Answer:
The work done in pulling the bucket to the top of the well is 3,360 ft-lb
Explanation:
Given
Weight = 6 lb
Depth = 80ft
Weight of Water = 40lb
Rate = 2ft/s
Leak Rate = 0.2ft/s
Calculating Workdone to lift the bucket
Work = Force * Distance
Work = 6 * 80
Work = 480ft-lb
At time t, the bucket is xi = 2t above the original depth of 80ft.
t = ½xi
But it now holds 40lb - 0.2t lb of water
= 40 - 0.2(½xi)
= 40 - 0.1xi.
This is the size of the water when it is x ft above the original depth.
To move this amount of water, we need (40 - 0.1xi)∆x
So, W = ∫(40 - 0.1xi)∆x {1,n}
Where n = 80
W = ∫(40 - 0.1x)dx {0,80}
W = 40x - ½(0.1x²) {0,80}
W = 40x - x²/20 {0,80}
W = 40(80) - 80²/20
W = 3200 - 320
W = 2880 ft-lb
The work done in pulling the bucket to the top of the well = 2880 + 480
= 3,360 ft-lb
