Question

The radius of a circle is so increased that its circumference increases by 5%. The area of the circle then increases by:

• A. 12.5%
• B. 10.25%
• C. 10.5%
• D. 11.25%

Hint:

Circumference is proportional to radius, but area is proportional to the square of radius: square the 1.05 growth factor.

Answer 1

The Correct Option is B

Step 1: Link circumference growth to radius growth.
Circumference \( C = 2\pi r \) is directly proportional to the radius \(r\). So a 5% rise in circumference means the radius also rises by exactly 5%, giving new radius \( r' = 1.05r \).

Step 2: Write the area in terms of the new radius.
Area is \( A = \pi r^2 \), so the new area is \( A' = \pi (1.05r)^2 = \pi (1.1025 r^2) = 1.1025 A \).

Step 3: Convert to percentage change.
The area becomes 1.1025 times the old area, a rise of \( (1.1025 - 1) \times 100 = 10.25\% \).

Step 4: Rule out the other options.
Option A (12.5%) wrongly doubles the 5% rise plus adds extra, ignoring the square relation. Option C (10.5%) and option D (11.25%) come from rounding or squaring mistakes and do not equal \( 1.05^2 \).

Final Answer:
Area rises by exactly 10.25% when radius rises by 5%. \[ \boxed{10.25\%} \]