Aheavy rope, 80 ft long and weighing 32 lbs, hangs over the edge of a building 100 ft high. how much work w is done in pulling the rope up 20 ft?

The first thing you should know for this case is that work is defined as the product of force by the distance traveled in the direction of force.
 We have then:
 W = Fd
 The distance varies, so we must integrate:
 from 0 to 20:
 W = ∫F (x) dx
 W = ∫32xdx
 W = 32∫xdx
 W = 32 (x ^ 2/2) = (16) (20 ^ 2) = 6400 ft * lbs
 
 6400 ft * lbs is work done pulling the rope up 20 ft