The helix is defined parametrically as:
Now, we need to determine a vector field that represents to force exerted by the farmer in lifting his own weight plus the weight of the can of paint. I'll work on that part now, but I'll see if you can figure that out.
Once we get the vector field, we can find the work done:
is the force vector field, and
is the vector that represents the path of the field (in this case, the path around the helix)
Note that the Force vector field will be different for both parts. In part A, the weight doesn't change as it follows the path. In part B, the weight decreases at a constant rate, and that is something that will need to be taken into consideration when finding the force field.
Does this make sense? Ask questions if there is something you don't understand.