Question

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be $\frac{1}{27}$ of the volume of the given cone, then the height above the base at which the section has been made, is

(a) 10 cm

(b) 15 cm

(c) 20 cm

(d) 25 cm

Answer 1

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 $∆VOD~∆VO\text{'}B$

Therefore,

$$\begin{gathered} \frac{OV}{O\text{'}V}=\frac{OD}{O\text{'}B} \\ \Rightarrow \frac{h_1}{30}=\frac{r_2}{r_1} \end{gathered}$$

But,

$$\begin{gathered} VolumeofconeVCD=\frac{1}{27}VolumeofconeVAB \\ \Rightarrow \frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_2\right)}^2{\mathrm{h}}_1=\frac{1}{27}\left(\frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^{2}30\right) \\ \Rightarrow {\left(\frac{{\mathrm{r}}_2}{{\mathrm{r}}_1}\right)}^2{\mathrm{h}}_1=\frac{10}{9} \\ \Rightarrow {\left(\frac{{\mathrm{h}}_1}{30}\right)}^2{\mathrm{h}}_1=\frac{10}{9} \\ \Rightarrow {\mathrm{h}}_1=10 \end{gathered}$$

Hence,

Required height

Hence, the correct answer is choice (c).