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

A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high.

(a) How much work is done in pulling the rope to the top of the building?

(b) How much work is done in pulling half the rope to the top of the building?

(a) $25 ft-lb$

(b) $\frac{1878}{4}$ ft lb

Applications of Integration

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

So first ones, too calculated the work. Listen, a polo the road from the bottom to top of a building recall. The length of the rope is faith. The feet hot buildings 1 20 feet and the pulling force is a constant equals deserve 15 ounce burger beat. So that's all all over initial data and the work. It will be done bye, bringing summons limits and goes to infinity. And the over I schools, too. Here is from one to in German. Five X I. That's off force here. And don't eggs. That's over. Adams of the Heights. The change in height and by the definition of breathing, so be crisp on an integral equals to 0 to 50 hissed at sea. Um, that's the length of the rope is there 15 X e X and which is equal to 0125 bags square from 0 to 50 and that instant to 65 feet. Thanks. Okay. And Harvey is too. That's half of the rope from the mountain tops. Before this part, we're gonna be white. The work by two ports that we wanted double too. So the first half that one response to the have the rope is full to the top of beauty. And that's the work exactly gonna mating those some this part of the top of the rope to the building. So that's that's calling the polling police work. Okay. And what's up to the devil too, is like, Think of it. This this this figure, that's our building. And that's why we wrote we're gonna leave thing the top parts. Oh, but, uh, the bottom the bottom half of the bun had still needs to be lifted. So double to is the the work that's gonna be done to the bottom half, holy work or top. And that's living work for bothering. So though the bottom the bottom are the rope will not be poached to the top of the building, but it's still need to lift it up to this part. That's the start. And that's the ending. You still need to live it, live it up up here. And that's so, uh, this spot. So we haven't tablets from several. And for that one among the top five, it's simple, really. That's what over four. And that's in part by part, eh? Right. And what way have already noticed that this this answer is, um, 620 flight over four guns at the bottom one. This is equals two 25 to 50. That's the least thing. And, uh, 25 over to the eds. So which he goes to 6 25 or to feed friends and overto the work w equals two devil. Impossible to which happens to 1875 04 feet pounds. That's it.

University of Illinois at Urbana-Champaign

Applications of Integration