Calculus

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[Exponential growth & decay]

The half-life of radioactive strontium-90 is approximately 29 years. In 1965, radioactive strontium-90 was released into the atmosphere during testing of nuclear weapons, and was absorbed into people's bones. How many years does it take until only 14 percent of the original amount absorbed remains?

time= ____ years

  • Calculus -

    Many math-teachers use base e for most exponential problems of this type, but since "half-life" is involved, let's use 1/2 as the base

    amount = a(1/2)^(t/k) or
    amount = a(2)^(-t/k) where a is the initial amount, t is the time in years, and k will be 29

    so we have to solve
    .14 = 1(2)^(-t/29(
    take the ln of both sides
    ln .14 = -t/29(ln 2), hope you know your log rules
    -t/29 = ln .14/ln 2
    -t/29 = -2.8365
    t = 82.26 years.

    check: after 29 years, we have 50% left
    after 58 years we have 25% left
    after 87 years we would have 12.5% left.

    our answer appears reasonable.

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