Question

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?

Answer 1

Let V be the volume of the sphere. Then,

V = \[\frac{4}{3}\pi r^3\] 

\[\Rightarrow \frac{dV}{dr} = 4\pi r^2\]

Let S be the total surface area of sphere. Then,

S = \[4\pi r^2\]

\[\Rightarrow \frac{dS}{dr} = 8\pi r\]

\[\therefore \frac{dV}{dS} = \frac{\frac{dV} {dr}}{\frac{dS}{dr}}\]

\[ \Rightarrow \frac{dV}{dS} = \frac{4\pi r^2}{8\pi r} = \frac{r}{2}\]

\[ \Rightarrow \left( \frac{dV}{dS} \right)_{r = 2} = \frac{2}{2}\]

\[ = 1 cm\]