2018 RI Promo Q2

An equilateral triangle with sides of length \(a\) cm is used to construct the base of a triangular prism container of height \(36\) cm (see Fig. \(1\)). Initially this triangular prism container is half filled with water. This amount of water is then poured at a constant rate into a hollow cone of semi-vertical angle \({{60}^{\circ }}\) held with its axis vertical and vertex downwards (see Fig. \(2\)), without overflowing. After \(t\) seconds, the depth of water in the conical container is \(h\) cm.

Given that it takes \(3\) minutes to empty the triangular prism container, find in terms of \(a\) and \(\pi \), the rate at which \(h\) is increasing at the instant when \(h=2.5\).[5]

[It is given that the volume of a circular cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi {{r}^{2}}h\).]