Question

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

Answer 1

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