PROBLEM $1(25$ PTS$)$: A slab with a thickness of $10$ mm and width of $1$ m is to be cold rolled to a thickness of $6$ mm using $2-$high rolls with roll diameter of $1$ m $.$ The rolls have a coefficient of friction of $0\mathrm{.}0775$ and they are rotating with a speed of $15$ RPM$.$ The slab has an entrance speed of $\backslash (60\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$min$\}\backslash )$ and a width that does not change as a result of drawing. If the strength coefficient $\backslash (\backslash$mathrm$\{$K$\}=240\backslash$mathrm$\{$MPa$\}\backslash )$ and the metal will cause the average flow stress to be $0\mathrm{.}8696$ of the final flow stresses after deformation. Answer the following:

a$)$ Is the proposed reduction possible? If not, then propose a way to perform this operation without changing the coefficient of friction and$/$or roll radius.

b$)$ Calculate the rolling forward slip assuming a draft of $2$ mm $.$

c$)$ Calculate the rolling force $\backslash ((\backslash$mathrm$\{$N$\})\backslash )$ assuming a draft of $2$ mm and $2$ passes during the rolling process.

d$)$ Find the power $($W$)$ consumed during the $2$ passes assuming a draft of $2$ mm for each pass.