The rate of change of volume of a sphere with respect to its surface…

The correct answer is Option (4) → 3 cm³/cm²

Let r be the radius. The volume and the surface area are

    \(\displaystyle V = \frac{4}{3}\pi r^3\)

    \(S = 4\pi r^2\)

Both depend on r, so by the chain rule

    \(\displaystyle \frac{dV}{dS} = \frac{dV/dr}{dS/dr}\)

    \(\displaystyle = \frac{4\pi r^2}{8\pi r}\)

    \(\displaystyle = \frac{r}{2}\)

At r = 6 cm:

    \(\displaystyle \frac{dV}{dS} = \frac{6}{2} = 3\) cm³/cm²