Answer
$82.3^\circ$
Work Step by Step
1. We know that the two gears should travel the same arc length.
2. For the smaller gear, we have the arc length traveled as $L_1=6.84\times150^\circ\times\frac{\pi}{180^\circ}\ in$
3. For the larger gear, we have the arc length traveled as $L_2=12.46\times \theta^\circ\times\frac{\pi}{180^\circ}\ in$
4. Let $L_2=L1$, we can find $\theta=\frac{6.84\times150}{12.46}\approx82.3^\circ$
