**Find the rate of change of the volume, \(V\) of a sphere with respect to its rad...**

### Question

Find the rate of change of the volume, \(V\) of a sphere with respect to its radius, \(r\) when \(r = 1.\)

### Options

A) \(12\pi\)

B) \(4\pi\)

C) \(24\pi\)

D) \(8\pi\)

The correct answer is B.

### Explanation:

Volume of sphere, \(V = \frac{4}{3} \times \pi r^3\)

Rate of change of \(V = \frac{dv}{dr}\)

Thus if, \(V = \frac{4}{3} \times \pi r^3,\)

\(\Rightarrow \frac{dv}{dr} = 4\pi^2\)

At \(r = 1,\) Rate \(= 4 \times \pi \times 1 = 4\pi\)

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Volume of sphere, \(V = \frac{4}{3} \times \pi r^3\)

Rate of change of \(V = \frac{dv}{dr}\)

Thus if, \(V = \frac{4}{3} \times \pi r^3,\)

\(\Rightarrow \frac{dv}{dr} = 4\pi^2\)

At \(r = 1,\) Rate \(= 4 \times \pi \times 1 = 4\pi\)