Differentiation - Problems Involving Maximisation and Minimisation
Differentiation - Problems Involving Maximisation and Minimisation
A cylinder of radius \(r\) cm is placed upright inside a cone so that the top of the cylinder is \(4\) cm above the top of the cone as shown in the figure above. The cone has a radius of \(6\) cm and a height of \(18\) cm. The part of the cylinder inside the cone is \(h\) cm deep.
- Show that \(h + 3r = 18\).
- Find an expression in terms of \(r\) for the volume of the cylinder.
- Hence find the value of \(r\) for which the volume has a stationary value. Find the corresponding value of \(h\) and determine whether the volume is a maximum or a minimum.
Solution:
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Answer:(a) \(h+3r=18\). (b) \(V=\pi r^2(22-3r)\ \text{cm}^3\). (c) \(r=\frac{44}9\) cm, \(h=\frac{10}3\) cm; maximum volume.
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