Question

Find the rate of change of the volume of a sphere with respect to its surface area when the radius is varying.

Answer 1

We know that $Surfacearea=2\pi r^2+2\pi rh$

$Volumev=\frac{4}{3}\pi r^3$

$\frac{ds}{dr}=2\pi (2r+h)$

$\frac{dv}{dr}=4\pi r^2$

$\frac{dv}{ds}=\frac{4\pi r^2}{2\pi (2r+h)}$

$=>\frac{dv}{ds}=\frac{2r^2}{(2r+h)}$