Volume of cone = \(\frac{1}{3}\)πr2h, r is the radius of the cone.
Let h and r be height and radius of the required cone.
R be the radius of the sphere.
Now, it must be understood that for the cone to have maximum volume, the axis of cone and sphere must be the same.
Let OD = x
In ΔBOD,
BD = \(\sqrt{R^2-x^2}\)
= \(\sqrt{144-x^2}\)
AD = AO + OD = 12 + x
The roots of this quadratic equation is - 12 and 4.
As - 12 is not possible, we have x = 4.
Therefore, the volume is maximum when the x = 4.
Therefore the height of the cone = R + x
= 12 + 4 = 16cm.
