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

Introduction / Context:
As circumference C = 2πr is linear in r, a 5% increase in circumference means a 5% increase in radius. Area A = πr^2 then increases by the square of the radius factor.

Given Data / Assumptions:

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

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

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

Verification / Alternative check:
Small-change estimate gives ~10%; adding the square term (0.25%) yields the precise 10.25%.

Why Other Options Are Wrong:
10% and 10.5% are approximations; 8.75% and 9.5% are unrelated to the exact square factor.

Common Pitfalls:
Forgetting that area scales with r^2, not r.

Final Answer:
10.25 %