In open die forging, a disc of diameter $$200mm$$ height $$60mm$$ is compressed without any barreling effect. The final diameter of the disc is $$400mm.$$ The true strain is

The thickness of metallic sheet is reduced from an initial value of $$16mm$$ to a final value of $$10mm$$ in one single pass rolling with a pair of cylindrical rollers each of diameter of $$400mm.$$ The bite angle in degrees will be

A $$200$$ $$mm$$ long down sprue has an area of cross section of $$650$$ $$m{m^2}$$ where the pouring basin meets the down sprue (i.e. at the beginning of the down sprue). A constant head of molten metal is maintaining by the pouring basin. The molten metal flow rate is $$6.5 \times {10^5}\,\,\,m{m^3}/s.$$ considering the end of down sprue to be open to atmosphere and an acceleration due to gravity of $${10^4}mm/{s^2},$$ the area of the down sprue in $$m{m^2}$$ at its end (avoiding aspiration effects) should be

Volume of a cube of a side $$''l''$$ and volume of sphere of radius $$' r '$$ are equal. Both the cube and the sphere are solid and same material. They are being cast. The ratio of the solidification time of the cube to the same of the sphere is