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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 bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

$W=3200 \mathrm{ft}-\mathrm{lb}$ ; See solution for Riemann sum approximation

Applications of Integration

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Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

the total work you have to get the job done equals to the work, this convict, the bucket and the work that's gonna lift the water inside pocket. So call this that one. Plus that too. The workers is for the workman like the market and double to is the working maybe to make the water instead of rockets for the oneness for its simple, it is equals to the mass or say the wait, wait terms the distance or the height. So it is equals 24 pounds and a fitz. So that's 320 for double to we're gonna use. We're gonna approach this by a Roman son. So think of it. I don't see the market is two feet above it's already eight feet Death, So attempt T velocity is too. So it's two t feet above its original death. This zero this 8 80 here at MT. This is the actual flight of rubber pockets. And this part is to t the Paris to t x x I because to to see. But the thing is, another holds all the water. Some order this lose. Were you lifting bracket and that attempt t the water that we hold. So we're still inside. The pocket is and t equals 2 40 minus zoop onto T pounds of water that's doing said the ball, kids. So we're plugging about this son. We plug in the first equation, go second, which I'm gonna have the block. It holds 40 months their home too. Times 1/2 x, I pounds of water and x I stands for extent for the depth that that is a bomb over original 80. Feats of the off the wall, the death. Okay, so his aborigine salmon is going to set it up as limits and ghost affinity Either's through one too in and 40 minus. So this part is there 11 x i that x. So that's a reverse up and converting to its ah, Chris money Women into girl We're gonna have 0 to 88 is a death wall 40 minus 011 x, the ETS and this Andrew Value. It's too Fourie Ex miners at square over, if only from 0 to 80 which seems to 28 80. So the total work is gonna be done. W equals two deadly one possible two is 3200 feet pounds

University of Illinois at Urbana-Champaign

Applications of Integration