Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 5A / a / Solution

If $z(t)$ is the water depth and $A(z)=\pi r(z{)}^2$ is the horizontal cross-sectional area, Torricelli's law gives the volume flux through the hole asConservation of volume givesFor a prescribed constant fall rate $\dot{z}=-\alpha$, $\alpha >0$, this requiresConsequentlyso the container radius must be proportional to the fourth root of height.