Effect of increasing circumference by 50%: If a circle’s circumference increases by 50%, by what percentage does its area increase?

Introduction / Context:
Area depends on radius squared, while circumference depends linearly on radius. A percentage change in circumference maps to the same percentage change in radius, and area responds quadratically.

Given Data / Assumptions:

• C = 2πr; A = πr^2
• New circumference C′ = 1.5 * C → r′ = 1.5 * r

Concept / Approach:
Since r scales by 1.5, area scales by (1.5)^2 = 2.25. The fractional increase is 2.25 − 1 = 1.25 → 125% increase.

Step-by-Step Solution:
Scale factor for r: k = 1.5Area scale factor: k^2 = 2.25Percentage increase: (2.25 − 1) * 100% = 125%

Verification / Alternative check:
Take r = 2 → C ≈ 12.566; increase 50% → r′ = 3; A from ~12.566 to ~28.274, a rise of ~125%.

Why Other Options Are Wrong:
50% confuses linear with area; 100% is doubling, not correct here; 225% is the final value vs. original, not the increase; 75% is arbitrary.

Common Pitfalls:
Not squaring the scale factor; mixing “increase to” vs. “increase by.”

Final Answer:
125%