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


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) \(8\pi\)

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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Dicussion (1)

  • 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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