The height of a cone is 30 cm.

Solution:

Let VAB be cone of height 30 cm and base radius r1 cm.

Suppose it is cut off by a plane parallel to the base at a height h2 from the base of the cone.

Clearly $\Delta V O D \sim \Delta V O^{\prime} B$

Therefore,

$\frac{O V}{O^{\prime} V}=\frac{O D}{O^{\prime} B}$

$\Rightarrow \frac{h_{1}}{30}=\frac{r_{2}}{r_{1}}$

But,

Volume of cone $V C D=\frac{1}{27}$ Volume of cone $V A B$

$\Rightarrow \frac{1}{3} \pi\left(\mathrm{r}_{2}\right)^{2} \mathrm{~h}_{1}=\frac{1}{27}\left(\frac{1}{3} \pi\left(\mathrm{r}_{1}\right)^{2} 30\right)$

$\Rightarrow\left(\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}\right)^{2} \mathrm{~h}_{1}=\frac{10}{9}$

$\Rightarrow\left(\frac{\mathrm{h}_{1}}{30}\right)^{2} \mathrm{~h}_{1}=\frac{10}{9}$

$\Rightarrow \mathrm{h}_{1}=10$Hence,

Required height 

$=30-10$

$=20 \mathrm{~cm}$

Hence, the correct answer is choice (c).