Schematic diagram of pouring basin and sprue of a gating system is shown in the Figure. Depth of molten metal in the pouring basin is $$100$$ $$mm$$ and the height of the sprue is $$1,500$$ $$mm.$$

Considering the cross-section of the sprue is circular, the ratio $${d_1}:{d_2}$$ to avoid aspiration is

Two cast iron blocks $$P$$ and $$Q,$$ each of $$500$$ $$mm$$ length, have the same cross-sectional area. Block $$P$$ has rectangular cross-section of $$100\,\,mm \times 200\,\,mm.$$ Block $$Q$$ is of square cross-section. Both $$P$$ and $$Q$$ were cast under the same conditions with all their surfaces enclosed within the mould. The solidification time of a casting is proportional to the square of the ratio of its volume to its surface area. The ratio of solidification time of the block $$P$$ to that of the block $$Q$$ is _________________

In a metal casting process, molten copper alloy is poured into a sand mould. The level of molten metal in the pouring basin is at a height of $$300$$ $$mm$$ from the runner having diameter of $$10$$ $$mm.$$ If the density and melting temperature of molten copper alloy are $$9000$$ $$kg/{m^3}$$ and $${1000^ \circ }C.$$ respectively then the rate of flow of molten metal into the mould neglecting friction and other losses, in $$c{m^3}/s,$$ is ______________.

For a given volume of a riser, if the solidification time of the molten metal in riser needs to be quadrupled, the surface area of the riser should be made