Circle — The radius is increased so that the circumference increases by 5%. By what percentage does the area increase?

Introduction / Context:
Circumference C = 2πr scales linearly with r. If the circumference increases by 5%, then r also increases by 5%. Area then scales with the square of the linear factor.

Given Data / Assumptions:

• C increases by 5% ⇒ r increases by 5%
• Area ∝ r^2

Concept / Approach:
If r → 1.05r, then area A → (1.05)^2 * A. The increase is (1.05^2 − 1) * 100%.

Step-by-Step Solution:
1.05^2 = 1.1025Percentage increase = (1.1025 − 1) * 100% = 10.25%

Verification / Alternative check:
Small-change rule: 2 * 5% = 10% plus the square term 0.25% gives the precise 10.25%.

Why Other Options Are Wrong:
12.5% corresponds to 1.118… factor; 10.5% and 11.25% are off; 9.75% is an underestimate.

Common Pitfalls:
Assuming a simple 10% (doubling 5%) without adding the 0.25% correction from the square.

Final Answer:
10.25%