The radius of a bigger sphere is 16 cm more than the radius of a smaller…

Concept: The surface area of a sphere of radius r is S = 4πr2. When two spheres' radii differ, the difference of their surface areas factors using the difference-of-squares identity: R2 − r2 = (R − r)(R + r).

Applying to this problem:

1. Let the smaller sphere's radius be r, so the bigger sphere's radius is R = r + 16 (given “16 cm more”), i.e. R − r = 16.

2. The surface-area difference is 4π(R2 − r2) = 1408π, so R2 − r2 = 352.

3. Factor using the difference-of-squares identity: (R − r)(R + r) = 352.

4. Substitute R − r = 16: 16(R + r) = 352, so R + r = 22.

5. Now solve the pair R − r = 16 and R + r = 22 together — adding them eliminates r and gives 2R = 38, so R = 19 cm (and r = 3 cm).

6. The diameter of the bigger sphere is 2R = 38 cm.

Cross-check: With R = 19 cm and r = 3 cm, 4π(R2 − r2) = 4π(361 − 9) = 4π(352) = 1408π cm², which matches the given difference — confirming the diameter is 38 cm.