Answer
The tension in the rope in part b) is $184N$.
Work Step by Step
As you can see in the free-body diagram, the magnitude of the tension in the rope in (a) is half that of the tension in the rope in (b).
The reason is that in (a), since the bucket does not move, $$\sum F=0$$ $$mg=T_1+T_2=2T_a$$ $T_1$ and $T_2$ each responds to the tension force in each of the half of the rope. The rope is equally balanced between the two halves so $T_1=T_2=T_a=92N$
In (b), since the bucket is pulled up at a constant velocity, $$\sum F=0$$ $$mg=T_b$$
This time, there is only one half of the rope opposing $mg$. The forces still balance each other, so the magnitude of the tension doubles.
$$T_b=2T_a=2\times92=184N$$
