Answer
The required volume is\[V=\frac{1}{12}\pi {{h}^{3}}\]
Work Step by Step
The radius of the cone is $r=6\text{ feet}$ and the height is $12\text{ feet}$.
Equate the small cone to the bigger cone:
$\frac{r}{h}=\frac{6}{12}$
So,
$\begin{align}
& r=\frac{6}{12}h \\
& r=\frac{1}{2}h \\
\end{align}$
Then the volume of cone is:
$\begin{align}
& V\left( h \right)=\frac{1}{3}\pi {{r}^{2}}h \\
& =\frac{1}{3}\pi {{\left( \frac{1}{2}h \right)}^{2}}h \\
& =\frac{1}{3}\pi \left( \frac{1}{4}{{h}^{2}} \right)h \\
& =\frac{1}{12}\pi {{h}^{3}}
\end{align}$
Therefore, the required volume is $V=\frac{1}{12}\pi {{h}^{3}}$.
