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Math Help - Need help finding exponential equation

  1. #1
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    Need help finding exponential equation

    A bacteria culture grows with constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000.

    (a) Find the initial population.

    (b) Find an expression for the population after t hours.

    (c) Find the rate of growth after 3 hours.

    Please help me with this problem. Also explain your steps and show your work. Thank you.
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  2. #2
    MHF Contributor harish21's Avatar
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    Your population growth is given by:

     P(t) = xt^n

    according to the question:

    P(2)= x2^n = 400........(I)

    P(8) = x8^n = 50,000.....(II)

    Divide (II)/(I) to find n, and then x

    after that, you can do (a),(b), and (c)
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  3. #3
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    Thanks for helping me. So I could you correctly, you mean divide it like this?

    50000 = x8^n
    400 = x2^n

    50000/400 = (8/2)^n?

    Don't we need to use the formula y(t) = P0e^kt in here?
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  4. #4
    MHF Contributor harish21's Avatar
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    Sorry I should have used that equation y(t) = P_{0}e^{kt}

    for this, you first need to find k, your rate..

    suppose P_{0} = 400

    then you have:

     50,000 = 400 e^{k(8-2)}

    using this, you can find k.....



    After finding k, you can find y_{0} using:

     50,000 = y_{0}e^{8k}
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  5. #5
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    Ah I see thank you for your help. May I ask one last question, how would we solve (c) Find the rate of growth after 3 hours? Do we need to find the derivative of y(t) = 80e^(0.80472t) in order to solve c? Thanks.
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  6. #6
    MHF Contributor Unknown008's Avatar
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    Yes

    Then you put t = 3
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