Scientists have estimated that the earth's crust was formed \(4.3\) billion years ago. The radioactive nuclide \({ }^{176} \mathrm{Lu}\), which decays to \({ }^{176} \mathrm{Hf}\), was used to estimate this age. The half- life of \({ }^{176} \mathrm{Lu}\) is 37 billion years. How are ratios of \({ }^{176} \mathrm{Lu}\) to \({ }^{176} \mathrm{Hf}\) utilized to date very old rocks?

The ratios of \({ }^{176} \mathrm{Lu}\) to \({ }^{176} \mathrm{Hf}\) are utilized to date very old rocks by first calculating the number of half-lives that have elapsed since the Earth's crust was formed. Then, the fraction of remaining \({ }^{176} \mathrm{Lu}\) is calculated through the formula, Fraction remaining = \((\frac{1}{2})^{\text{number of half-lives}}\). This ratio is then compared to the measured ratio in the rock samples, allowing scientists to deduce the time when the rock was formed and thus date the formation of the Earth's crust accurately. The formula for the \({ }^{176} \mathrm{Lu}\) to \({ }^{176} \mathrm{Hf}\) ratio is: Ratio = \(\frac{(\frac{1}{2})^{\frac{4.3}{37}}}{1 - (\frac{1}{2})^{\frac{4.3}{37}}}\)