A cube of side 12 cm is painted on all faces and then cut into smaller cubes of side 2 cm each. How many of the smaller cubes have exactly one face painted?

A right circular cone is cut by a plane parallel to its base into two parts. The ratio of the volumes of the smaller cone and the frustum is 1:7. What is the ratio of the height of the smaller cone to the height of the original cone?

A sphere of radius 5 cm is placed inside a right circular cylinder such that it touches both the base and curved surface. What is the ratio of the volume of the sphere to the volume of the cylinder?

A solid metallic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. What is the height of the cone?