$10-1.$ The solid shaft of radius $r$ is subjected to a torque $T.$

Determine the radius $r^{\text{'}}$ of the inner core of the shaft that

resists one$-$half of the applied torque $(\frac{T}{2}).$ Solve the

problem two ways: $($a$)$ by using the torsion formula, $($b$)$ by

finding the resultant of the shear$-$stress distribution.