Given half life of ${}^{14}\mathrm{C}= 5730$ years.

$\mathrm{\lambda t}=\ln \frac{{\left({}^{14}\mathrm{C}/{}^{12}\mathrm{C}\right)}_{\text{atmosphere }}}{{\left({}^{14}\mathrm{C}/{}^{12}\mathrm{C}\right)}_{\text{wood sample }}}$

As per the question,

$\frac{{\left({}^{14}\mathrm{C}/{}^{12}\mathrm{C}\right)}_{\mathrm{wood}}}{{\left({}^{14}\mathrm{C}/{}^{12}\mathrm{C}\right)}_{\mathrm{atmosphere}}}=\frac{1}{8}$

So, $\mathrm{\lambda t}=\ln 8$

$\frac{\ln 2}{{\mathrm{t}}_{1/2}}\mathrm{t}=\ln 8$

$$\begin{gathered} \mathrm{t}=3\times {\mathrm{t}}_{1/2} \\ =3\times 5730 \\ =17190 \text{years} \end{gathered}$$