Question

The volume of the largest cone that can be inscribed in a sphere of radius $9$ cm is

• A. $\frac{32}{3}\pi cc$
• B. $72\pi cc$
• C. $288\pi cc$
• D. $\frac{288}{3}\pi cc$

Answer 1

Answer: (C)

$288\pi cc$

From figure, we have

$\tan \theta =\frac{r}{h}$

$\cot \theta =\frac{r}{2R-h}$

$r^2=h(2R-h)$

Volume $=\frac{1}{3}\pi r^{2}h$

$\Rightarrow \frac{dv}{dh}=\left(2h\right(2R-h)-h^2)\frac{\pi }{3}$

$\Rightarrow 0=h(4R-3h)$

$\Rightarrow h=\frac{4R}{3}$ $r^2=\frac{8R^2}{9}$

Volume $=\frac{1}{3}\pi r^{2}h$

$=288\pi$