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The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 76.17% of their carbon-14. How old were the bones at the time they were discovered?
Solution
Hi.
The way to solve this problem is to use a very special formula:
A = a0(0.5)t/h
A = amount now
a0 = starting amount
t = time
h = half-life
If 76.17% of the parent isotope (carbon-14) had decayed to the daughter element, then 23.83% of the parent isotope remained.
0.2383a0 = a0(0.5)t/5750 divide both sides by a0
0.2383 = (0.5)t/5750 take natural log of both sides
ln 0.2383 = (t/5750) ln 0.5 calculate
-1.434 ≈ (t/5750)(-0.693) divide both sides by -0.693
2.07 ≈ t/5750 multiply both sides by 5730
11,900.1 ≈ t
The mastodon bones are approximately eleven thousand nine hundred point one years old.
By the way, that calculation of the bones’ age would make sense because mastodons lived beginning during the Miocene and Pliocene epochs and until they went extinct at the end of the Pleistocene epoch, about 10,000-11,000 years ago.