Question

Choose the correct answer of the following question:

A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is

(a) 1 : 2 (b) 1 : 4 (c) 1 : 6 (d) 1 : 8 [CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radii}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{and}\mathrm{given}\mathrm{cones}\mathrm{be}r\mathrm{and}R,\mathrm{respectively};\mathrm{and}\mathrm{their}\mathrm{heights}\mathrm{be}h\mathrm{and}H,\mathrm{respectively}. \\ \mathrm{We}\mathrm{have}, \\ H=2h.....\left(\mathrm{i}\right) \\ \mathrm{In}∆\mathrm{AQD}\mathrm{and}∆\mathrm{APC}, \\ \angle \mathrm{QAD}=\angle \mathrm{PAC}\left(\mathrm{Common}\mathrm{angle}\right) \\ \angle \mathrm{AQD}=\angle \mathrm{APC}=90° \\ \mathrm{So},\mathrm{by}\mathrm{AA}\mathrm{criteria} \\ ∆\mathrm{AQD}~∆\mathrm{APC} \\ \Rightarrow \frac{\mathrm{AQ}}{\mathrm{AP}}=\frac{\mathrm{QD}}{\mathrm{PC}} \\ \Rightarrow \frac{h}{H}=\frac{r}{R} \\ \Rightarrow \frac{h}{2h}=\frac{r}{R}\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow \frac{1}{2}=\frac{r}{R} \\ \Rightarrow R=2r.....\left(\mathrm{ii}\right) \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone}\mathrm{to}\mathrm{the}\mathrm{whole}\mathrm{cone}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{whole}\mathrm{cone}} \\ =\frac{\left({\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{2}h\right)}{\left({\displaystyle \frac{1}{3}}\mathrm{\pi }{R}^{2}H\right)} \\ ={\left(\frac{r}{R}\right)}^2\times \left(\frac{h}{H}\right) \\ ={\left(\frac{r}{2r}\right)}^2\times \left(\frac{h}{2h}\right)\left[\mathrm{Using}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\right] \\ ={\left(\frac{1}{2}\right)}^2\times \left(\frac{1}{2}\right) \\ =\frac{1}{8} \\ =1:8 \end{gathered}$$

Hence, the correct answer is option (d).