A playground merry$-$go$-$round has radius $2\mathrm{.}40$ m and moment of inertia about a vertical axle through its center, and it turns with negligible friction. $($a$)$ A child applies an $18\mathrm{.}0$ N force tangentially to the edge of the merry$-$go$-$round for $15\mathrm{.}0$ s$.$ If the merry$-$go$-$round is initially at rest, what is its angular speed after this $15\mathrm{.}0$ s interval? $($b$)$ How much work did the child do on the merry$-$go$-$round? $($c$)$ What is the average power supplied by the child?