The diagram shows a sphere with a fixed radius $a$ cm and centre $O$ which is inscribed exactly

Question

The diagram shows a sphere with a fixed radius $a$ cm and centre $O$ which is inscribed exactly inside a right cone of base radius $r$ cm and height $h$ cm Express $r$ in terms of $a$ and $h$. The volume of cone is $V$ cm$^3$. Show that $V = \frac{\pi a^2 h^2}{3(h - 2a)}$. Find the value of $h$ for which $V$ has a turning value and determine the nature of this turning value. Hence find this volume, leaving your answer in terms of $\pi$.

Accepted Answer

(i) $r=a\sqrt{\dfrac{h}{h-2a}}$ (ii) $V=\dfrac{\pi a^2h^2}{3(h-2a)}$ (iii) $h=4a$, minimum (iv) $\dfrac{8\pi a^3}{3}\text{ cm}^3$

Differentiation - Problems Involving Maximisation and Minimisation

Differentiation - Problems Involving Maximisation and Minimisation

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