From the time on this Roman-numeral clock, when will the hands next be 110° apart?

At 2:00 the minute hand is $60^{\circ }$ behind the hour hand. It overtakes and must get $110^{\circ }$ ahead: a gain of $170^{\circ }$ at $5.5^{\circ }$/min, i.e. $\frac{170}{5.5}=30\frac{10}{11}$ min past $2.(32\frac{8}{11}$ is the time they are opposite.)