A bucket that weighs 6 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

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