A large horizontal circular platform $($M$=102\mathrm{.}8$ kg$,$ r$=3\mathrm{.}62$ m$)$ rotates about a frictionless vertical axle. A student $($m$=70\mathrm{.}86$ kg$)$ walks slowly from the rim of the platform toward the center. The angular velocity of the system is $3\mathrm{.}79$ rad$/$s when the student is at the rim.

Find the moment of inertia of platform through the center with respect to the z$-$axis.

Find the moment of inertia of the student about the center axis $($while standing at the rim$)$ of the platform.

Find the moment of inertia of the student about the center axis while the student is standing $1\mathrm{.}22$ m from the center of the platform.

Find the angular speed when the student is $1\mathrm{.}22$ m from the center of the platform.