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ODE bacterial growth problem

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data

    A cell of some bacteria divides into two cells every 40 minutes. The initial population is 2 bacteria.

    a)Find the size of the population after t hours
    b) Find the size of the population after 6 hours.
    c) When will the population reach 12?

    2. Relevant equations

    None given.

    3. The attempt at a solution

    taking t = time in hours, y(t) = population,

    we say that

    dy/dt = (3/2)y --- the 3/2 converts from hours to the 40 min growth period
    dy/y = (3/2)dt
    ln(y) = (3/2)t + C --- C is arbitrary constant
    y = [e^( (3/2)t )] * L -- L is arbitrary constant analogous to e^C.

    we know that y(0) = 2, so to solve initial value problem,

    2 = 1*L

    therefore L = 2

    and the solution for part a) should be y(t) = 2 e^[(3/2)t]

    but this solution is unfortunately incorrect and I don't see how to fix it. Of course it holds for y(0) as it should, but it falls apart when I check other values. a friend mentioned something about "adding or subtracting a 1" somewhere but I don't see where I would do that, and even if I did I wouldn't understand the justification.

    Any help is appreciated, thanks!
  2. jcsd
  3. Feb 8, 2010 #2
    Look at your ODE. It's missing something. The population _DOUBLES_ every 40 minutes. You seem to have forgotten about this fact.

    A suggestion: convert to hours at the very end.
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