Answer
about $213^{o}$
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.
----------------
For the smaller gear, the length of the arc that turns is
$ s=r\theta$
$\theta=315^{o}$ needs to be converted to radians (case 1):
$ s=r\displaystyle \cdot 315\cdot\frac{\pi}{180}=4.8\cdot 315\cdot\frac{\pi}{180}=8.4\pi$
The bigger gear rotates for the same arc length, so
$ s=r\theta\qquad$ ( ... solve for $\theta$)
$ 8.4\pi=7.1\theta$
$\displaystyle \theta=\frac{8.4\pi}{7.1} \qquad$(...radians to degrees ...)
$\displaystyle \theta=\frac{8.4\pi}{7.1}\cdot\frac{180^{\mathrm{o}}}{\pi}=\frac{8.4\cdot 180}{7.1}\approx$212.957746479
The larger gear rotates for about $213^{o}.$