A cube of wax is melted down and poured into a spherical mold. The wax perfectly fills half of the spherical mold (therefore taking the shape of a hemisphere).
Would the original cube of wax have fit inside the spherical mold (without changing the shape of the cube)?
The volume of the wax cube = S^3 where S is the side of the cube
The length of the 3D diagonal vertex of the cube = sqrt (3) S ≈ 1.73S
The diameter of the sphere must be greater than this for the cube to fit inside the sphere
The volume of the hemisphere = (2/3)pi * r^3
Volume of the cube =Volume of the hemisphere
S^3 = (2/3)pi * r^3 solve for r
S^3 / [ (2/3)pi ] = r^3 take the cube root
S / [ (2/3)pi] ^(1/3) = r ≈ .782 S
So the diameter of the sphere = 2(.782) S = 1.564S
So....the wax cube would not fit