The life of carbon-14 isotope is about 5 thousand years. But we still are able to detect traces of it in fossils which are older, more than 10 thousand years. Why?
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The half-life of $^{14}$C is $5730\pm40$ years. That means that after 5730 years, we would have 0.5 the original amount of it. After 11460 years we should have 0.25, and after 17190 years we should have 0.125, and after 22920 years we should have 0.0625. Since 6% of original should still be pretty observable, I am not sure what you are having problems with.
Nobody is saying that exactly after 5730 years, there will be no more $^{14}$C. Instead, we should have quite a lot left.
naturallyInconsistent
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