# Show that the volume of the greatest cylinder that can be inscribed in a cone of height ‘h’ and semi-vertical angle ‘α’ is.

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Show that the volume of the greatest cylinder that can be inscribed in a cone of height ‘h’ and semi-vertical angle ‘α’ is.

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Let cylinder of base radius r and height h1 is included in a cone of height h and semi-vertical angle α.

Then,

AB = r,

OA = (h - h1),

In right angle triangle OAB,

Differentiating with respect to h1,we get

For maximum volume V,
$\frac{dV}{dh_1}=0$

Again differentiating with respect to h1, we get