Answer
The ball makes in the air $24.5$ revolutions.
Work Step by Step
1) Find the time $t$ the ball hangs in the air.
The ball's vertical throwing speed is $v_y=19\times\sin55=15.56m/s$
Since the ball is caught at the same height as it is released, the change in height $\Delta h=0$. On the vertical, its initial speed $v_0=15.56m/s$ and its deceleration as the ball goes up $g=-9.8m/s^2$
$$\Delta h=v_0t+\frac{1}{2}gt^2$$ $$15.56t-4.9t^2=0$$ $$15.56-4.9t=0$$ $$t=3.18s$$
2) The ball's spinning speed is given as $\omega=7.7rev/s$
The number of revolutions it makes in the air is $$\theta=\omega t=24.5 rev$$
