Show that the volume of greatest culinder that can be inscribed in a cone of height h and semi vertical angle alpha is 4/27 pi h cube tan square alpha

Dear Student!

Here is the answer to your query.

Let VAB be a given cone of height *h*, semi-vertical angle α and let *x* be the radius of the base of the cylinder A´ B´ DC. Which is inscribed in the cone VAB.

Then,

OO´ = height of the cylinder

⇒ OO´ = VO – VO´ = *h* – *x* cot α

Let V be the volume of the cylinder. Then

V = π*x*^{2} (*h* – *x* cot α) ...... (1)

for maximum of minimum V we must have

Hence V is maximum when

The maximum volume of the cylinder is given by

Cheers!

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