A bucket that weighs 4 lb and a rope of negligible weight are used to draw water

Question

A bucket that weighs 4 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 42 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 W in pulling the bucket to the top of the well. The work can be computed in two parts: part 1) the work for lifting an empty bucket and part 2) the work for lifting the water. Find the work for part 1. Express the work for part 2 as an integral. Find the total work done in this problem.

Explanation / Answer

wd pulling empty bucket = mgh = 4*80 = 320 lb-ft

now, weight of water as function of distance x is to calculate

water remain in water after time t = 42-.2t

water remain after distance x = 42-.2x/2

wd = integral F.dx

wd = integral 0 to 80 [42-x/10] = 3040

Total wd = 3040 + 320 = 3360 lb-ft

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