Answer
39,616 rotations.
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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In 1 hour, the wheel rotates with arc length of 55 miles. We need to find the total angle of rotation.
First, convert 55 mi to inches:
$s=55$ mi$=55 ($5280 $)$ ft $=290,400$ ft
$=290,400(12)$ in. $=3,484,800$ in.
Solving for total angle $\theta$,
$ s=r\theta$
$3,484,800$ in. $= ($14 in. $) \theta$
$\displaystyle \theta=\frac{3,484,800}{14}\approx 248,914.29$ radians
One rotation = $ 2\pi$ radians.
Total rotations:
$\displaystyle \frac{\theta}{2\pi}=\frac{248,914.29}{2\pi}\approx 39,615.94\approx$
39,616 rotations.
