COMEDK · Maths · 27. Application of Derivatives
\(\text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is }\)
- A \(\dfrac{2}{3} \sqrt{\dfrac{S}{\pi}}\)
- B \(\sqrt{\dfrac{S}{\pi}}\)
- C \(\dfrac{1}{4} \sqrt{\dfrac{S}{\pi}}\)
- D \(\dfrac{1}{2} \sqrt{\dfrac{S}{\pi}}\)
Answer & Solution
Correct Answer
(C) \(\dfrac{1}{4} \sqrt{\dfrac{S}{\pi}}\)
Step-by-step Solution
Detailed explanation
Let \(r\) be the radius of the sphere. The volume \(V\) and surface area \(S\) are given by \(V = \dfrac{4}{3} \pi r^3\) and \(S = 4 \pi r^2\). From the expression for surface area, \(r^2 = \dfrac{S}{4 \pi}\), which implies…
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