Question

A solid right circular cone is cut into two parts at the middle of its height by a planeparallel 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

Answer 1

Let the height of the cone be H and radius of the cone be R.
Now, the cone is divided into two parts by the parallel plane
∴ OC = CA = $\frac{H}{2}$
Now, In ∆OCD and OAB
∠OCD = OAB (Corresponding angles)
∠ODC = OBA (Corresponding angles)
By AA-similarity criterion ∆OCD ∼ ∆OAB
$$\begin{gathered} \therefore \frac{\mathrm{CD}}{\mathrm{AB}}=\frac{\mathrm{OC}}{\mathrm{OA}} \\ \Rightarrow \frac{\mathrm{CD}}{R}=\frac{H}{2\times H} \\ \Rightarrow \mathrm{CD}=\frac{R}{2} \end{gathered}$$
$$\begin{gathered} \frac{\mathrm{Volume}\mathrm{of}\mathrm{smaller}\mathrm{cone}}{\mathrm{Volume}\mathrm{of}\mathrm{whole}\mathrm{cone}}=\frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(\mathrm{CD}\right)}^2\mathrm{OC}}{\frac{1}{3}\mathrm{\pi }{\left(\mathrm{AB}\right)}^2\mathrm{OA}} \\ =\frac{{\left({\displaystyle \frac{R}{2}}\right)}^2{\displaystyle \frac{H}{2}}}{R^{2}H} \\ =\frac{1}{8}\frac{R^{2}H}{R^{2}H} \\ =\frac{1}{8} \end{gathered}$$
Hence, the correct answer is option (d).