$3.$ A $10-$inch $\backslash (\backslash$times $10\backslash )-$inch $\backslash (\backslash$times $0\mathrm{.}75\backslash )-$inch plate casting is produced with a steel alloy with $\backslash (\backslash$rho$=0\mathrm{.}28\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$in$\}^\{3\}\backslash )(\backslash (=7\mathrm{.}75\backslash$mathrm$\{$~g$\}/\backslash$mathrm$\{$cm$\}^\{3\}\backslash )).$ Two third of the steel plate casting volume should be added more into the original one, and all molten metal should be poured in $10$ seconds. The cylindrical, tapered sprue height is $4$ inches. The sprue has a cylindrical base and is connected to two runners. Each runner with a square vertical cross$-$section is connected to two gates. The gate has a rectangular vertical cross$-$section, and its width is three times the height.

The ratio of the cylindrical sprue base's horizontal cross$-$sectional area to the sum of two runners' vertical cross$-$sectional areas to the sum of four gates' vertical cross$-$sectional areas is $1$:$2$:$3.$

Area of a sprue base's horizontal cross$-$section

$1$

The sum of areas of two runners' vertical cross$-$sections

$2$

The sum of areas of four gates' vertical cross$-$sections

Note that drawings are not to actual scale or ratios.

The height of the cylindrical riser is $1\mathrm{.}75$ times its diameter, so it takes $\backslash (30\backslash \%\backslash )$ longer for the riser to solidify compared to the solidification time of the steel plate casting. The dynamic viscosity of steel is $\backslash (4\backslash$times $10^\{-3\}\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$S$\}\backslash$left$(=6\backslash$mathrm$\{$cP$\}=6\backslash$times $10^\{-3\}\backslash$mathrm$\{$~Pa$\}\backslash$cdot $\backslash$mathrm$\{$~S$\}\backslash$right$)\backslash ),$ and the mold constant $\backslash (=16\backslash$mathrm$\{$~min$\}/\backslash$mathrm$\{$in$\}^\{2\}\backslash ).$ The gravitational acceleration is $\backslash (32\mathrm{.}2\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$

Determine the following, showing your work clearly:

$($a$)$ Based on the problem statement, sketch an isometric view of a pattern with all the components, such as sprue, sprue base, runners, gates, risers, etc., required for this casting process. Add a label for each component. Note that the sketch does not need to have actual scale or ratios.

$($b$)$ The horizontal cross sectional diameter of the cylindrical sprue base. $($Unit: inches$)$

$($c$)$ The dimensions of the vertical cross$-$section of each runner. $($Unit: inches$)$

$($d$)$ The dimensions of the vertical cross$-$section of each gate. $($Unit: inches$)$

$($e$)$ The dimensions of the cylindrical riser. $($Unit: inches$)$