The diagram shows a sphere with fixed radius $a$ which is inscribed in a cone with radius $r$ and he

Question

The diagram shows a sphere with fixed radius $a$ which is inscribed in a cone with radius $r$ and height $h$.

Show that ${{r}^{2}}=\frac{{{a}^{2}}h}{h-2a}$ and find the volume of cone, $V$ in terms of $a$ and $h$.

As $h$ varies, find the value of $h$ that gives the minimum $V$, leaving your answer in terms of $a$.

(You do not need to verify that $V$ is the minimum.)

[Volume of cone, $V=\frac{1}{3}\pi {{r}^{2}}h$]

Accepted Answer

$r^2 = \frac{a^2 h^2}{h(h-2a)}$ (shown); $h = 4a$ when $V$ is minimum\)

2014 CJC P1 Q2