Respuesta :
Answer:
We can conclude that the work applied is 3440ft-lb
Explanation:
Water leaks out of te bucket at rate of [tex]\frac{0.2lb/s}{2ft/s} = 0.1 lb/ft[/tex]
We define now that the Force here is just the water x feet above the bottom and the combined weight of the bucket.
In an equation we can arrange that,
[tex]F(x) = (42-0.1x) + 5[/tex]
The integral of a Force is the Work.
[tex]W = \int\limit^b_a F(x) dx[/tex]
[tex]W = \int\limit^{80}_0 (47-0.1x)dx[/tex]
[tex]W = 47x-0.05x^2 \big |^{80}_0[/tex]
[tex]W = 3440 ft-lb[/tex]
We can conclude that the work applied is 3440ft-lb.
The work done is required in pulling of the bucket to the top of the well is 3440 ft-ib.
What is Riemann sum?
Riemann sum is the method of finding the area under the graph or curve by the application of integration.
As the water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Convert this rate into the ib/ ft as,
[tex]Q=\dfrac{0.2}{2}\\Q=0.1\rm ib/ft[/tex]
Thus, the water leaks out of a hole in the bucket at a rate of 0.1 lb/ft.
Let x be the height in feet above the bottom of the well. Thus the total water leaks out from the hole is,
[tex]0.1x[/tex]
The bucket is filled with 42 lb of water initially and the 0.1x water is leaked out. Thus the water remains in the bucket is,
[tex]42-0.1x[/tex]
The weighs of the bucket is 5 lb. Thus the total weighs of the bucket, when it comes at the top of the well is,
[tex](42-0.1x)+5[/tex]
As the well is 80 ft deep. Thus the limit for the integration should be from 0 to 80 feet. Take the above function of x and solve it by the Riemann sum method as,
[tex]W=\int\limits^{80}_0 {(42-0.1x)+5} \, dx \\W= |{(0-0.1\dfrac{x^2}{2})+0}|^{80}_0 \\W=0.05(80)^2-0\\W=3440\rm ft-ib[/tex]
Thus, the work done is required in pulling the bucket to the top of the well is 3440 ft-ib.
Learn more about the Riemann sum here;
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