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Math Help - Work compute with integral

  1. #1
    Member RedBarchetta's Avatar
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    Work compute with integral

    A tank is full of water. Find the work W required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft3.
    a = 5 b = 9 c = 12


    So for this one, you have to set up an amount of rectangles.

    The biggest rectangle is 5(or y) by 9, in relation to a smaller rectangle that is w by 9. I can also setup a relationship between the two to find that y/w=9/9 or y=w. So therefore we can produce an equation for the area of one rectangle.

    dA=9y

    Now the thickness of this rectangle is extremely small so the thickness is dy.

    dV=9y dy

    The density of water is 62.5 lb/ft^3, so this one rectangular-box of water is:

    dM=562.5y dy

    To find the force, we multiply by the force of gravity, I used 32.2 ft/sec^2.

    dF=18112.5y dy

    So our work is dF times the distance. This is where I become lost. Would the distance be vertical or horizontal? would it be (5-y) or (12-y)?
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  2. #2
    Member RedBarchetta's Avatar
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    Well, after working the problem I came up with 50625 ft-lb. Wrong. Ugh.

    I tried using a linear relationship. y=12/5y+12

    dV=9(12/5y+12) dy
    dV=108/5*y + 108

    I multiplied by the density of water(62.5 lb/ft^3). d=m/v then d*v=m.

    dV=1350y+6750 dy

    Now I integrate from zero to five.

    [675x^2+6750x](zero to five)=50625 ft-lb...
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  3. #3
    Member RedBarchetta's Avatar
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    Anyone?

    Thank you.
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  4. #4
    Member RedBarchetta's Avatar
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    ^^ I just entered your answer into my homework checker and it told me I was wrong.
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  5. #5
    Eater of Worlds
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    I am sorry, I didn't notice the spout. Let me get back. I am at work now and busy.
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  6. #6
    Eater of Worlds
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    I think what we forgot to do was multiply by x.

    By similar triangles, \frac{5}{12}=\frac{5-x}{y}, \;\ y=\frac{12}{5}(5-x)

    (62.5)(12/5)(9)=1350

    1350\int_{0}^{5}x(5-x)dx=28125 \;\ ft/lbs
    Last edited by galactus; May 14th 2008 at 12:08 PM.
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  7. #7
    Member RedBarchetta's Avatar
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    Yes! Finally, with one try left I get it. Thanks Galactus. It's quite the problem.
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  8. #8
    Eater of Worlds
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    That was correct then?. I thought so. I just forgot the x last time.
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