Show that semi-vertical angle of right circular cone of given surface area and maximum volume is .

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Solution

Let r be the radius, l be the slant height and h be the height of the cone of given surface area, S.

Also, let α be the semi-vertical angle of the cone.

Then S = πrl + πr²

Let V be the volume of the cone.

Differentiating (2) with respect to r, we get

For maximum or minimum, put

Differentiating again with respect to r, we get

Thus, V is maximum when S= 4πr²

As S = πrl + πr²

⇒ 4πr² = πrl + πr²

⇒ 3πr² = πrl

⇒ l = 3r

Now, in ΔCOB,

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