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Differential Equations - Related Rate

  1. Sep 9, 2008 #1
    1. The problem statement, all variables and given/known data

    Assume that a typical raindrop is spherical. Starting at some time, which we designate as t = 0, the raindrop of radius r sub o falls from rest from a cloud and begins to evaporate.

    a) If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates - that is, the rate at which it loses mass - is proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t).

    There is a b part but I think I understand how that is done.

    2. Relevant equations
    Surface Area of a Sphere is 4*pi*r2

    3. The attempt at a solution
    -d(m)/dt is proportional to 4*pi*r2


    dm/dt = -k*4*pi*r2

    Where k is a constant to remove the proportionality.

    if dr/dt = constant represented by c

    then after we integrate that equation we get

    r(t) = c1*t + c2

    After that I did some really weird math that I don't think is possible. Any ideas?
  2. jcsd
  3. Sep 9, 2008 #2


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    Homework Helper

    Mass is density (rho) times volume. So you have (rho*(4/3)*pi*r(t)^3)'=k*4*pi*r(t)^2. Can you show that implies r'(t) is constant?
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