To calculate the age of the rock containing Lokuim and its daughter isotope DOL, you can use the following steps based on the principles of radioactive decay and half-lives:
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Understand the Decay Relationship: Lokuim (L) decays into DOL. When the rock formed, it would have had only Lokuim, and over time, some of it decayed into DOL.
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Determine Initial and Current Amounts: Since the sample now has 25% Lokuim and 75% DOL, it indicates that for every part of Lokuim remaining, there has been a corresponding amount of DOL produced. Since DOL is produced by the decay of Lokuim, the remaining 25% L is equivalent to the initial amount of Lokuim minus the amount that has decayed.
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Use the Decay Formula: The relationship between remaining Lokuim (N), initial amount (N₀), and time (t) can be expressed as: \[ N = N_0 \cdot \left(0.5\right)^{t/T} \] where T is the half-life (3500 years). To find the initial amount (N₀) in this case, note that if 25% remains, we can say: \[ 0.25 N_0 = N \]
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Calculate the Age: Since 75% of the original Lokuim has decayed, we can set up the equation: \[ \dfrac{75}{100} = \left(0.5\right)^{t/3500} \] Simplifying gives: \[ 0.75 = \left(0.5\right)^{t/3500} \] Taking the logarithm of both sides: \[ \log(0.75) = \frac{t}{3500} \log(0.5) \] Rearranging gives: \[ t = 3500 \cdot \frac{\log(0.75)}{\log(0.5)} \] Using logarithm values, you can compute t, which will yield the age of the rock in years.
By following these steps, you will calculate the time (age) since the rock formed based on the remaining percentage of Lokuim and the half-life of isotopes involved.
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