Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is

Question

Philoid · Accepted Answer

Let ABC be cone having slant height l and semi-vertical angle θ.

Let V be the volume of the cone,

⇒

Now,

⇒

⇒ sin θ = 0 or sin²θ - 2cos²θ =0

⇒ sin θ = 0 or (1- cos²θ)-2cos²θ =0

⇒ θ = 0 or 1- 3cos²θ =0

⇒ θ = 0 or

Now,

⇒

At θ = 0,

And at ,

Hence, V is maximum for