...Well I think got the function for the first problem:

The integral from 0 to 20 of : 4(40 - X) + 30.

The integral's interval corresponds to the 20 foot distance the problem is asking for, and 4(40 - X) + 30 is the ropes weight (4lbs) multiplied by the length left (40 - X) plus the weight of the bucket (30lbs). Does this look right to anyone? thanks