Find the work done

Question

A bucket begins weighing 15 pounds, including the sand it holds. The bucket is to be lifted to the top of a 30 foot tall building by a rope of negligible weight. However, the bucket has a hole in it, and leaks 0.3 pounds of sand each foot it is lifted.

Find the work done lifting the bucket to the top of the building in foot-pounds.

Answer 1

Work = F•d, in this case since "F" is the weight (W) and "d" is height, let's say Work = W•h however W is not constant. W = 15 - 0.3h (at h = 0, the bucket weight is 15 lbs and at h = 30, the bucket weight is 15 - 9 = 6 lbs)

Work = ₀∫³⁰Wdh

Work = ₀∫³⁰(15 - 0.3h)dh

Work = (15h - 0.15h²) evaluated between 0 and 30

Work = (15(30) - 0.15(30)²) - (15(0) - 0.15(0)²)

Work = 450 - 135 - 0

Work = 315 ft lbs