The height of a cone is …

According to the question, The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base.
Let the radii of smaller cone and original cone be r₁ and r₂ respectively and the height of smaller cone be h.
{tex}\triangle A B C \sim \triangle A P Q{/tex} 
{tex}\Rightarrow \quad \frac { h } { 30 } \sim \frac { r _ { 1 } } { r _ { 2 } }{/tex} ...(1)
Volume smaller cone {tex}= \frac { 1 } { 27 } \times{/tex} Volume of original cone
{tex}\Rightarrow \quad \frac { 1 } { 3 } \pi r _ { 1 } ^ { 2 } \times h = \frac { 1 } { 27 } \times \frac { 1 } { 3 } \pi r _ { 2 } ^ { 2 } \times 30{/tex} 
{tex}\Rightarrow \quad \left( \frac { r _ { 1 } } { r _ { 2 } } \right) ^ { 2 } \times \frac { h } { 30 } = \frac { 1 } { 27 }{/tex} 
{tex}\Rightarrow \quad \left( \frac { h } { 30 } \right) ^ { 2 } \times \frac { h } { 30 } = \frac { 1 } { 27 }{/tex} 
{tex}\left( \mathrm { Using } \frac { h } { 30 } = \frac { r _ { 1 } } { r _ { 2 } } \text { From } ( \mathrm { i } ) \right){/tex} 
{tex}\Rightarrow \quad \left( \frac { h } { 30 } \right) ^ { 3 } = \frac { 1 } { 27 }{/tex} 
{tex}\Rightarrow \quad h ^ { 3 } = \frac { 30 \times 30 \times 30 } { 27 }{/tex} 
h = 10 cm
Hence, required height = (30 - 10) = 20 cm