Answer
12.66 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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The weight will be raised for the length of the arc of the pulley as it rotates through the given angle.
The angle needs to be in radians
$\displaystyle \theta=(51.6)\cdot\frac{\pi}{180}=\frac{51.6\pi}{180}$
Solve for r, inserting values for s and $\theta:$
$ s=r\theta$
$11.4=r\displaystyle \cdot\frac{51.6\pi}{180}$
$ r=\displaystyle \frac{11.4\cdot 180}{51.6\pi}\approx$12.6583698924
... about 12.66 cm.