show that the cone of greatest volume which can be inscribed in a given sphere has an altitude =2/3RD of diameter of sphere.
Let be the centre of the sphere of radius r.
Also let ACB be the height of the cone
ACB = h units
BC = h – r
Let s be the radius of the cone
Now, volume of the cone,
Putting value of s2 in equation (1)
For maximum volume,
where, d = diameter of the sphere
Hence, cone of greatest volume which can be inscribed in a given sphere has an altitude (diameter) of the sphere.