Question

The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius __________.

Answer 1

The largest right circular cone that can be fitted in a cube of given edge is such that the diameter of the base of the cone is equal to the edge of the cube and the height of the cone is equal to the edge of the cube.

It is given that the edge of cube is 2r.

Let R be the radius and H be the height of the largest right circular cone that can be fitted in the given cube.

∴ Diameter of the base of the cone = Edge of the cube

⇒ 2R = 2r

⇒ R = r

Height of the cone = Edge of the cube

⇒ H = 2r

∴ Volume of the largest cone that can be fitted in the given cube

$=\frac{1}{3}\mathrm{\pi }{R}^{2}H$

$=\frac{1}{3}\mathrm{\pi }\times r^2\times 2r$

$=\frac{2}{3}\mathrm{\pi }{r}^3$

= Volume of the hemisphere of radius r

Thus, the volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.

The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius ____r____.