Answer:
The total work done in in pulling the bucket to the top of the well is 3040ft-lb
Step-by-step explanation:
Given
Weight of bucket = 4lb
Depth of well = 80ft
So, Work done = Force * Distance
So, Work done to lift the bucket = Weight of buckets * Depth of the well
W1 = 4 * 80
W1 = 320ft-lb
Calculating the work done in pulling the bucket to the top of the well
At time t, the bucket is 1.5 ft above its original depth of 80ft
Also at time t, it now holds (38 - 0.15t) lbs of water.
Mathematically, we have
xi = 1.5t
xi = 3/2t
t = ⅔xi
In terms of distance, the bucket now holds
38 - 0.15 * ⅔xi lb off water when it is xi ft above its original depth of 80ft
Moving this water along a distance of ∆x requires a work done of
38 - 0.15(⅔xi) ∆x
= (38 - 0.3/3xi) ∆x
= (38 - 1/10xi) ∆x
= (38 - xi/10)∆x
Integrating the above we have
W2 = Integral of (38 - x/10) dx. Upper bound= 80 and lower bound = 0
(i.e from the depth (80ft) to the top (0ft))
Integrating, we have.
W2 = 38x - x²/20 (80----0)
=>
W2 = [38(80) - 80²/20] - [38(0) - 0²/20)]
W2 = [3040 - 320] - (0)
W2 = 2720 - 0
W2 = 2720ft-lb
Total work done in in pulling the bucket to the top of the well.
Total Work = W1 + W2
Total Work = 320ft-lb + 2720ft-lb
Total Work = 3040ft-lb
