#### Answer

about $81.6^{o}$

#### Work Step by Step

Arc length s (for central angle $\theta$):$ \quad s=r\theta$, where $\theta$ is in radians
Converting between Degrees and Radians
1. Multiply a degree measure by $\displaystyle \frac{\pi}{180}$ radian and simplify to convert to radians.
2. Multiply a radian measure by $\displaystyle \frac{180^{\mathrm{o}}}{\pi}$ and simplify to convert to degrees.
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For the smaller gear, the length of the arc that turns is
$ s=r\theta$
$\theta=150^{o}$ needs to be converted to radians (case 1):
$s=r\displaystyle \cdot 150\cdot\frac{\pi}{180}=6.84\cdot 150\cdot\frac{\pi}{180}=\frac{6.84\cdot 5\cdot\pi}{6}$
The bigger gear rotates for the same arc length, so
$ s=r\theta\qquad$ ( ... solve for $\theta$)
$\displaystyle \frac{6.84\cdot 5\cdot\pi}{6}=12.46\theta$
$\displaystyle \theta=\frac{6.84\cdot 5\cdot\pi}{6\cdot 12.46} \qquad$(...radians to degrees ...)
$\displaystyle \theta=\frac{6.84\cdot 5\cdot\pi}{6\cdot 12.46}\cdot\frac{180^{\mathrm{o}}}{\pi}=\frac{6.84\cdot 5\cdot 30^{o}}{12.46}\approx$81.568471337$6^{o}$
The larger gear rotates for about $81.6^{o}.$