Two spheres — ratio of diameters 3 : 5 — find ratio of surface areas: If the diameters of two spheres are in the ratio 3 : 5, what is the ratio of their surface areas?

Introduction / Context:
Surface area of a sphere depends on the square of its radius (or diameter). Thus, a ratio of diameters translates to the square of that ratio for surface areas.

Given Data / Assumptions:

• d1 : d2 = 3 : 5 ⇒ r1 : r2 = 3 : 5.
• S ∝ r^2 (or ∝ d^2).

Concept / Approach:
Square the linear ratio to obtain the surface area ratio.

Step-by-Step Solution:
S1 : S2 = (3^2) : (5^2) = 9 : 25

Verification / Alternative check:
Pick r1 = 3, r2 = 5; S1 = 36π, S2 = 100π ⇒ 36π : 100π = 9 : 25.

Why Other Options Are Wrong:
9 : 10 and 3 : 5 are linear ratios; 27 : 125 applies to volumes (cube).

Common Pitfalls:
Confusing surface-area scaling (square) with volume scaling (cube).

Final Answer:
9 : 25