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Differential calculus question

  1. Jul 8, 2014 #1
    1. The problem statement, all variables and given/known data
    The number of termites in a colony is increasing at a rate proportional to the number present on any day. If the number of termites increases by 25% in 100 days, how much longer (to the nearest day) will it be until the population is double the initial number?

    2. The attempt at a solution
    [itex]\frac{dT}{dt}[/itex]=at
    T=∫at
    =[itex]\frac{at^{2}}{2}[/itex] + c
    T(x)=T
    T(0)*[itex]\frac{5}{4}[/itex] = T(100)
    => a=[itex]\frac{c}{20000}[/itex]

    I think I am approaching this wrong. Help is appreciated,
    thanks.

    **UPDATE: I overlooked a something, I think I figured it out now. I got t=200, does that seem right?
     
    Last edited: Jul 8, 2014
  2. jcsd
  3. Jul 8, 2014 #2
    Your initial equation seems to be wrong. The rate of change of the population is proportional to the population (on the same day).
     
  4. Jul 8, 2014 #3

    Ray Vickson

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    Your expression ##T(t) = a t^2/2 + c## is incorrect; it does NOT satisfy the condition that the derivative ##dT/dt## is proportional to ##T## itself.
     
  5. Jul 8, 2014 #4

    HallsofIvy

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    ManiFresh and Ray Vickson's point is that this is not the correct equation. Saying that "the rate is proportional to the number present" is [itex]\frac{dT}{dt}= aT[/itex].
    The "T" on the right is the number of termites, not the time in days.

     
  6. Jul 8, 2014 #5
    So how do I solve it? I mean, are you saying the rate of change is dT/dt = aT?
     
  7. Jul 8, 2014 #6

    Ray Vickson

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    I cannot see how to answer this question without doing your problem for you. I suggest you look through your textbook or course notes to find an answer. You could also do a Google search under "growth models", for example.
     
  8. Jul 8, 2014 #7
    What is the derivative of ln(y) with respect to t, if y is a function of t?

    Chet
     
  9. Jul 8, 2014 #8

    HallsofIvy

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    You need to get both "dT" and "T" on the same side of the equation, "dt" on the other side.
     
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