An isotope has a half-life of 5,000 years, How long will it take to decay to 15% of its original quantity?

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An isotope has a half-life of 5,000 years, How long will it take to decay to 15% of its original quantity?

Mathematics
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\[f(t) = Ae ^{kt}\]\[k = \frac {-ln 2}{t _{\frac{1}{2}}} = \frac {-ln 2}{5000} = 0.000138629436 \]We'll let A = 100, so its easy to see the percent change.\[f(t) = 100e^{0.000138629436t} = 15\]\[.15 = e^{-0.000138629436t}\]\[t = \frac {\ln .15}{-0.000138629436} \approx 13685 years\]
The constant is negative in the exponential decay equation, I made a mistake when I first wrote the equation. \[f(t) = Ae^{-kt}\]
okay thank you. may i ask though how you found k?

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Exponential decay is first order, so there is a formula that relates the half life and the constant.\[t _{\frac{1}{2}}^{} = \frac {-\ln 2}{k}\]
okay, thank you very much!
Alright, no problem, good luck with your studies :)

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