Below is a gating system for an aluminum AA$356$ casting that weighs $241$ bs $.$ The gating system will be as shown below with a round sprue and a gating ratio of $1$:$4$:$4.$ If the effective sprue height is $12$ in $,$ determine the cross sectional areas of the gating system $($runner$($s$)$ and gate$($s$)$ sprue$)$ necessary to pour this casting. Assume rectangular cross sections for the runner$($s$)$ and gates. If the width to the thickness ratio is $1$:$2$ for runners and gates, what will be the dimensions of each? $(10$pts$)$