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Math Help - half life, growth rates

  1. #1
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    half life, growth rates

    A population doubles every 15 years. Determine the yearly growth rate

    A radioactive substance decays from 400 grams to 120 grams in 320 years. Determine it's half life



    Hey guys, as far as i know the above problems should be solvable with the formula A=I*r^(t/p)

    A being the final value
    I being the initial value
    r being the rate of growth/decay
    t being the time
    and p being the growth/decay period per rate

    It would be great if anyone can help. Thanks in advance
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by imppy725 View Post
    A population doubles every 15 years. Determine the yearly growth rate

    A radioactive substance decays from 400 grams to 120 grams in 320 years. Determine it's half life



    Hey guys, as far as i know the above problems should be solvable with the formula A=I*r^(t/p)

    A being the final value
    I being the initial value
    r being the rate of growth/decay
    t being the time
    and p being the growth/decay period per rate

    It would be great if anyone can help. Thanks in advance
    I believe the equation you should be using is:
    A = I*e^(rt)

    Where e is the natural number: e = 2.71828182846...

    For the first problem, we know that A = 2I when t = 15:
    2I = I*e^(15r)
    2 = e^(15r)
    ln2 = lne^(15r)
    ln2 = 15r
    r = ln2/15 = 0.0462

    For the second problem, we know that I = 400, A = 120 when t = 320:
    120 = 400*e^(320r)
    1/3 = e^(320r)
    ln(1/3) = 320r
    r = ln(1/3)/320

    Now, we want A = 1/2(I) = 1/2(400) = 200:
    200 = 400*e^(t*ln(1/3)/320)
    1/2 = e^(t*ln(1/3)/320)
    ln(1/2) = t*ln(1/3)/320
    t = 320*ln(1/2)/ln(1/3) = 201.8975
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