Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is cot-1√2.

Let r ,h,l and θ be the radius ,height, slant height and semi-vertical angle of the cone respectively.

We know that ,volume of cone

⇒

Let S be the curved Surface area of cone.

⇒

⇒

⇒

⇒

⇒

⇒

Now ,

⇒

⇒

⇒

⇒

At ,

Hence, for , surface area is minimum.

Now,

⇒

⇒

⇒

⇒ cotθ = √2

⇒ θ = cot^-1(√2)