Question

Find the surface area of a sphere when its volume is changing at the same rate as its radius ?

Answer 1

\[\text{Let r be the radius and V be the volume of the sphere at any time t . Then }, \]

\[V = \frac{4}{3}\pi r^3 \]

\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \left( \frac{dr}{dt} \right)\]

\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \left( \frac{dV}{dt} \right) \left[ \because \frac{dV}{dt} = \frac{dr}{dt} \right]\]

\[ \Rightarrow 4\pi r^2 = 1 \]

\[ \Rightarrow \text { Surface area of sphere =1 square unit }\]