Question

The height of a right circular cone is 20 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be $\frac{1}{8}$ of the volume of the given cone, then at what height above the base is the section made? [HOTS] [CBSE 2014]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{cone},H=20\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{cone}\mathrm{be}R, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone}\mathrm{be}h\mathit{}\mathrm{and} \\ \mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone}\mathrm{be}r. \\ \mathrm{Now},\mathrm{in}∆\mathrm{AQD}\mathrm{and}∆\mathrm{APC}, \\ \angle \mathrm{QAD}=\angle \mathrm{PAC}\left(\mathrm{Common}\mathrm{angle}\right) \\ \angle \mathrm{AQD}=\angle \mathrm{APC}=90° \\ \mathrm{So},\mathrm{by}\mathrm{AA}\mathrm{criteria} \\ ∆\mathrm{AQD}~∆\mathrm{APC} \\ \Rightarrow \frac{\mathrm{AQ}}{\mathrm{AP}}=\frac{\mathrm{QD}}{\mathrm{PC}} \\ \Rightarrow \frac{h}{H}=\frac{r}{R}.....\left(\mathrm{i}\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{smaller}\mathrm{cone}=\frac{1}{8}\times \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{cone} \\ \Rightarrow \frac{\mathrm{Volume}\mathrm{of}\mathrm{smaller}\mathrm{cone}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{cone}}=\frac{1}{8} \\ \Rightarrow \frac{\left({\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{2}h\right)}{\left({\displaystyle \frac{1}{3}}\mathrm{\pi }{R}^{2}H\right)}=\frac{1}{8} \\ \Rightarrow {\left(\frac{r}{R}\right)}^2\times \left(\frac{h}{H}\right)=\frac{1}{8} \\ \Rightarrow {\left(\frac{h}{H}\right)}^2\times \left(\frac{h}{H}\right)=\frac{1}{8}\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow {\left(\frac{h}{H}\right)}^3=\frac{1}{8} \\ \Rightarrow \frac{h}{H}=\sqrt[3]{\frac{1}{8}} \\ \Rightarrow \frac{h}{20}=\frac{1}{2} \\ \Rightarrow h=\frac{20}{2} \\ \Rightarrow h=10\mathrm{cm} \\ \therefore \mathrm{PQ}=H-h=20-10=10\mathrm{cm} \end{gathered}$$

So, the section is made at the height of 10 cm above the base.