$3\mathrm{.}3-9$ A high$-$strength steel drill rod used for boring a hole in the earth has a diameter of $12$ mm $($see figure$).$ The allowable shear stress in the steel is $300$ MPa and the shear modulus of elasticity is $80$ GPa $.$

$($a$)$ What is the minimum required length of the rod so that one end of the rod can be twisted $30$ with respect to the other end without exceeding the allowable stress?

$($b$)$ If the shear strain in part $($a$)$ is limited to $3\mathrm{.}2{10}^{-3},$ what is the minimum required length of the drill rod?