Prove that the semi - vertical angle of the right circular cone of given volume and least curved surface is .  (CBSE 2014)

Let ‘r’ be the radius of the base circle of the cone and ‘l’ be the slant length and ‘h’ be the height of the cone:

Let us assume ‘’ be the semi - vertical angle of the cone.

We know that Volume of a right circular cone is given by:

⇒

Let us assume r²h = k(constant) …… (1)

⇒

⇒ …… (2)

We know that surface area of a cone is

⇒ …… (3)

From the cross - section of cone we see that,

⇒

⇒ …… (4)

Substituting (4) in (3), we get

⇒

From (2)

⇒

⇒

⇒

⇒

⇒

Let us consider S as a function of R and We find the value of ‘r’ for its extremum,

Differentiating S w.r.t r we get

⇒

Differentiating using U/V rule

⇒

⇒

⇒

⇒

⇒

⇒

Equating the differentiate to zero to get the relation between h and r.

⇒

⇒

Since the remainder is greater than zero only the remainder gets equal to zero

⇒ 2r⁶ = k²

From(1)

⇒ 2r⁶ = (r²h)²

⇒ 2r⁶ = r⁴h²

⇒ 2r² = h²

Since height and radius cannot be negative,

⇒ …… (5)

From the figure

⇒

From(5)

⇒

⇒

∴ Thus proved.