2023 JPJC Promo Q4

Water leaks out at a rate of \(2\)cm\(^{3}\) per second from a container in the form of an open cone. The semi-vertical angle of the cone is \(\alpha \), where \(\tan \alpha =\sqrt{2}\). At time \(t\) seconds after the start, the radius of the water surface is \(r\) cm (see diagram). Find the rate of change of the depth of water when the volume of water in the container is \(\mathrm{1000}\)cm\(^{3}\).[5]

[The volume of a cone of base radius \(r\) and height \(h\) is given by \(V=\frac{1}{3}\pi {{r}^{2}}h\).]