SHOW THAT THE SEMI-VERTICAL ANGLE OF RIGHT CIRCULAR CONE OF GIVEN SURFACE AREA AND MAX VOLUME IS SIN INVERSE(1/3).

Let r be the radius of the cone, l be the slant height and h be the height of the cone.

let surface area of the cone be S. therefore

Let V be the volume of the cone.

Differentiating (2) w.r.t. r, we get–

For maximum or minimum,

⇒ either r = 0 or S – 4πr ² = 0

Differentiating (3), we have –

Now if α be the semi-vertical angle of the cone, then