Question

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

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

Answer 1

The Correct Option is B

To solve the problem, we first need to understand the relationship between the radius, circumference, and area of a circle.

The circumference \( C \) of a circle is given by the formula:

\(C = 2\pi r\)

where \( r \) is the radius of the circle. If the circumference increases by 5%, the new circumference is:

\(C_{new} = 1.05 \times C = 1.05 \times 2\pi r\)

Setting the equations equal gives:

\(2\pi r_{new} = 1.05 \times 2\pi r\)

Solving for \( r_{new} \), we have:

\(r_{new} = 1.05 \times r\)

The area \( A \) of a circle is given by:

\(A = \pi r^2\)

The new radius \( r_{new} \) leads to a new area:

\(A_{new} = \pi (r_{new})^2 = \pi (1.05 \times r)^2\)

Simplifying the expression within the square, we have:

\(A_{new} = \pi \times 1.1025 \times r^2\)

This shows that the new area is 1.1025 times the old area:

\(A_{new} = 1.1025 \times A\)

The percentage increase in area is then \(1.1025 - 1\) times 100%:

Increase = \(0.1025 \times 100\%\)

= 10.25%

Hence, the area of the circle increases by 10.25%.