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Show that semi-vertical angle of right circular cone of given surface area and maximum volume is 
.
Asked by Raveena Last Modified
Answer 
Harshit Bhardwaj
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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