IB AA HL May 2026 P2 TZC Q8

A bowl is in the shape of part of a sphere with radius \(15\text{ cm}\). Water in the bowl has depth \(h\text{ cm}\) and surface radius \(r\text{ cm}\), as shown in the diagrams.

1. Show that \(r=\sqrt{30h-h^2}\).[2]
2. Water is poured into the bowl at a constant rate of \(20\text{ cm}^3\text{ min}^{-1}\).
   The volume of water at depth \(h\) is \(V=\frac{\pi}{3}(45h^2-h^3)\).
   Find the rate of change of \(r\) when \(h=5\text{ cm}\).[5]