In a playground, there is a small merry$-$go$-$round of radius $1\mathrm{.}20$ m and mass $160$ kg$.$ Its radius of gyration is $91\mathrm{.}0$ cm$.($Radius of gyration k is defined by the expression I$=$Mk$2.)$ A child of mass $44\mathrm{.}0$ kg runs at a speed of $2\mathrm{.}00$ m$/$s along a path that is tangent to the rim of the initially stationary merry$-$go$-$round and then jumps on$.$ Neglect friction between the bearings and the shaft of the merry$-$go$-$round. Calculate the following:

$($a$)$ the rotational inertia of the merry$-$go$-$round about its axis of rotation

kg$*$m$2$

$($b$)$ the magnitude of the angular momentum of the child, while running, about the axis of rotation of the merry$-$go$-$round and

kg$*$m$2/$s

$($c$)$ the angular speed of the merry$-$go$-$round and child after the child has jumped on$.$

rad$/$s