2025 PHS P2 Q10

[The volume of a cone with base radius \(r\) cm and vertical height \(h\) cm is given by \(\dfrac13\pi r^2h\).]

To make drip coffee, coffee is poured into an inverted funnel of radius 5 cm and height 13 cm. The coffee flows in at a constant rate of \(\alpha\text{ cm}^3/\text{s}\) and dripped out into a cup at a constant rate of \(\beta\text{ cm}^3/\text{s}\) where \(\alpha>\beta\). Initially, the funnel is empty.

1. Express \(\dfrac{\mathrm{d}V}{\mathrm{d}t}\) in terms of \(\alpha\) and \(\beta\).[1]

At time \(t\) s, the coffee surface has radius \(r\) cm and is at a vertical height of \(x\) cm above the vertex.

2. Show that the volume of coffee in the cone at any time, \(t\), is given by \(V=\dfrac{25}{507}\pi x^3\), where \(x\) is the depth of the water.[2]
3. Given that \(\alpha=6\) and \(\beta=2\), find the rate of change of the vertical height of the coffee surface when \(x=5.2\).[5]