Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin^−1(1/3) .

Let  be the radius and  be the height of the cone and semi-vertical angle be 

  Total Surface area of cone (S) = 

   (say)

 

 

 

 

    ……….(i)

Volume of cone (V) = 

= 

= 

 

=   [Using quotient rule]

        ……….(ii)

Now 

  

 

 

 

   [height can’t be negative]

   is the turning point.

Since, , therefore, Volume is maximum at 

 From eq. (i), 

  

Now Semi-vertical angle of the cone =