Answer:
The age of the artifact is around 4384 years
Step-by-step explanation:
There is indeed missing information on the directions, there can be different values of the count per rate, but you can replace that information on the last step of the following procedure to get the result that you seek, so for the exercise we can use the following to complete the exercise:
"A wooden artifact from a 14C Chinese temple has a activity of 34.2 counts per minute as compared with an activity of 58.2 counts per minute for a standard of zero age."
For exponential decay models we can use the general equation,
[tex]N= N_0 e^{-kt}[/tex]
And the equation for the half life which is
[tex]t_{1/2}= \cfrac{\ln 2}{k}[/tex]
Thus solving for the decay rate k we get
[tex]k = \cfrac{\ln 2 }{t_{1/2}}[/tex]
Replacing the given information we have
[tex]k = \cfrac{\ln 2 }{5715 \,years}[/tex]
So the decay rate is
[tex]k=1.21286 \times 10^{-4} \,years^{-1}[/tex]
Determining the age of the artifact.
We can then use main decay exponential model and solve for time t.
[tex]N= N_0 e^{-kt}[/tex]
Dividing both sides by [tex]N_0[/tex]
[tex]\cfrac{N}{N_0}= e^{-kt}[/tex]
And we can apply logarithms to both sides
[tex]\ln \left( \cfrac{N}{N_0}\right)=\ln e^{-kt}[/tex]
And we can simplify the logarithm with the exponential function.
[tex]\ln \left( \cfrac{N}{N_0}\right)=-kt[/tex]
And solving for the time.
[tex]t= \cfrac{\ln \left( \cfrac{N}{N_0}\right)}{-k}[/tex]
We can finally replace the given information about the counts per minute as well as the decay constant k previously found.
[tex]t= \cfrac{\ln \left( \cfrac{34.2}{58.2}\right)}{-1.21286\times 10^{-4}\, years^{-1}}[/tex]
Thus we get
[tex]\bold{t = 4384 \, years}[/tex]
The artifact is around 4384 years old.
