Answer:
The wooden arrow is [tex]t = 4793 \ years[/tex] old
Explanation:
From the question we are told that
The ratio of carbon -14 to carbon- 12 is [tex]n = 56.0[/tex]%
The half - life of carbon 14 is [tex]t_h = 5730 \ years[/tex]
Generally half-life is mathematically evaluated as
[tex]t_h = \frac{ln 2}{\lambda }[/tex]
Where [tex]\lambda[/tex] is the decay constant
making [tex]\lambda[/tex] the subject of the formula
[tex]\lambda = \frac{ln2 }{5730}[/tex]
Now the age of the wooden arrow can be mathematically obtained
[tex]t =\frac{1}{\lambda } * ln (\frac{Z_o}{Z} )[/tex]
The initial amount of [tex]Carbon -14[/tex] is [tex]Z_0 = 1[/tex]
The amount [tex]Carbon -14[/tex] remaining in the wooden arrow is
[tex]Z = 0.56[/tex]%
substituting values
[tex]t = \frac{5730}{ln 2} * ln(\frac{1}{0.56} )[/tex]
[tex]t = 4793 \ years[/tex]
