Question

The volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle $\alpha$

• A. $\frac{4\pi h^3}{27}{\tan }^2\alpha$
• B. $4\pi h^2{\tan }^2\alpha$
• C. $\frac{4\pi h^3}{9}{\tan }^2\alpha$
• D. $\frac{4\pi h^3}{27}{\tan }^3\alpha$

Answer 1

The correct option is A $\frac{4\pi h^3}{27}{\tan }^2\alpha$

Let $h^1$ height of cylinder
r of cylinder $=(h-h^1)tan\alpha$

Volume of cylinder $=\pi h^1(h-h^1{)}^{2}ta{n}^2\alpha$
$\frac{dv}{d{h}^1}=\pi \left(\right(h-h^1{)}^2-2h^1(h-h^1))ta{n}^2\alpha$
$0=(h-h^1)(h-3h^1)$
$h^1=\frac{h}{3}$

$v=\pi \frac{h}{3}(h-\frac{h}{3}{)}^{2}ta{n}^2\alpha$
$=\frac{4\pi }{27}{h}^{3}ta{n}^2\alpha$