#### Answer

The pulley must be rotated through an angle of $37^{\circ}04'$

#### Work Step by Step

If the weight rises a height of 6 inches, then a point on the outside of the pulley moves through an arc length $S$ of 6 inches. We can find the angle $\theta$ in radians:
$S = \theta ~r$
$\theta = \frac{S}{r}$
$\theta = \frac{6~in}{9.27~in}$
$\theta = 0.647~radians$
We can convert the angle $\theta$ to degrees:
$\theta = (0.647~rad)(\frac{180^{\circ}}{\pi~rad})$
$\theta = 37.07^{\circ}$
$\theta = 37^{\circ}+(0.07)(60)'$
$\theta = 37^{\circ}04'$
The pulley must be rotated through an angle of $37^{\circ}04'$