2026 HCI P1 Q6

[The volume of a cone of base radius \(r\) and height \(h\) is given by \(V=\frac13\pi r^2h\).]

The diagram below shows a right circular hollow cone of negligible thickness with fixed height \(h\) cm and fixed semi-vertical angle \(\theta\) radians standing on a horizontal ground. A smaller inverted right circular cone of height \(a\) cm and base radius \(x\) cm is inscribed in the cone as shown.

1. Show that the volume, \(V\text{ cm}^3\), of the smaller inverted cone is given by \(\displaystyle V=\frac{\pi x^2}{3}(h-x\cot\theta)\).[2]
2. As \(x\) varies, use differentiation to find the stationary value of \(V\) in terms of \(\pi\), \(h\) and \(\theta\) and determine its nature.[5]
3. Determine the exact rate at which \(V\) is changing when \(\theta=\frac\pi6\), \(h=10\) and \(x=3\), given that \(x\) is increasing at a rate of 0.2 cm per second.[3]