Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is.

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Solution

A sphere of fixed radius (r) is given.

Let R and h be the radius and the height of the cone respectively.

The volume (V) of the cone is given by,

Now, from the right triangle BCD, we have:

∴h

∴ The volume is the maximum when

Hence, it can be seen that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is.

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Prove that the altitude of the right circular cone of maximum volume that can be inscribed 8 in a sphere of radius r is $\frac{4 r}{3}$ . Also show that the maximum volume of the cone is $\frac{8}{27}$ of the volume of the sphere.

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