If the ratio of the volumes of the two spheres is 125 : 64, find the ratio of their surface areas.

Given,
ratio of the volumes of the two spheres is 125 : 64.

$\therefore \dfrac{\text{Vol. of Sphere 1}}{\text{Vol. of Sphere 2}} = \dfrac{125}{64} \\[1em] \Rightarrow \dfrac{\dfrac{4}{3}π(r$1)^3}{\dfrac{4}{3}π(r2)^3} = \dfrac{125}{64} \\[1em] \Rightarrow \dfrac{(r1)^3}{(r2)^3} = \dfrac{5^3}{4^3} \\[1em] \Rightarrow \dfrac{r1}{r2} = \dfrac{5}{4}.

Surface area of sphere = 4πr².

$\therefore \dfrac{\text{Surface area of Sphere 1}}{\text{Surface area of Sphere 2}} = \dfrac{4π(r$1)^2}{4π(r2)^2} \\[1em] = \Big(\dfrac{r1}{r2}\Big)^2 \\[1em] = \Big(\dfrac{5}{4}\Big)^2 \\[1em] = \dfrac{25}{16}.

Hence, the ratio of the surface areas of two spheres is 25 : 16.