Carbon-14 is used to determine the age of organic material. The procedure is based on the…

\(\text { Carbon-14 }\) is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\({ }_{7}^{14} \mathrm{~N}+{ }_{0} \mathrm{n}^{1} \rightarrow{ }_{6}^{14} \mathrm{C}+{ }_{1} \mathrm{p}^{1}\) 
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls due to the decay which C-14 underoges
\({ }_{6}^{14} \mathrm{C} \rightarrow{ }_{7}^{14} \mathrm{~N}+\beta^{-}\) 
The half-life period of \({ }^{14} \mathrm{C}\) is \(5770 \mathrm{yr}\).
The decay constant (λ) can be calculated by using the following formula \[\ddot{e}=\frac{0.693}{t_{\frac{1}{2}}}\]
The comparison of the \(\hat{\mathrm{a}}^{-}\) activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than \(30,000 \mathrm{yr}\). The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\). (2006)
A nuclear explosion has taken place leading to increase in concentration of C¹⁴ in nearby areas. C¹⁴ concentration is C_l in nearby areas and C₂ in areas far away. If the age of the fossil is determined to be T₁ and T₂ at the places respectively then