Gray cast iron blocks of size $$100\,\,mm \times 50\,\,mm \times 10\,\,mm$$ with a central spherical cavity of diameter $$4$$ mm are sand cast. The shrinkage allowance for the pattern is $$3\% $$. The ratio of the volume of the pattern to volume of the casting is ___________________

A cube and a sphere made of cast iron (each of volume $$1000\,\,c{m^3}$$) were cast under identical conditions. The time taken for solidifying the cube was $$4$$ $$s.$$ The solidification time (in $$s$$) for the sphere is _____________________

The solidification time of a casting is proportional to $${\left( {{V \over A}} \right)^2},$$ Where $$V$$ is the volume of the casting and $$A$$ is the total casting surface area losing heat. Two cubes of same material and size are cast using sand casting process. The top face of one of the cubes is completely insulated. The ratio of the solidification time for the cube with top face insulated to that of the other cube is

Ratio of solidification time of a cylindrical casting (height $$=$$ radius) to that of a cubic casting of side two times the height of cylindrical casting is ____________