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The approximate age of the fossil is 507.36 million years.
The amount of an isotope ([tex]n[/tex]) decays exponentially in time ([tex]t[/tex]), in years, whose formula is presented below:
[tex]n = n_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (1)
Where:
The time constant ([tex]\tau[/tex]), in years, is calculated in terms of half-life ([tex]t_{1/2}[/tex]), in years:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (2)
If we know that [tex]\frac{n}{n_{o}} = 0.06[/tex] and [tex]t_{1/2} = 125000000\,yr[/tex], then the age of the fossil is:
[tex]\tau = \frac{125000000\,yr}{\ln 2}[/tex]
[tex]\tau \approx 180336880.1\,yr[/tex]
[tex]t = -\tau \cdot \ln \frac{n}{n_{o}}[/tex]
[tex]t = -180336880.1\cdot \ln 0.06[/tex]
[tex]t \approx 507361711.1\,yr[/tex]
The approximate age of the fossil is 507.36 million years. [tex]\blacksquare[/tex]
The statement is incomplete, complete form is shown below:
A radioactive isotope has a half-life of 125,000,000 years. If 6 % of the original radioactive isotope X remains in a sample of your fossil, approximately how old is the fossil?
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