Differentiation - Problems Involving Maximisation and Minimisation
Differentiation - Problems Involving Maximisation and Minimisation
The diagram below shows a right circular cone of base radius \(r\) cm and height \(h\) cm fits exactly into a sphere of internal radius
\(6\) cm, with centre, \(O\).
- Express \(r\) in terms of \(h\).
- Show that the volume of the cone, \(V = 4\pi h^2 - \frac{1}{3}\pi h^3\) cm\(^3\).
- Find the value of \(r\) for which \(V\) has a turning value and determine whether the value of \(V\) is a maximum or minimum.
- Hence, find this volume.
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Answer:(a) \(r=\sqrt{12h-h^2}\) (b) \(V=4\pi h^2-\frac13\pi h^3\) (c) \(r=4\sqrt2\text{ cm}\), giving a maximum (d) \(V_{\max}=\frac{256\pi}{3}\text{ cm}^3\)
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