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23 September, 00:24

A certain radioactive isotope decays at a rate of 2% per 100 years. if t represents time in years and y represents the amount of the isotope left then the equation for the situation is y=y0e-0.0002t. in how many years will there be 89% of the isotope left? round to the nearest year.

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  1. 23 September, 00:35
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    We rewrite the equation:

    y = y0 * e ^ ( - 0.0002 * t)

    In this equation:

    I = represents the initial amount of the isotope

    We have then:

    89% of the isotope left:

    0.89 * y0 = y0 * e ^ ( - 0.0002 * t)

    We clear the time:

    e ^ ( - 0.0002 * t) = 0.89

    Ln (e ^ ( - 0.0002 * t)) = Ln (0.89)

    -0.0002 * t = Ln (0.89)

    t = Ln (0.89) / ( - 0.0002)

    t = 582.6690813

    round to the nearest year:

    t = 583 years

    Answer:

    There will be 89% of the isotope left in а bout:

    t = 583 years
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