2019 NYJC P1 Q7

A spherical tank with negligible thickness and internal radius \(a\)cm contains water. At time \(t\)s, the water surface is at a height \(x\)cm above the lowest point of the tank and the volume of water in the tank, \(V\)cm\(^{3}\), is given by \(V=\frac{1}{3}\pi {{x}^{2}}\left( 3a-x \right)\). Water flows from the tank, through an outlet at its lowest point, at a rate \(\pi k\sqrt{x}\)cm\(^{3}\) s\(^{-1}\), where \(k\) is a positive constant.

1. Show that \(\left( 2ax-{{x}^{2}} \right)\frac{\mathrm{d}x}{\mathrm{d}t}=-k\sqrt{x}\).[2]
2. Find the general solution for \(t\) in terms of \(x\),\(a\) and \(k\).[3]
3. Find the ratio \({{T}_{1}}:{{T}_{2}}\), where \({{T}_{1}}\) is the time taken to empty the tank when initially it is completely full, and \({{T}_{2}}\) is the time taken to empty the tank when initially it is half full.[4]