Please show all work to receive max amount of points. A bucket that weighs 6 lb

Question

Please show all work to receive max amount of points.

A bucket that weighs 6 lb and a rope or negligible weight are used to draw water from a well that is 50 ft deep. The bucket is filled with 40 lb or 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. Show how to approximate the required work by a Riemann sum. (Enter xi * as xi .) Express the work as an integral. Evaluate the integral. ft-lb

Explanation / Answer

Weight of water(t) = 40 - 0.2t

x = 0 + 2t

= 2t

hence

t = x/2

now

Weight in terms of x ll be

W(x) = 40 - 0.2*x/2

= 40 - 0.1x

At x = 50

amount of water left in bucket is = (40-0.1*50) = 35 lbs

Hence

total work done ll be

W = int (Net Weight)dx {limit value is from 0 to 50}

= int (Weight of rope + weight of bucket + weight of water)dx

= int (0 + 6 + (40-0.1x))dx

= int (46 - 0.1x)dx

= 46x - 0.1x^2/2

= 46x - 0.05x^2   

= 46*50 - 0.05*(2500)

= 2175 lb-sec   

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