If the radius of a circle is reduced by 40%, its circumference is reduced by-

Question: If the radius of a circle is reduced by 40%, its circumference is reduced by-

• 30%

• 35%

• 40%

• 60%

The correct answer is 40%.

Reasoning:

To understand why the circumference is reduced by 40% when the radius is reduced by 40%, we should start by recalling the formula for the circumference of a circle:

Formula:

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

$\( C = 2\pi r \)$

where \( r \) is the radius of the circle and \( \pi \) (pi) is a constant approximately equal to 3.14159.

Step-by-Step Explanation:

1. Let's denote the original radius as \( r \).

2. When the radius is reduced by 40%, the new radius $\( r' \)$ is given by:

3. $\( r' = r - 0.40r = 0.60r \)$

4. The original circumference \( C \) is:

   $\( C = 2\pi r \)$

5. The new circumference \( C' \) with the reduced radius is:

   $\( C' = 2\pi r' = 2\pi (0.60r) = 1.20 \pi r \)$

6. To find the reduction in circumference, we need to calculate the difference between the original and the new circumference, then express it as a percentage of the original circumference:

7. $\( \text{Reduction in Circumference} = C - C' = 2\pi r - 1.20 \pi r = 0.80 \pi r \)$

   Reduction percentage = $\( \frac{0.80\pi r}{2\pi r} \times 100 = \frac{0.80}{2} \times 100 = 0.40 \times 100 = 40% \)$

Therefore, the circumference is reduced by 40% when the radius is reduced by 40%.

Conclusion:

This explanation matches our given choices, confirming that the correct answer is indeed 40%.

For further reading on circle properties and circumference calculation, you could refer to resources like Khan Academy (www.khanacademy.org) or the Math section of the National Center for Biotechnology Information (NCBI) at www.ncbi.nlm.nih.gov.