Answer:
Workdone = 3200 lb.ft
Step-by-step explanation:
We are told that the bucket is filled with 40 lb of water but water leaks out of a hole in the bucket at a rate of 0.2lb/s
Thus,
Weight of water at any given time (t) would be;
w(t) = 40 - 0.2t - - - - (1)
We are told the bucket is pulled up at a rate of 2ft/s.
Thus, height at time (t); y = 0 + 2t = 2t
Since y = 2t,
Then,t = y/2
Put y/2 for t in eq 1
Thus; w(y) = 40 - 0.2(y/2)
w(y) = 40 - 0.1y
Now, at y = 80 ft, we have;
w(80) = 40 - 0.1(80)
w(80) = 40 - 8 = 32 lb
Since 32 lbs are left, it means there is always water in the bucket.
Thus, work done is;
W = 80,0[∫(Total weight).dy]
W = 80,0[∫[(weight of rope) + (weight of bucket) + (weight of water)]dy]
W = 80,0[∫[0 + 4 + 40 - 0.1y]dy]
Integrating, we have;
W = [44y - y²/20] at boundary of 80 and 0
So,
W = [44(80) - 80²/20] - [0 - 0²/20]
W = 3200 lb.ft
