#### Answer

about 18.7 cm

#### 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$
$\theta=80^{o}$ needs to be converted to radians (case 1):
$s=r\displaystyle \cdot 80\cdot\frac{\pi}{180}=11.7\cdot 4\cdot\frac{\pi}{9}=\frac{11.7\cdot 4\cdot\pi}{9}$
This arc length is the same as the arc length of the larger wheel,
rotating through the angle of $50.0^{o}.$
$ s=r\theta\qquad$ ( ... solve for $r$)
$\theta = 50.0\displaystyle \cdot\frac{\pi}{180}=\frac{5\pi}{18}$
$\displaystyle \frac{11.7\cdot 4\cdot\pi}{9}=r\cdot\frac{5\pi}{18}$
$ r=\displaystyle \frac{11.7\cdot 4\cdot\pi\cdot 18}{9\cdot 5\pi}=\frac{11.7\cdot 4\cdot 2}{5}\approx$18.72
The radius of the bigger wheel is about 18.7 cm