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, at what height above the base, the section has been made?

Answer 1

Suppose that the smaller cone has radius r cm, height h cm and the original cone has radius R cm.
Height of the original cone = 30 cm
Considering $△$ABC & $△$ADE, we have:
$$\begin{gathered} \angle BAC=\angle DAE\left(\mathrm{Common}\right) \\ \angle ABC=\angle ADE(90°\mathrm{each}) \\ \mathrm{i}.\mathrm{e}.,△ABC~△ADE(\mathrm{AA}\mathrm{similarity}\mathrm{condition}) \end{gathered}$$
Therefore, we can say that
$$\begin{gathered} \frac{r}{R}=\frac{h}{30} \\ r=\frac{hR}{30} \end{gathered}$$
Also, volume of a small cone = $\frac{1}{27}$ volume of the original cone
$$\begin{gathered} \Rightarrow \frac{1}{3}\mathrm{\pi }{r}^{\mathit{2}}h\mathit{}=\frac{1}{27}\times \frac{1}{3}\mathrm{\pi }{R}^2\times 30 \\ \Rightarrow r^{\mathit{2}}h\mathit{}=\frac{1}{27}\times 30R^2 \\ \Rightarrow {\left(\frac{hR}{30}\right)}^{2}h=\frac{10}{9}{R}^2 \\ \Rightarrow h^{\mathit{3}}=\frac{10\times 900}{9}=1000 \\ \Rightarrow h=10\mathrm{cm} \end{gathered}$$
i.e., height of the small cone = 10 cm.

∴ The section has been made at a height of (30 − 10) cm = 20 cm above the base of original cone.