Question

The radius of a circle is increasing at the rate of 2 m/s. The rate of change of its circumference is:

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

Hint:

To find the rate of change of a circle’s circumference, differentiate the formula \( C = 2\pi r \) with respect to time.

Answer 1

The Correct Option is A

Step 1: Understanding the relationship between radius and circumference.
The circumference \( C \) of a circle is given by: \[ C = 2\pi r \] where \( r \) is the radius of the circle. Step 2: Differentiating with respect to time.
To find the rate of change of the circumference, we differentiate \( C = 2\pi r \) with respect to time \( t \): \[ \frac{dC}{dt} = 2\pi \frac{dr}{dt} \] We are given that \( \frac{dr}{dt} = 2 \) m/s, so: \[ \frac{dC}{dt} = 2\pi \times 2 = 4\pi \, \text{m/s} \] Step 3: Conclusion.
Thus, the rate of change of the circumference is \( 4\pi \) m/s, corresponding to option (A).