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Application of Integrals

  1. Jun 29, 2009 #1
    1. The problem statement, all variables and given/known data

    A tank in the form of a hemispherical bowl of radius 4 m is full of water. Find the work required to pump all of the water to a point 2 m above the tank.


    2. Relevant equations

    w = fd = mgd
    density = 1000L/m2
    1L = 1kg

    3. The attempt at a solution

    x = height

    w = V*d*g*h
    = (π)(r^2)(x)*(1000)*(9.8)*(x)*dx

    I think my problem is trying to show how the height a strip of cylinder varies with the radius. I've tried using similar triangles but it was obviously incorrect. Is there a similar property I can use for hemispheres?
     
  2. jcsd
  3. Jun 29, 2009 #2

    Mark44

    Staff: Mentor

    Think about this in terms of horizontal layers of water. It's probably most convenient to put the origin at the top level of the hemispherical tank, so that [itex]\Delta y[/itex] ranges from -4 to 0. Each layer has to be lifted (pumped) from its position in the tank to a point 2 m. above the top of the tank. You are using x in your integral expression; I would use y, and I would also find the equation of the semicircular cross section of the tank so that I could exploit a relationship between x and y at a point (x, y) on the semicircular boundary of the tank. A sketch will be very helpful if you haven't already done one. A drawing of a half-circle will do just fine, as long as you realize that the water layers are three-dimensional.
     
  4. Jul 1, 2009 #3
    w = V*d*g*h <--new

    V = pi*r^2*dh
    r^2 = 8h-h^2
    d = 1000
    g = 9.8
    h = 6-h

    V = 9800pi ∫ [(8h-h^2)*(6-h)] dh
    V = 9800pi*(24h^2 - (14/3)h^3 + (1/4)h^4)
    V ~ 1463466pi Joules

    This is the first time I've done a question like this and the energy seems a little high to me, is it correct?

    I know I can't check it with the potential energy equation because you need different amounts of energy to lift the water at different heights, is there anyway I can check that this is correct?
     
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