Question

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^-1(2)^1/2.

Answer 1

HELLO DEAR

Let α be the semi vertical angle of a cone of slant height l

Then VO = l cos α and OA = l sin α

Let V be the volume of the cone Then
V =1/3π(OA)²(VO)

V = 1/3π(l²sin²α )(lcosα)

V = 1/3πl³(sin²αcosα)

dV/da = 1/3πl²(-sin³α + 2cos²αsinα)

dV/da = πl²/3sinα(-sin²α + 2cos²α)

now, dV/da = 0,

πl²/3sinα(-sin²α + 2cos²α) = 0

2cos²α = sin²α

tan²α = 2

The V is maximum when tan α =√2 or α = tan^{–1}√2

 i.e. when the semi vertical angle of the cone is tan^{–1}√2

I HOPE ITS HELP YOU DEAR,
THANKS