To determine the age of the rock based on the given sample containing the elements Lokuim and its daughter product DOL, you can use the principles of radiometric dating, specifically the decay of Lokuim into DOL.
Steps to Find the Age of the Rock:
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Understand the Relationship Between Parent and Daughter Isotopes: Lokuim decays into DOL over time. In your sample, you have 25% Lokuim (the parent isotope) and 75% DOL (the daughter isotope). Since Lokuim transforms into DOL, you know that the amount of Lokuim has decreased as it has decayed into DOL.
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Determine the Ratios: Let’s define:
- \( N_0 \): Initial amount of Lokuim
- \( N \): Current amount of Lokuim = 25% of the sample
- \( D \): Current amount of DOL = 75% of the sample
The relationship can be expressed as: \[ D = N_0 - N \] Given that \( D = 75% \) and \( N = 25% \): \[ 75% = N_0 - 25% \] Thus, \[ N_0 = 100% \]
So, the initial amount of Lokuim was 100%, and it has now decayed to 25%. Therefore, 75% of the original Lokuim has decayed into DOL.
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Calculate the Amount of Time Passed: To find the age of the rock, we need to find out how many half-lives have passed. Since 75% of the original Lokuim has decayed, that means that 25% remains. In terms of half-lives, we can relate this to the formula for radioactive decay: \[ N = N_0 \left( \frac{1}{2} \right)^n \] Where \( n \) is the number of half-lives.
Rearranging this gives: \[ \left( \frac{1}{2} \right)^n = \frac{N}{N_0} \] Substituting in the percentages gives us: \[ \left( \frac{1}{2} \right)^n = \frac{0.25}{1} = 0.25 \]
Recognizing that \( 0.25 = \left( \frac{1}{2} \right)^2 \), we see that: \[ n = 2 \] This means two half-lives have passed.
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Calculate the Age of the Rock: Since the half-life of Lokuim is given as 3,500 years, we multiply the number of half-lives by the length of a half-life to find the age of the rock: \[ \text{Age} = n \times \text{Half-life} = 2 \times 3500 \text{ years} = 7000 \text{ years} \]
Therefore, the age of the rock is 7000 years.
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