Question

A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 90 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 1.5 ft/s, but water leaks out of a hole in the bucket at a rate of 0.15 lb/s. Find the work done in pulling the bucket to the top of the well.

Answer 1

The work on the bucket is a function of height--->h 
dwork = F(h)*dh 
dwork = weight(h)*dh 

weight(t) = 44 - 0.2t 
but t = h/v 
weight(h) = 44 - 0.2(h/v) --->[44 - 8] = 36 lbs at top 

work = ∫ weight(h) * dh 
work = ∫[44 - 0.2(h/v)] dh <0 to 80>ft 
work = ∫[44 - 0.1h] dh <0 to 80>ft 

work = 44h - 0.1*0.5*h^2 <0 to 80>ft 
work = 44*80 - (0.1*0.5*80^2) = 3200 ft-lbs