Find the radius and the height

Formula for the volume of a cone is $V=\dfrac{1}{3} \pi r^2 h$ .

$V_{cone}=\dfrac{1}{3} \pi (2^2)(6)=8\pi$ $;$ $V_{water}=\dfrac{1}{2}(8\pi)=4 \pi$

By ratio and proportion in the diagram, we have: $\dfrac{r}{h}=\dfrac{2}{6}$ $\implies$ $r=\dfrac{1}{3}h$

Now we substitute,

$4\pi =\dfrac{1}{3} \pi \left(\dfrac{1}{3}h\right)^2(h)$

$h^3=108$

$h=\sqrt[\leftroot{-1}\uproot{2}\scriptstyle 3]{108}=3\sqrt[\leftroot{-1}\uproot{2}\scriptstyle 3]{4}$

It follows that, $r=\dfrac{1}{3} \times 3\sqrt[\leftroot{-1}\uproot{2}\scriptstyle 3]{4}=\sqrt[\leftroot{-1}\uproot{2}\scriptstyle 3]{4}$ . So $h=3r$ .

$h:r=6:2$

Therefore, $h=3r$ .

Also, the smaller cone is half the volume of the larger one, $2^{3}=2r^{3}$

Therefore, $r=\sqrt[3]{4}$