# Show that the vol

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 :

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