A bucket that weighs $4.6$ pounds and a rope of negligible weight are used to draw water from a well that is $80$ feet deep. The bucket is filled with $35$ pounds of water and is pulled up at a rate of $2.7$ feet per second, but water leaks out of a hole in the bucket at a rate of $0.3$ pounds per second. Find the work done pulling the bucket to the top of the well.
So far I got $ \ \int_0^{80} \ x[39.6-(.11x^2)] \ dx \ \ $ but my webwork keeps telling me it's wrong. What am I doing wrong?
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$\begingroup$It looks as if you are working with height. It doesn't matter much, but let $x=0$ be the bottom of the well. Let us calculate the work done in lifting the bucket from height $x$ to height "$x+dx$." The weight of the bucket when we are at height $x$ is $39.6 -\frac{0.3}{2.7}x$. So the work done in moving from $x$ to $x+dx$ is approximately $\left(39.6 -\frac{0.3}{2.7}x\right)\,dx$ foot pounds. "Add up," (integrate) from $x=0$ to $x=80$. We get that work done is $$\int_0^{80} \left(39.6 -\frac{0.3}{2.7}x\right)\,dx.$$
Added: Integrate. An antiderivative is $39.6x-\frac{0.3}{2(2.7)}x^2$. Now calculate.
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