Answer:
t = 23136 years old
Explanation:
The intensity of radiation obeys the exponential decay law.
[tex]I(t)=I_{0}e^{-\lambda t}[/tex] (1)
Here:
λ is the radiation decay constant.
I is the intensity of radiation after t time.
I₀ is the initial intensity of radiation.
t is the time
Now, we know that the intensity of radiation is a 6% of what it would be in a freshly cut piece of wood, in other words:
[tex]I(t)=0.06I_{0}[/tex] (2)
Combining (2) with (1), we have:
[tex]0.06I_{0}=I_{0}e^{-\lambda t}[/tex]
[tex]0.06=e^{-\lambda t}[/tex] (3)
Now, we just need to solve (3) for t.
[tex] t=\frac{-ln(0.06)}{\lambda}[/tex] (4)
and [tex]\lambda = \frac{ln(2)}{t_{1/2}}[/tex] (5)
[tex]t_{1/2}[/tex] is the half-life.
In our case, we have ¹⁴C, so the [tex]t_{1/2} = 5700 y[/tex]
Finally, we can find t putting (5) in (4):
[tex] t=\frac{-t_{1/2}ln(0.06)}{ln(2)} = 23136 y[/tex]
I hope it helps you!
