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

Given: Height of cone is ‘h’ and semi-vertical angle of the cone is ‘a’

To prove: the volume of the greatest cylinder that can be inscribed in a cone is

Let PQR is cone and PO = h

Let x be the radius of cylinder ABCD inscribed in a cone PQR

Height of cylinder = OO’ = PO – PO’

In Δ APO’

⇒ PO’ = x cot a

Volume of cylinder, V

= πr²h

= πx²(h – x cot a)

= πx²h – πx³ cot a

We need to maximize volume

V = πx²h – πx³ cot a

Differentiating both sides with respect to x:

Now, Put :

⇒ 2πxh – 3πx² cot a = 0

⇒ πx(2h – 3xcot a) = 0

⇒ 2h – 3xcot a = 0

⇒ – 3xcot a = -2h

Differentiating again both sides with respect to x:

From (1) :

Since,

⇒ V is maximum at

V = πx²h – πx³ cot a

Put :