## Related questions with answers

Use the radiometric dating formula to answer the following question. Uranium- $238$ has a half-life of $4.5$ billion years. a. You find a rock containing a mixture of uranium-$238$ and lead. You determine that $60 \%$ of the original uranium- $238$ remains; the other $40 \%$ decayed into lead. How old is the rock? b. Analysis of another rock shows that it contains $55 \%$ of its original uranium-$238$; the other $45 \%$ decayed into lead. How old is the rock?

Solution

Verified$(a)$ Using the percentage of uranium which stayed in the original form and the half-life of uranium, we can calculate how old is the rock using the general expression for exponential decay

$\begin{align*} U&=U_0\cdot\dfrac12^{\frac{t}{T_{\text{half}}}}\\ 0.6U_0&=U_0\cdot\dfrac12^{\frac{t}{4.5}}\\ 0.6&=\dfrac12^{\frac{t}{4.5}}\\ \dfrac{t}{4.5}&=\log_{\frac12}0.6\\ t&=4.5\cdot\log_{\frac12}0.6\\ t&=3.3163\ \text{billion years} \end{align*}$

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