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Math Help - Norwegian Integral Highschool task

  1. #1
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    Norwegian Integral Highschool task

    " A germpopulation have on a given time 10^5 germ. N = germnumber after h, hours and assume that the growingspeed is given with N`= 0,15N(1-10^-6N)."

    a) Show that the diff.equation is separable.

    b) Find N(h) and draw the graph.

    c) Find the germnumber after 5 hour.

    d) When do the germnuber grow fastest?

    e) What are the name of this type of modell? (function/growingmoddel)
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  2. #2
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    dN/dh = 0,15N(1-10^-6N)

    It is fairly simple to separate

    dN/(0,15N(1-10^-6N)) = dh

    Integrate by parts.

    Once you Have N(h) c is easy

    use dN/dh to answer d

    This is an example of a logistic differential equation
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  3. #3
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    Quote Originally Posted by DELLARMADA View Post
    " A germpopulation have on a given time 10^5 germ. N = germnumber after h, hours and assume that the growingspeed is given with N`= 0,15N(1-10^-6N)."

    a) Show that the diff.equation is separable.

    b) Find N(h) and draw the graph.

    c) Find the germnumber after 5 hour.

    d) When do the germnuber grow fastest?

    e) What are the name of this type of modell? (function/growingmoddel)
    k = 1.5 \times 10^{-7}

    M = 10^6


    \frac{dN}{dh} = kN(M - N)

    \frac{dN}{N(M-N)} = k \, dh

    \frac{1}{M} \left(\frac{1}{N} + \frac{1}{M-N}\right) \, dN = k \, dh

    \int \left(\frac{1}{N} - \frac{-1}{M-N}\right) \, dN = Mk \int \, dh

    \ln{N} - \ln(M-N) = MKh + C

    \ln\left(\frac{N}{M-N}\right) = Mkh + C

    \frac{N}{M-N} = Ae^{Mkh}

    N = \frac{M}{1 + Be^{-Mkh}}

    10^5 = \frac{10^6}{1 + B}

    N = \frac{M}{1 + 9e^{-Mkh}}
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  4. #4
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    Thanks !

    but can you please show me d) ?
    Last edited by DELLARMADA; April 22nd 2009 at 04:17 AM.
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  5. #5
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    Quote Originally Posted by DELLARMADA View Post
    Thanks !

    but can you please show me d) ?
    Find the t-coordinate of the point of inflection of N.
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  6. #6
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    N`= 0,15N(1-10^-6N)

    N`= 0,15N-1,5*10^-7N^2

    N``= 0,15-3*10^-7N

    N``= 0

    0,15 = 3*10^-7N

    N = 500000

    Grow fastes when N(t) = 500000


    Is this right?
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