Answer:
The wood was cut approximately 8679 years ago.
Step-by-step explanation:
At first we assume that examination occured in 2020. The decay of radioactive isotopes are represented by the following ordinary differential equation:
(Eq. 1)
Where:
- First derivative of mass in time, measured in miligrams per year.
- Time constant, measured in years.
- Mass of the radioactive isotope, measured in miligrams.
Now we obtain the solution of this differential equation:
![\int {\frac{dm}{m} } = -\frac{1}{\tau}\int dt](https://tex.z-dn.net/?f=%5Cint%20%7B%5Cfrac%7Bdm%7D%7Bm%7D%20%7D%20%3D%20-%5Cfrac%7B1%7D%7B%5Ctau%7D%5Cint%20dt)
![\ln m = -\frac{1}{\tau} + C](https://tex.z-dn.net/?f=%5Cln%20m%20%3D%20-%5Cfrac%7B1%7D%7B%5Ctau%7D%20%2B%20C)
(Eq. 2)
Where:
- Initial mass of isotope, measured in miligrams.
- Time, measured in years.
And time is cleared within the equation:
![t = -\tau \cdot \ln \left[\frac{m(t)}{m_{o}} \right]](https://tex.z-dn.net/?f=t%20%3D%20-%5Ctau%20%5Ccdot%20%5Cln%20%5Cleft%5B%5Cfrac%7Bm%28t%29%7D%7Bm_%7Bo%7D%7D%20%5Cright%5D)
Then, time constant can be found as a function of half-life:
(Eq. 3)
If we know that
and
, then:
![\tau = \frac{5730\,yr}{\ln 2}](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%5Cfrac%7B5730%5C%2Cyr%7D%7B%5Cln%202%7D)
![\tau \approx 8266.643\,yr](https://tex.z-dn.net/?f=%5Ctau%20%5Capprox%208266.643%5C%2Cyr)
![t = -(8266.643\,yr)\cdot \ln 0.35](https://tex.z-dn.net/?f=t%20%3D%20-%288266.643%5C%2Cyr%29%5Ccdot%20%5Cln%200.35)
![t \approx 8678.505\,yr](https://tex.z-dn.net/?f=t%20%5Capprox%208678.505%5C%2Cyr)
The wood was cut approximately 8679 years ago.