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Math Help - Expression for population in h hours

  1. #1
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    Expression for population in h hours

    Hi!

    Suppose that a population insects doubles every 5 hours. If you start with 6000 insects:

    Find an expression for the number of insects present in h hours.

    n(h) = n_0 e^\text{rh}

    n(h) = 6000e^{5\text{h}}

    \frac{n(h)}{6000} = e^{5\text{h}}

    \ln\left(\frac{n(h)}{6000}\right) = 5\text{h}

    \frac15\ln\left(\frac{n(h)}{6000}\right) = h
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  2. #2
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    Re: Expression for population in h hours

    If you start with 6000, and every 5 hours it doubles, then when h = 5, n = 12 000. So

    \displaystyle \begin{align*} 12\,000 &= 6000 \, e^{5r} \\ 2 &= e^{5r} \\ \ln{(2)} &= 5r \\ \frac{1}{5}\ln{(2)} &= r \end{align*}

    So the equation which gives you the number of insects present after h hours is

    \displaystyle \begin{align*} n(h) &= 6000 \, e^{\frac{h\ln{(2)}}{5}} \\ &= 6000 \, e^{\ln{\left( 2^{\frac{h}{5}} \right)}} \\ &= 6000 \left( 2^{\frac{h}{5}} \right) \end{align*}.
    Last edited by Prove It; April 29th 2013 at 02:49 AM.
    Thanks from Unreal
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