Answer
about 146 in.
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.
-------------------
A pedal rotation of $\theta=180^{o}$ converted to radians:
$\displaystyle \theta=180(\frac{\pi}{180}$ radian $)=\pi$ radians.
The arc length of the pedal is $ s=r\theta=4.72\pi$.
The chain moves this same distance, $ 4.72\pi$ in.
The small gear rotates with this arc length through an angle $\theta$...(solve for $\theta$) .
$ s=r\theta$
$\displaystyle \theta=\frac{s}{r}$
$\displaystyle \theta=\frac{4.72\pi}{1.38}\approx 3.42\pi$
The wheel, of which the small gear is a part of,
rotates through this same angle of $ 3.42\pi$ radians.
So, the arc the wheel will cover is .
$s=r\theta\Rightarrow r=13.6(3.42\pi)\approx 146.12$
The bicycle will move about 146 in.