2026 NYGH P2 Q12

An inverted right solid cone \(PQR\) with a base diameter of \(x\text{ cm}\) and vertical height of \(h\text{ cm}\) is inscribed in a right cone \(ABC\) such that \(PQ\) is parallel to \(BC\). The vertex \(R\) touches the centre of the base of the cone \(ABC\). It is also given that \(BC=10\text{ cm}\), \(AC=13\text{ cm}\) and the volume of the cone \(PQR\) is \(V\text{ cm}^3\).

[Volume of cone \(=\dfrac13\times\text{base area}\times\text{perpendicular height}\)]

1. Show that
   1. \(h=\dfrac65(10-x)\),[2]
   2. \(V=\pi x^2-\dfrac1{10}\pi x^3\).[1]
2. Hence, find
   1. the value of \(x\) for which \(V\) is a maximum,[4]
   2. the maximum value of \(V\), leaving your answer in terms of \(\pi\).[1]