How long would it take the same cutter to travel the distance $AB$ downstream with its motor shut off?

$t_3=12\ \mathrm{h}$.

The equations of motion of the cutter from $A$ to $B$ and $B$ to $A$ are:

$$S=\left(v_1+v_2\right)t_1$$

and

$$S=\left(v_1-v_2\right)t_2$$

where $t_1=3\ \mathrm{h}$ is the time it takes the cutter to travel downstream, $t_2=6\ \mathrm{h}$ is the time it takes the cutter to travel upstream, $v_1$ is the speed of the cutter relative to the water, $v_2$ is the speed of the current, and $S$ is the distance between points $A$ and $B$.

The equation of motion of the cutter with its motor switched off is

$$S=v_2t_3$$

Solving these equations for $t_3$, we obtain

$$t_3=\frac{2t_1t_2}{\left(t_2-t_1\right)}$$

$$t_3=12\ \mathrm{h}$$