2026 IGCSE Additional Mathematics May/June 0606/13 Q6

In this question time is in seconds and lengths are in centimetres.

You are given that \(\cos30^\circ=\dfrac{\sqrt3}{2}\), \(\sin30^\circ=\dfrac12\) and \(\tan30^\circ=\dfrac1{\sqrt3}\).

A water tank is shaped as a cone with its axis vertical.

The angle between the vertical axis and the sloping edge of the cone is \(30^\circ\).

At time \(t\) the depth of the water is \(h\), the volume of water in the tank is \(V\), and the radius of the circle formed by the surface of the water is \(r\).

1. Show that \(V=\dfrac19\pi h^3\).[2]
2. Water is flowing into the tank at a constant rate of \(50\text{ cm}^3\text{ s}^{-1}\).
   Find the rate at which \(h\) is increasing at the instant when \(h=60\).
   Give your answer in terms of \(\pi\).[4]