Question

The rate of change of the surface area of a sphere of radius \( r \), when the radius is increasing at the rate of \( 2 \, \text{cm/s} \), is proportional to

• A. \( \frac{1}{r} \)
• B. \( \frac{1}{r^2} \)
• C. \( r^2 \)
• D. \( r^3 \)

Hint:

The rate of change of surface area of a sphere can be found by differentiating the surface area formula with respect to time.

Answer 1

The Correct Option is B

Step 1: Understanding the surface area change.
The surface area \( A \) of a sphere is given by \( A = 4\pi r^2 \). The rate of change of surface area is proportional to \( r^2 \). Since the radius is increasing at a rate of \( 2 \, \text{cm/s} \), the rate of change of surface area is proportional to \( \frac{1}{r^2} \).

Step 2: Conclusion.
The rate of change of the surface area is proportional to \( \frac{1}{r^2} \), corresponding to option (2).