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Suppose there is a hole in the can of paint in Exercise 43 and 9 lb of paint leaks steadily out of the can during the man's ascent. How much work is done?

Work done $=16,245$ ft-lbs

Vector Calculus

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So in this question again were given. A man who weighs up ÂŁ60 is lifting a ÂŁ25 Pinkett. So that means that, uh, the work done so the So That means the man is doing work against the force of gravity. Uh, since he's lifting, this paint can 90 feet, right? So now the total weight of the man and they can and he is harder than 80. So now we're told that this paint can just ÂŁ25. It is actually leaking right, and it's leaking nine. Uh, it's leaking ÂŁ9 of pink. So in the best case scenario, the weight of the man and the can is 185. In the worst case scenario, when the man is going up the stairs is steadily leaks ÂŁ9 of paint. So we're gonna have that the weight of the banner That care is 176. And now we're asked to find the work done by the man and lift they the work done and the bad going go uh, 90 90 feet on the staircase while lifting this pain. That's, uh, that's that's readily that steadily, uh, losing paint again. The work is just the integral from 0 to 90 of f dot ph. All right, so now our foresee our forces in constant does our forces 185. However, this force has really keep going higher and higher. Well, we'll keep decreasing. Has pain is leaking outside of the the camp. So it's 185 minus each divided. What? 10. So zero. So if we plug in zero, this would be 185 minus zero divided like that. So he's so at times. The height zero starts with 185 at a time. 90 so well when he started at a time. Height zero He is the man and the paint is 100 and 85 at height 90 which is the top of the staircase. The weight of the man and the paint camp is 176. Right, which is exactly so. Now if we take the been to grill of 185 minus eight, divided by 10 teach that's just 180 friend H minus H squared, divided by 20. And our height varies from zero tonight. So now we Just plenty of 9700 and 85 times 90 minus 90 square, divided by 20 minus that we're gonna plug in zero for ages of zero minus here. So we work out being out to where we get a 16,650 minus 405. So are the work done and lift in the work done in the man going from 0 to 90 uh, feet in height while the paint is leaking steadily is 16 1245. Fate tubs pounds.