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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

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.

A thick cable, 60 ft long and weighing 180 lb, hangs from a winch on a crane. Compute in two different ways the work done if the winch winds up 25 ft of the cable.

(a) Follow the method of Example 4.

(b) Write a function for the weight of the remaining cable after x feet has been wound up by the winch. Estimate the amount of work done when the winch pulls up x ft of cable.

To determine

(a)

To find:

Work done if the winch winds up 25 ft of the cable

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) Given:

i) Length of cable 60 ft

ii) Weights of cable 180 lb

3) Definition 4:

W=limnt=1nfxi* x=abfx dx

4) Calculation:

Use an argument similar to the one that led to Definition 4

Let’s place the origin at the top of the crane and 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 such interval, then all the points in the interval are lifted by approximately the same amount,xi*.

Total length of  the cable 60 ft

Weights of cable 180 lb

Therefore, the cable weights 180/60=3 pounds per foot

If xi*<25ft, then the i th part has to be lifted roughly xi* ft. If xi*25 ft, then the i th part has to be lifted 25ft.

The weight of the ith part is 3lb/ftx ft=3 x lb

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

3 x·xi*=3xi* x, if xi

To determine

(b)

To estimate:

Work done when the winch pulls up x ft of the cable

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