The radius of a circle is increased by 1%. What is the resulting percentage increase in its area?

Introduction / Context:
Area of a circle is proportional to r^2. A small percentage change in radius causes approximately double that percentage change in area, with a slight adjustment due to compounding. This problem asks for the exact percentage using scale factors.

Given Data / Assumptions:
Radius increase = 1% ⇒ radius factor = 1.01.

Concept / Approach:
Area factor = (radius factor)^2. Thus new area = old area * (1.01)^2. The percentage increase is [(1.01)^2 − 1] * 100%. Multiplying out gives 1.0201, which equates to a 2.01% increase, slightly more than a simple doubling because of compounding.

Step-by-Step Solution:

Verification / Alternative check:
For small p%, area increase ≈ 2p% for a circle. Here 2 * 1% = 2% is the approximation; exact value 2.01% confirms the small compounding adjustment.

Why Other Options Are Wrong:
1% and 1.1% treat area as linear. 2% ignores compounding. 2.1% overshoots the exact squared factor.

Common Pitfalls:
Forgetting that area scales with r^2, or ignoring the compounding effect of the square when percentages are not infinitesimally small.

Final Answer:
2.01%