Differentiation - Problems Involving Maximisation and Minimisation
Differentiation - Problems Involving Maximisation and Minimisation
The diagram shows a solid body which consists of a cone fixed to the top of a right circular cylinder of radius \(r\) cm and height \(h\) cm. The slant edge of the cone is \(3r\) cm.
- Given that the volume of the solid body is \(108\pi\) cm\(^3\), express \(h\) in terms of \(r\).
- Show that the total surface area, \(A\) cm\(^2\) of the solid is given by \(A = 4\pi \left( r^2 - \frac{\sqrt{2}}{3}r^2 + \frac{54}{r} \right)\).
- Given that \(r\) and \(h\) can vary,
- find the value of \(r\) for which \(A\) has a stationary value,
- determine whether this stationary value is a maximum or minimum.
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Answer:(a) \(h=\frac{108}{r^2}-\frac{2\sqrt{2}}{3}r\) (b) \(A=4\pi\left(r^2-\frac{\sqrt{2}}{3}r^2+\frac{54}{r}\right)\) (shown) (c)(i) \(r=\sqrt[3]{\frac{81(3+\sqrt{2})}{7}}\text{ cm}\approx3.71\text{ cm}\) (ii) Minimum
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