0606 P23 - Jun 2021 - Q10 - 8 marks
8006
In this question all lengths are in centimetres.
The volume and curved surface area of a cone of base radius \(r\), height \(h\) and sloping edge \(l\) are \(\frac13\pi r^2h\) and \(\pi rl\) respectively.
The diagram shows a cone of base radius \(x\), height \(y\) and sloping edge \(\sqrt{x^2+y^2}\). The volume of the cone is \(10\pi\).
(a) Find an expression for \(y\) in terms of \(x\) and show that the curved surface area, \(S\), of the cone is given by
\(S=\frac{\pi\sqrt{x^6+900}}{x}.\)
(b) Given that \(x\) can vary and that \(S\) has a minimum value, find the exact value of \(x\) for which \(S\) is a minimum.
