If the ratio of the radii of two spheres is 3 : 7, find :

(i) the ratio of their volumes.

(ii) the ratio of their surface areas.

Let the radii of two spheres be 3a and 7a.

(i) Volume of sphere = $\dfrac{4}{3}πr^3$.

$\dfrac{\text{Vol. of Sphere 1}}{\text{Vol. of Sphere 2}} = \dfrac{\dfrac{4}{3}π(3a)^3}{\dfrac{4}{3}π(7a)^3} \\[1em] = \dfrac{\dfrac{4}{3}π \times 27a^3}{\dfrac{4}{3}π \times 343a^3} \\[1em] = \dfrac{27}{343}.$

Hence, the ratio of the volumes of two spheres is 27 : 343.

(ii) Surface area of sphere = 4πr².

$\dfrac{\text{Surface area of Sphere 1}}{\text{Surface area of Sphere 2}} = \dfrac{4π(3a)^2}{4π(7a)^2} \\[1em] = \dfrac{4π \times 9a^2}{4π \times 49a^2} \\[1em] = \dfrac{9}{49}.$

Hence, the ratio of the surface areas of two spheres is 9 : 49.