• # question_answer Half lives of two isotopes X and Y of a material are known to be $2\times {{10}^{9}}$ years and $4\times {{10}^{9}}$ years respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be A) $2\times {{10}^{9}}\,years$ B) $4\times {{10}^{9}}\,years$ C) $6\times {{10}^{9}}\,years$ D) $8\times {{10}^{9}}\,years$

 ${{N}_{x}}=\frac{{{N}_{0}}}{2}.{{e}^{-{{\lambda }_{1}}t}}=0.2{{N}_{0}}\,\,;\,\,{{N}_{y}}=\frac{{{N}_{0}}}{2}.{{e}^{-{{\lambda }_{2}}t}}=0.8{{N}_{0}}$ ${{e}^{({{\lambda }_{1}}-{{\lambda }_{2}})}}=4$    $\Rightarrow t=8\times {{10}^{9}}\,years$