Answer
27 h.
Work Step by Step
The spaceship applies a torque on the asteroid, changing the direction of its angular momentum. Torque is the time rate of change of angular momentum.
$$\tau=\frac{\Delta L}{\Delta t}\approx\frac{L \Delta \theta}{\Delta t}$$
$$\Delta t=\frac{\Delta L}{\tau}=\frac{I \omega \Delta \theta}{Fr}$$
$$\Delta t=\frac{0.4mr^2 \omega \Delta \theta}{Fr}=\frac{0.4mr \omega \Delta \theta}{F}$$
$$ \Delta t =\frac{0.4(2.25\times10^{10}kg)(123m) (8 \pi rad/86400s) ((5.0/360)2 \pi rad)}{285N}$$
$$ \Delta t =9.860\times10^{4}s=27h$$
