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 :

1. 1 : 4

2. 4 : 1

3. 8 : 1

4. 1 : 8

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.

$\Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{\dfrac{4}{3}πr^3}{\dfrac{1}{3}πr^2h} \\[1em] \Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{4 \times 3 \times πr^3}{3 \times πr^2h} \\[1em] \Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{4r}{h} \\[1em] \Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{4 \times 2x}{x} \\[1em] \Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{8x}{x} \\[1em] \Rightarrow \dfrac{\text{Vol. of sphere}}{\text{Vol. of cone}} = \dfrac{8}{1} = 8 : 1.$

Hence, Option 3 is the correct option.