Question

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^-1√2

Answer 1

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 .