Answer:
14873 years
Step-by-step explanation:
The exponential decay of the ¹⁴C can be expressed as follows:
[tex] N_{t} = N_{0}e^{(-\lambda t)} [/tex] (1)
where Nt: is the disintegration per minute per gram of carbon that has not yet decayed after a time t = 2.25, N₀: is the initial disintegration per minute per gram of carbon = 13.6 (at the time of death), λ: is the decay constant and t: is time.
The half-life [tex]t_{1/2}[/tex] of carbon-14 is related to the decay constant by the following equation:
[tex] t_{1/2} = \frac{ln(2)}{\lambda} [/tex]
[tex] \lambda = \frac{ln(2)}{t_{1/2}} [/tex] (2)
Hence, by entering equation (2) into (1) we have:
[tex] N_{t} = N_{0}e^{(-\frac{ln(2)}{t_{1/2}} t)} [/tex] (3)
Now, by solving equation (3) for t, we can find the age of the remnants:
[tex] ln(\frac{N_{t}}{N_{0}}) = -\frac{ln(2)}{t_{1/2}} t [/tex]
[tex] t = -\frac{ln(\frac{N_{t}}{N_{0}})}{ln(2)} t_{1/2} [/tex]
[tex] t = -\frac{ln(2.25/13.6)}{ln(2)} (5730 y) [/tex]
[tex] t = 14873 y [/tex]
Therefore, the age of the remnants is 14873 years.
I hope it helps you!
