$18\mathrm{.}30(10$ points per section$)$ A cup$-$shaped part is backward extruded from an aluminum slug that is $50$ mm in diameter. The final dimensions of the cup are: outside diameter $=50mm,$ inside diameter $=30mm,$ height $=100mm,$ and thickness of base $=5mm.$ Determine $($a$)$ extrusion ratio, $($b$)$ shape factor, and $($c$)$ If the metal has flow curve parameters $K=400$MPa and $n=0\mathrm{.}29,$ and the constants in the Johnson extrusion strain equation are: $a=0\mathrm{.}8$ and $b=1\mathrm{.}5,$ determine the extrusion force $($please use the "average" flow stress in your calculation, $\frac{?}{\text{b}\text{a}\text{r}}(Y{)}_f).$