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# A 160 -lb man carries a $25-$ lb can of paint up a helical staircase that encircles a silo with a radius of 20 $\mathrm{ft}$ . If the silo is 90 $\mathrm{ft}$ high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?

## Work done $=16,650 \mathrm{ft}-\mathrm{lbs}$

Vector Calculus

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all right. So and this question were given were given 100 and £60 man carries a £25 can't have painted a care and he goes up a helical. Their case fortunes 90 feet high and as a radius of 20 and he goes around three revolutions and then were asked to determine the work. And we know that the line integral can be interest interpreted as work along the path. So this is the physical interpretation. But here is the thing from physics. We should no, that that that the work done is four stamps displacement. However, it's the force in the direction of the displacement. So remember this guy is doing so. The force is the guys. Wait. So 160 bus the cant that the paint cans wait so it's 150 fun. So the work that he's doing is lifting £185 up The staircase there is doing work against the force of graft, so the radius doesn't matter with us. How many revolutions don't matter because displacement is just the final point, minus the initial point. The final kind of the initial point is here are The final point is 90 feet, so 90 minus zero. It's just nice. So we're any sort of know from physics that our answer is just gonna be 100 and 84. However, let us try to use the line integral to term in the world again. The line integral is just the drug roll over the paths on the prophecy of f dot de each The H is the differential hunt. All right, so we go from 0 to 90 of 185 th What we know is that this 185 is a constant. We can pull it to the outside and now the integral off the D H. It's simply each and the limits there are from zero today. So now we're gonna do is it's gonna be under than 85 times 99 0 which is 116,650 Be the beat pants. All right, so it's just so we saw that from physics. We know that work is force Times distance, uh, four stuns distance. However, the distance has to be the sorry force. Times displacement. However, the force and a displacement must be the same direction. We know that the forces 185 going up. And we know that displacement going up since the force and a displacement have to be the same direction of their bullets, vector quantities just 185 times. But here we also showed that you using what we know about line into girls. We end up with safe.

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