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Answer:
The sample is about 34380 years old.
Explanation:
The amount of Carbon-14 mass diminishes exponentially in time, whose model is described below:
[tex]\frac{m(t)}{m_{o}} = e^{-\frac{t}{\tau} }[/tex] (1)
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (2)
Where:
[tex]m_{o}[/tex] - Initial mass, in grams.
[tex]m(t)[/tex] - Current mass, in grams.
[tex]t[/tex] - Time, in years.
[tex]\tau[/tex] - Time constant, in years.
[tex]t_{1/2}[/tex] - Half-life, in years.
If we know that [tex]\frac{m(t)}{m_{o}} = \frac{1.5625}{100}[/tex] and [tex]t_{1/2} = 5730\,yr[/tex], then the age of the sample is:
[tex]\tau = \frac{5730\,yr}{\ln 2}[/tex]
[tex]\tau = 8266.643\,yr[/tex]
[tex]t = - \tau \cdot \ln \frac{m(t)}{m_{o}}[/tex]
[tex]t = - (8266.643\,yr)\cdot \ln \frac{1.5625}{100}[/tex]
[tex]t \approx 34380.002\,yr[/tex]
The sample is about 34380 years old.