#### To determine

**To find:**

Work done in pulling the bucket to the top of the well

#### Answer

W=3200 ft-lb

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Calculation:**

Weight of bucket 4 lb

Depth of well from the water 80 ft

The work needed to lift the bucket itself is 4 lb·80 ft=320 ft-lb

Let’s place the origin at the top of the well and x-axis pointing downward as in the figure. Divide the rope into small parts with length ∆x

If xi* is a point in the ith such interval, then all points in the interval are lifted by approximately the same amount,xi*.

At time t(in seconds) the bucket is xi*=2t ft above its original 80 ft depth

Therefore, t=xi*/2

The bucket is filled with 40 lb of water but it now holds only 40-0.2t lb of water. Since water leaks out of a hole in the bucket at the rate of 0.2 lb/s

Thus, the bucket holds 40-0.212xi* lb of water when it is xi* ft above its original 80 ft depth.

Moving this amount of water at a distance ∆x requires 40-110xi* ∆x ft-lb of work.

Thus, the work needed to lift the water is

W=limn→∞∑i=1n40-110xi* ∆x=∫08040-110xdx

=40x-120x2080

=4080-120802

=3200-640020

=3200-320 ft-lb

Work done in pulling the bucket to the top of the well is work needed to lift the water plus work needed to lift the bucket itself

Work done in pulling the bucket to the top of the well 3200-320+320=3200 ft-lb

**Conclusion:**

The work done in pulling the bucket to the top of the well is 3200 ft-lb