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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?

To determine

To find:

Work required to raise the elevator from the basement to the third floor.

Explanation

1) Concept:

Work done in moving an object from a to b

2) Formula:

Work done in moving an object from a to b

abfxdx

Work=Force ×Distance 

3) Given:

Weight of elevator is 1600 lb.

Length of cable is 200 ft.

Weight of cable is 10 lb/ft.

4) Calculations:

Let’s place the origin at the top and the x-axis pointing downward as in the figure.

Divide the cable into small parts with length x.

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

Now, the weight of the cable is 10 pounds per foot.

So the weight of the ith part is (10 lb/ft)(x ft)=10x lb.

Thus, work done on the ith part in foot-pounds is

10x·xi*=10xi*x

The total work done by adding all these approximations and letting the number of parts become large (so  x<

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