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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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 \(= 4 \times \pi \times 1 = 4\pi\)