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Diff. Eq. Problem

  1. Sep 28, 2005 #1
    A symmetrical tank of height 15 feet, as shown in the figure on the next page, is leaking water through an aperture in its base. The aperture is a uniform circular hole with radius r_0 = 3 inches. The top and base radii of the tank are r_1 and r_2 respectively, where r_1 = r_2 = 2 feet. The curve passing through P1, P2 and P3 represents the arc of a parabola i.e. y = a + bx + cx^2 where a, b and c are suitable constants (to be found). The radius of the tank at P2 is r_3 and at P3 is r_4. It is also known that the vertical height of P2 from the top of the tank is h_4 such that h_4 = h_1/2.

    a) Derive the height h of the water in the tank at any time t.

    b) If the initial level of water in the tank is h_0, how long will it take to empty the whole tank?

    c) How long will it take for half the water to leak out from the tank given the initial level h_0?

    ASCII drawing of the tank:

    ************------------ <-- P1, h_1 = height from P1 to P3, rad.=r_1
    ***********/**********\ This region is a parabolic curve outwards.
    *********/**************\<-- P2, ht. h_4 = 0.5 * h_1, rad. = r_3
    ***---**********************--- <-- P3, rad. = r_4, ht. from top h_1
    ***|*************************| ^
    ***|*************************| | <-- height is h_2
    ***|*************************| V
    ****\***********************/*| <-- height is h_3
    *****\*********************/**| This region is a linear inwards.
    *******---------****---------- <-- bottom radius = r_2
    **************^^^ radius of hole/aperture in bottom = r_0

    The height values and radius values are:
    h_0 = 10 feet
    h_1 = 11 feet
    h_2 = 2 feet
    h_3 = 2 feet
    r_3 = 3.5 feet
    r_4 = 5 feet

    Any help and guidance on this problem would be greatly appreciated. :smile:
  2. jcsd
  3. Sep 29, 2005 #2


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    Science Advisor
    Homework Helper

    I'd write a computer program.

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