In this question all lengths are in metres.

A water container is in the shape of a triangular prism. The cross-section of the water in the container is an isosceles triangle \(ABC\), with \(\angle ABC=\angle BAC=30^\circ\). The length of \(AB\) is \(x\) and the depth of water is \(h\). The length of the container is \(5\).

(i) Show that \(x=2\sqrt3h\) and hence find the volume of water in the container in terms of \(h\).

(ii) The container is filled at a rate of \(0.5\text{ m}^3\) per minute. At the instant when \(h=0.25\text{ m}\), find

(a) the rate at which \(h\) is increasing,

(b) the rate at which \(x\) is increasing.