If the radius of a sphere is increased by 3%, by what percentage does its total surface area increase?

Introduction / Context:For similar shapes, areas scale with the square of the linear scale factor. A small percentage change in radius produces approximately double that percentage for area, but the exact value uses the square of the scale factor.

Given Data / Assumptions:

• Initial radius r.
• New radius r′ = 1.03 r (3% increase).
• Surface area of a sphere: S = 4πr^2.

Concept / Approach:Compute ratio S′/S = (4πr′^2)/(4πr^2) = (r′/r)^2 = (1.03)^2.

Step-by-Step Solution:

Verification / Alternative check:Binomial: (1 + 0.03)^2 = 1 + 2(0.03) + (0.03)^2 = 1 + 0.06 + 0.0009 = 1.0609 → 6.09%.

Why Other Options Are Wrong:

• 7%, 9%: Do not match exact square scaling.
• 5.06%: This would be for (1.025)^2 − 1, not 3%.

Common Pitfalls:Doubling the percent to 6% and ignoring the small square term 0.09%.

Final Answer:6.09%