A casting process involves pouring molten metal into a mold, letting the metal cool and solidify, and removing the part from the mold. The solidification time is a function of the casting volume and its surface area, known as Chvorinov’s rule.

Solidification time = K × (Volume = Surface area)²

where K is a constant that depends on the metal. Three parts are to be cast that have the same total volume of 0.015 m³, but different shapes. The first is a sphere of radius Rsph, the second is a cube with a side length L_cube, and the third is a circular cylinder with its height equal to its diameter (H_cyl = D_cyl = 2R_cyl). All the castings are to be made from the same metal, so K has the same value for all three parts.

Which piece will solidify the fastest?

The following table gives the equations for the volume and surface area of these parts.

Object Surface Area Volume Sphere Cube 4tRsph 6Laube 2nRoy + 2nRoyiHeyi = 6tRey (4/3)rReph Leube Cylinder