Question

The radius and the height of a cone are in the ratio 2 : 1. The ratio between the volumes of a sphere and this cone (both having equal radii) is:

Options

• 1 : 4

• 4 : 1

• 8 : 1

• 1 : 8

Answer 1

8 : 1

Explanation:

Given,

The radius and the height of a cone are in the ratio 2 : 1.

Let radius of cone be 2x and height of cone be x.

Let radius f sphere be r.

Given,

Radius of cone and sphere are equal.

∴ r = 2x.

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{\frac{4}{3}\, \pi r^3}{\frac{1}{3}\, \pi r^2 h}\]

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{4 \times 3 \times \pi r^3}{3 \times \pi r^2 h}\]

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{4r}{h}\]

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{4 \times 2x}{x}\]

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{8x}{x}\]

\[\Rightarrow \frac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \frac{8}{1} = 8 : 1.\]