0606 P23 - Jun 2020 - Q11 - 7 marks
In this question all lengths are in centimetres.
The volume, \(V\), of a cone of height \(h\) and base radius \(r\) is given by
\(V=\frac13\pi r^2h.\)
The diagram shows a large hollow cone from which a smaller cone of height \(180\) and base radius \(90\) has been removed. The remainder has been fitted with a circular base of radius \(90\) to form a container for water. The depth of water in the container is \(w\) and the surface of the water is a circle of radius \(R\).
(a) Find an expression for \(R\) in terms of \(w\) and show that the volume \(V\) of the water in the container is given by
\(V=\frac{\pi}{12}(w+180)^3-486000\pi.\)
(b) Water is poured into the container at a rate of \(10000\text{ cm}^3\text{ s}^{-1}\). Find the rate at which the depth of the water is increasing when \(w=10\).
