Problem $5$:

$($a$)$ Two previously undeformed cylindrical specimens of an alloy are to be strain hardened by reducing their cross$-$sectional areas $($while maintaining their circular cross sections$).$ For one specimen, the initial and deformed radii are $16$ mm and $11$ mm $,$ respectively. The second specimen, with an initial radius of $12$ mm $,$ must have the same deformed hardness as the first specimen; compute the second specimen's radius after deformation. $(\backslash (\backslash$mathbf$\{10\}\backslash )$ marks$)$

$($b$)$ Two previously undeformed specimens of the same metal are to be plastically deformed by reducing their cross$-$sectional areas. One has a circular cross section, and the other is rectangular; during deformation the circular cross section is to remain circular, and the rectangular is to remain as such. Their original and deformed dimensions are as follows:

Which of these specimens will be the hardest after plastic deformation, and why? $(10$ Marks$)$