If the radius of a circle is increased by 14%, by approximately what percentage does the area of the circle increase?

Introduction / Context:
This question tests understanding of how percentage changes in the radius of a circle affect its area. Because area depends on the square of the radius, the percentage change in area is not the same as the percentage change in radius. Recognising this square relationship is important in many quantitative reasoning problems.

Given Data / Assumptions:

• Original radius = R.
• Radius is increased by 14%.
• New radius = 1.14 * R.
• We must find the percentage increase in area.
• Area is proportional to the square of the radius.

Concept / Approach:
The area A of a circle is A = pi * r^2. If the radius changes from R to k * R, the new area becomes pi * (k * R)^2 = pi * k^2 * R^2, which is k^2 times the original area. Here k = 1.14. We compute k^2, subtract 1, and convert to a percentage to find the percentage increase in area.

Step-by-Step Solution:
Step 1: Original area A1 = pi * R^2. Step 2: New radius r2 = 1.14 * R. Step 3: New area A2 = pi * (1.14 * R)^2 = pi * 1.14^2 * R^2. Compute 1.14^2: 1.14 * 1.14 = 1.2996. So A2 = 1.2996 * pi * R^2. Step 4: Ratio of new area to old area = A2 / A1 = 1.2996. This means the new area is 129.96% of the old area. Step 5: Percentage increase in area = (1.2996 - 1) * 100% = 0.2996 * 100% = 29.96%.

Verification / Alternative check:
We can use an example. Let R = 10 units, so original area A1 = pi * 100. New radius is 1.14 * 10 = 11.4 units, so new area A2 = pi * (11.4)^2 = pi * 129.96. The factor by which area has increased is 129.96 / 100 = 1.2996, meaning an increase of 29.96%. This numerical check matches the algebraic result.

Why Other Options Are Wrong:
Option A (28%): This is close but underestimates the true increase and may come from approximating 1.14^2 poorly. Option C (14%): This assumes a linear relationship between radius and area and ignores the square dependence. Option D (14.98%): This is roughly half the correct increase and has no proper basis. Option E (19.6%): This may arise from incorrectly multiplying 14% by 1.4 or some similar shortcut.

Common Pitfalls:
A typical error is to think that if the radius increases by 14%, the area also increases by 14%, which is incorrect because of the square relationship. Another mistake is to approximate 1.14^2 as 1.28 or 1.26 instead of 1.2996. To avoid mistakes, write out the squared factor explicitly and do the multiplication carefully or use fraction based approximations when possible.

Final Answer:
The area of the circle increases by approximately 29.96%.