2003 RJC P1 Q16

At the instant when the radius of a sphere is \(r\)cm, its volume, \(V\)cm\(^{3}\), is increasing at the rate of
\(\pi \left( \lambda -{{r}^{3}} \right)\)cm\(^{3}\)min\(^{-1}\), where \(\lambda \) is a constant. Express \(\frac{\mathrm{d}V}{\mathrm{d}t}\) in terms of \(\frac{\mathrm{d}r}{\mathrm{d}t}\) and show that \(4{{r}^{2}}\frac{\mathrm{d}r}{\mathrm{d}t}=\lambda -{{r}^{3}}\).[3]

If the initial volume of the sphere is zero, find \(t\) when the volume is \(\frac{\pi \lambda }{6}\)cm\(^{3}\).[5]

[Volume of sphere\(=\frac{4}{3}\pi {{r}^{3}}\)]