2024 RI P1 Q2

[The volume of a sphere with radius \(r\) is given by \(\frac{4}{3}\pi {{r}^{3}}\) and the surface area of a sphere with radius \(r\) is given by \(4\pi {{r}^{2}}\).]

1. The volume of an expanding sphere is increasing at a constant rate of \(5\)cm\(^{3}\)s\(^{-1}\). Show that, at any instant, the rate of increase of the surface area is \(\frac{k}{r}\) cm\(^{2}\)s\(^{-1}\), where \(r\) is the radius of the sphere and \(k\) is a constant to be determined.[3]
2. Find the exact rate of change of surface area of the expanding sphere when the surface area is \(20\ \mathrm{c}{{\mathrm{m}}^{2}}\).[2]