Let θ be the semi-vertical angle of the cone.
It is clear that 
Let r, h, and l be the radius, height, and the slant height of the cone respectively.
The slant height of the cone is given as constant.
Now, r = l sin θ and h = l cos θ
The volume (V) of the cone is given by,
By second derivative test, the volume ( V) is the maximum when
Hence, for a given slant height, the semi-vertical angle of the cone of the maximum volume is . 
