Related Rates Math Example 4

Solution

1. 1
   By similar triangles, radius at height $h$: $\frac{r}{h} = \frac{3}{6} \Rightarrow r = \frac{h}{2}$.
2. 2
   Volume: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi\frac{h^2}{4}h = \frac{\pi h^3}{12}$.
3. 3
   $\frac{dV}{dt} = \frac{\pi h^2}{4}\frac{dh}{dt}$.
4. 4
   At $h=4$: $2 = \frac{\pi(16)}{4}\frac{dh}{dt} = 4\pi\frac{dh}{dt} \Rightarrow \frac{dh}{dt} = \frac{1}{2\pi}$ m/min.