The height of the cone is 30 𝑐𝑚. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is of the cone then at what height it is $\frac{1}{27}$ cut from the base.

We need to find the height at which it is cut provided that The height of the cone is 30 𝑐𝑚. 

Let the radii of the smaller and original cone be $r_1$ and $r_2$respectively and the height of the cone be ℎ.

It can be said that ∆𝐴𝐵𝐶~∆𝐴𝑃𝑄.

 It is known that the ratio of the corresponding sides of similar triangles is equal. Therefore,

$\Rightarrow \frac{h}{30}=\frac{r_1}{r_2}$

$\text{ It is given that the volume of smaller cone is }\frac{1}{27}\text{ of the cone. Therefore, }$$\begin{array}{l}\Rightarrow \text{ Volume of the smaller cone }=\frac{1}{27}\times \text{ Volume of original cone }\\ \Rightarrow \frac{1}{3}\pi r_1^2\times h=\frac{1}{27}\times \frac{1}{3}\pi r_2^2\times 30\\ \Rightarrow r_1^2\times h=\frac{1}{27}\times r_2^2\times 30\\ \Rightarrow {\left(\frac{r_1}{r_2}\right)}^2\times \frac{h}{30}=\frac{1}{27}\\ \Rightarrow {\left(\frac{h}{30}\right)}^2\times \frac{h}{30}=\frac{1}{27}\\ \Rightarrow {\left(\frac{h}{30}\right)}^3=\frac{1}{27}\\ \Rightarrow \frac{h}{30}=\frac{1}{3}\\ \Rightarrow h=10\mathrm{cm}\end{array}$

Hence, the required height is 30 − 10 = 20 cm