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 = πx2 (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!