Question

Radius of a circle is increasing at the rate of \( 2 \text{ m/s} \). Rate of change of its circumference is:

• A. \( 4\pi \text{ m/s} \)
• B. \( 2 \text{ m/s} \)
• C. \( 2\pi \text{ m/s} \)
• D. \( 4 \text{ m/s} \)

Hint:

For a circle, the rate of change of the circumference is always \( 2\pi \) times the rate of change of the radius, regardless of what the current radius is.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is an application of derivatives where we relate the rate of change of one variable to another using a geometric formula.
Step 2: Detailed Explanation:
Let \( r \) be the radius and \( C \) be the circumference of the circle.
The formula for circumference is \( C = 2\pi r \).
Differentiating both sides with respect to time \( t \):
\[ \frac{dC}{dt} = \frac{d}{dt}(2\pi r) = 2\pi \frac{dr}{dt} \]
Given that the radius is increasing at a rate of \( 2 \text{ m/s} \), we have \( \frac{dr}{dt} = 2 \).
Substituting this into the derived equation:
\[ \frac{dC}{dt} = 2\pi \times 2 = 4\pi \text{ m/s} \] Step 3: Final Answer:
The rate of change of circumference is \( 4\pi \text{ m/s} \).