Results 1 to 3 of 3

Math Help - help with work

  1. #1
    Senior Member
    Joined
    Jul 2006
    From
    Shabu City
    Posts
    381

    help with work: Integral applications

    A hemispherical tank, placed so that the top is a circular region of radius 6ft, is filled with water to a depth of 4 ft. Find the work done in pumping the water to the top of the tank.

    Ans. 50,185 lb-ft
    Last edited by ^_^Engineer_Adam^_^; October 4th 2007 at 04:06 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2006
    From
    Shabu City
    Posts
    381
    does anyon know how to solve this problem?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Work = weight * (delta height)

    weight = volume * density

    density of water is 62.4 lbs/cu.ft.

    Hemisphere.
    Diameter = 2(6)
    If the origin (0,0) is the center of the circle of the hemisphere, the equation of the circle is
    x^2 +y^2 = 6^2 --------------(i)

    Our infinitesimal volume, dV--your rectangular element-- is a horizontal disc whose radius is x, and whose thickness is dy.
    So, dV = pi(x^2) *dy
    Its weight, dW, is
    dW = (pi(x^2)*dy)(62.4) = (62.4pi)(x^2)dy

    To raise this dW to the top of the bowl, the work done is, say, d(work).
    d(work) = dW * (-y) <-----negative because the the y's below (0,0) are negative.

    So,
    d(work) = [(62.4pi)(x^2)dy]*(-y)

    From (i), x^2 = 36 -y^2, so,
    d(work) = [(62.4pi)(36 -y^2)dy]*(-y)
    d(work) = (62.4pi)[y^3 -36y]dy

    Because the water is 4ft deep, then the integration with dy is from y = -6 to y = -2.
    Hence,
    Work = (62.4pi)INT.(-6 --> -2)[y^3 -36y]dy
    Work = (62.4pi)[(1/4)(y^4) -18y^2]|(-6 --> -2)
    Work = (62.4pi)[{(1/4)(-2)^4 -18(-2)^2} - {(1/4)(-6)^4 -18(-6)^2}]
    Work = (62.4pi)[{4 -72} - {324 -648}]
    Work = (62.4pi)[4 -72 -324 +648]
    Work = (62.4pi)[256]
    Work = 50,185 ft-lbs. ---------------answer.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Work--please check my work for me
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 1st 2011, 09:02 AM
  2. Replies: 1
    Last Post: July 21st 2010, 06:02 PM
  3. Replies: 1
    Last Post: December 4th 2007, 03:45 PM
  4. work
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 20th 2007, 01:38 PM
  5. Please!!! Help me work it out!
    Posted in the Geometry Forum
    Replies: 3
    Last Post: April 30th 2007, 12:00 PM

Search Tags


/mathhelpforum @mathhelpforum