of an inverted circular cone:
    \begin{itemize}
      \item base radius $2m$ and the height is $4m$,
      \item water is pumped into the tank at a rate of $2\text{m}^3/\text{min}$.  
    \end{itemize}
    At what rate is the water rising when the water is $3$m deep?
    
    \begin{minipage}{.39\textwidth}
      \begin{tikzpicture}[default,scale=1]
        \draw [draw=none,fill=cblue!20] (0,0) -- (1,2) -- (-1,2) -- cycle;
        \draw [draw=cblue!30,fill=cblue!15] (0,2) ellipse (1 and .27);
        \begin{scope}[line width=.5pt]
        \draw (1.5,3) -- (0,0) -- (-1.5,3);
        \draw (0,3) ellipse (1.5 and .4);
        \draw[<->] (1.3,0) -- node[right] {$h$} (1.3,2);
        \draw[<->] (1.8,0) -- node[right] {$4$} (1.8,3);
        \draw (1.2,0) -- (1.4,0);
        \draw (1.2,2) -- (1.4,2);
        \draw (1.7,0) -- (1.9,0);
        \draw (1.7,3) -- (1.9,3);
        \draw[dashed] (0,0) -- (0,3) -- node[above] {$2$} (1.5,3);
        \draw[dashed] (0,2) -- node[above] {$r$} (1,2);
        \end{scope}
      \end{tikzpicture}
    \end{minipage}
    \begin{minipage}{.59\textwidth}
      \pause
      \begin{talign}
        V = \frac{1}{3}\pi r^2 h
      \end{talign}
      \pause
      How is $r$ related to $h$?
      \pause
      \begin{talign}
        \frac{r}{h} = \frac{2}{4}
        \mpause[1]{\quad\implies\quad r = \frac{1}{2}h} 
      \end{talign}\vspace{-2ex}
      \pause\pause
      \begin{talign}
        V = \frac{1}{3}\pi (\frac{1}{2}h)^2 h \mpause[1]{= \frac{1}{12}\pi h^3} 
      \end{talign}
    \end{minipage}
    \pause\pause\medskip
    
    We differentiate both sides with respect to $t$:
    \begin{talign}
      \frac{dV}{dt} = \frac{d}{dt} (\frac{1}{12}\pi h^3) 
      \mpause[1]{= \frac{1}{12}\pi 3h^2 \frac{dh}{dt} } 
      \mpause[2]{\;\implies\; \frac{dh}{dt} = \frac{4}{\pi h^2}\frac{dV}{dt} } 
      \mpause[3]{\stackrel{h=3}{=} \frac{4}{\pi 9} \cdot 2} 
    \end{talign}
    \pause\pause\pause\pause
    Thus the water rises with $8/(\pi 9)$m/min when its is $3$m deep.
  \end{exampleblock}  
\end{frame}