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Differential equation with sphere

  1. Apr 10, 2013 #1
    1. The problem statement, all variables and given/known data

    A spherical mothball of original radius 1.2 cm slowly
    evaporates such that 180 days later, its radius is only 1 cm.
    Physically, the rate of evaporation dr/dt is proportional to the
    surface area of the sphere. Determine a) the time required for
    the radius of a new mothball to shrink to 25 percent of its
    original radius, and b) the time required for the volume of a
    new mothball to become half of its original value.

    2. Relevant equations

    SA= 4 pi r^2

    3. The attempt at a solution

    I started by saying -dr/dt = c1 4 pi r^2

    where c1 is a constant of proportionality

    Then through separation of variables I found that

    1/r = 4 pi c1 t + c2

    after imposing the initial conditions I found c1=7.383*10^-5 and c2=.833

    so I have

    1/r = 4 pi (7.383*10^-5) t + .833

    and this gives me answers of
    1) 2694 yr
    2) 233.82 yr

    but these aren't right. Any ideas of where I went wrong?
  2. jcsd
  3. Apr 10, 2013 #2


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    2017 Award

    Staff: Mentor

    Your equation give the right values for t=0 and t=180, so I think you have some calculation error in the final steps.

    I seriously doubt that mothballs evaporate like that. It would mean that evaporation is proportional to the square of the surface area.
  4. Apr 10, 2013 #3
    Yeah, I think I might have the wrong initial equation, but I'm not sure of what it could be
  5. Apr 10, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper

    1/r = 4 pi (7.383*10^-5) t + .833 looks ok. Why are you giving the times in yrs? Don't you mean days? And I'd check those again.
  6. Apr 10, 2013 #5
    Wow. Don't I feel silly. I was working on several problems at once and I guess I forgot this one was in days, not years and it worked. Good eye! Also, Thanks a ton!
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