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 fo 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 and the radius of whole cone be H and R respectively.

The cone is divided into two parts by drawing a plane through the mid point of its height and parallel to the base.

$OC=CA=\frac{H}{2}cm$

Let the radius of the smaller cone be r cm.

In $△$OCD and $△$OAB,

$\angle OCD=\angle OAB\left({90}^o\right)$

$\angle COD=\angle AOB$ (Common)

∴$△OCD\sim △OAB$ (AA Similarity criterion)

$\Rightarrow \frac{OA}{OC}=\frac{AB}{CD}\Rightarrow \frac{H}{\frac{H}{2}}=\frac{R}{r}\Rightarrow R=2r$

Volume of smaller cone $=\frac{1}{3}\pi (CD{)}^2\times OC=\frac{1}{3}\pi (r{)}^2\times \frac{H}{2}=\frac{\pi r^{2}H}{6}$

Volume of whole cone $=\frac{1}{3}\pi (R{)}^2\times H=\frac{\pi (2r{)}^{2}H}{6}=\frac{4}{3}\pi r^{2}H$

$\frac{\text{Volume of smaller cone}}{\text{Volume of whole cone}}=\frac{\frac{\pi r^{2}H}{6}}{\frac{4}{3}\pi r^{2}H}=\frac{1}{8}=1:8$

Thus, the ratio of smaller cone to the whole cone is \( 1:8\)