If the surface areas of two spheres are in the ratio 16 : 49, what is the…

Concept

For similar three-dimensional solids, surface area varies as the square of a corresponding linear dimension, while volume varies as its cube.

Therefore, if the surface-area ratio is a2 : b2, the corresponding radius ratio is a : b and the volume ratio is a3 : b3.

Application

1. Let the radii of the two spheres be r1 and r2. Since the surface area of a sphere is 4πr2, we have r12 : r22 = 16 : 49.

2. Take the positive square root of both terms because radii are positive: r1 : r2 = √16 : √49 = 4 : 7.

3. The volume of a sphere is (4/3)πr3, so the common factor cancels and V1 : V2 = r13 : r23 = 43 : 73 = 64 : 343.

Cross-check

Squaring 4 : 7 gives 16 : 49, which reproduces the given surface-area ratio; cubing the same radius ratio gives 64 : 343.

Result

The required volume ratio is 64 : 343.