Question

find the radius of the pulley if the rotation of 45.9 raises the weight (w) to a height of 10.3cm?

Answer 1

it simply means the length of the arc subtending angle 45.9 degrees at the center is 10.3 cm.
Use the face that length of arc = R*theta , where R= radius and theta = angle in radians subtended by the arc at center.
Hope this helps

Answer 2

use the fact* that