you are given the half-life of carbon-14 to be 5730 yrs
it will take 5730 yrs to decay to half
the decayed half will take another 5730 to decay to half of its current weight (which one-fourth of the initial mass)
that is 5730 yrs + 5730 yrs = 11460 yrs
The Other Way
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N = N₀e^(-t)
half-life = 5730 yrs, the time it will take N / N₀ to be half
deduce that N / N₀ = ½ for t = 5730 years
N = N₀e^(-t)
e^(-t) = N/ N₀
e^(-5730) = ½
e^(-5730) = 0.5
apply natural logs
-5730 = ln(0.5)
-5730 = -ln(2)
= (1/5730) ln(2)
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time it takes to reach one-fourth
e^(-t) = N/ N₀
e^(-t) = ¼
e^(-(t/5730) ln(2) ) = ¼
e^(-(t/5730) ln(2) ) = ¼
apply natural logs
-(t/5730) ln(2) = -ln(4)
(t/5730) ln(2) = ln(4)
t = 5730 ln(4) / ln(2)
t = 5730 (2)
t = 11460 yrs
