Draining a sphere: height is not volume

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, $V$ cm³, is given by $V=\frac13\pi x^2(3a-x)$. Water flows from the tank, through an outlet at its lowest point, at a rate $\pi k\sqrt x$ cm³ s⁻¹, where $k$ is a positive constant.

(i) Show that $(2ax-x^2)\frac{dx}{dt}=-k\sqrt x$.2 marks

(ii) Find the general solution for $t$ in terms of $x,a,k$.3 marks

(iii) 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 marks

[2019/NYJC/Prelim/I/7]