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Thread: Exponential Function

  1. #1
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    Exponential Function

    Hi,

    If dX/dt=X/2 and X=6 million when t=5, how do you find X when t=10.

    Any help would be appreciated.

    Thank you!
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  2. #2
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    Re: Exponential Function

    $\dfrac{dx}{dt} = \dfrac x 2$

    $\dfrac {dx}{x} = \dfrac{dt}{2}$

    $\ln(x) = \dfrac t 2 + C$

    $x(t) = x_0 e^{t/2},~x_0= e^C$

    $x(5)=6000000 = x_0 e^{5/2}$

    $x_0 =6000000e^{-5/2}$

    $x(t) = 6000000e^{-5/2} e^{t/2} = 6000000e^{\frac{t-5}{2}}$
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  3. #3
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    Re: Exponential Function

    $\dfrac{dx}{dt} = kx \implies x=x_0e^{kt}$

    $x_{10}=x_0e^{10k}$
    ---------------
    $6 = x_0e^{5k}$

    $\dfrac{x_{10}}{6}=e^{5k} \implies x_{10}=6e^{5/2} \approx 73.1 \text{ million}$
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