Question

In a sphere, the rate of change of volume is:

• A. proportional to rate of change of radius
• B. proportional to rate of change of diameter
• C. surface area times rate of change of diameter
• D. surface area times rate of change of radius

Hint:

For spheres, derivative of volume always becomes surface area multiplied by rate of radius change.

Answer 1

The Correct Option is D

Concept: Volume of sphere: $V=\frac{4}{3}\pi r^3$

Step 1: {Differentiate volume w.r.t time.}
\[ \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \]

Step 2: {Identify surface area.}
\[ \text{Surface Area} = 4\pi r^2 \]

Step 3: {Final relation.}
\[ \frac{dV}{dt} = (\text{Surface Area}) \cdot \frac{dr}{dt} \]