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Answer :
To determine the age of the rock using the given data, we'll apply the concept of radioactive decay. Here’s a step-by-step explanation:
1. Understand the Decay Process:
- Potassium-40 (K-40) decays into Argon-40 (Ar-40). The half-life of this decay process is 1.3 billion years.
2. Identify the Given Amounts:
- We have 25 grams of Potassium-40.
- We have 75 grams of Argon-40.
3. Determine the Initial Amount of Potassium-40:
- The Potassium-40 that has decayed turned into Argon-40. Therefore, the initial amount of Potassium-40 must have been the current amount of Potassium-40 plus the amount that has decayed into Argon-40.
- Thus, the initial amount of Potassium-40 was [tex]\( 25 \, \text{grams} + 75 \, \text{grams} = 100 \, \text{grams} \)[/tex].
4. Calculate the Number of Half-Lives Passed:
- With the initial amount of Potassium-40 being 100 grams and the current amount being 25 grams, we need to find out how many half-lives have passed.
- The fraction of the original Potassium-40 remaining is [tex]\( \frac{25}{100} = 0.25 \)[/tex].
- To find the number of half-lives that have passed, we use the formula:
[tex]\[
\left(\frac{1}{2}\right)^n = 0.25
\][/tex]
where [tex]\( n \)[/tex] is the number of half-lives.
- [tex]\( 0.25 \)[/tex] is actually [tex]\( \left(\frac{1}{2}\right)^2 \)[/tex], so [tex]\( n = 2 \)[/tex].
5. Calculate the Age of the Rock:
- Each half-life is 1.3 billion years. Since two half-lives have passed, the age of the rock is:
[tex]\[
2 \times 1.3 \, \text{billion years} = 2.6 \, \text{billion years}
\][/tex]
Therefore, the age of the rock is approximately 2.6 billion years.
1. Understand the Decay Process:
- Potassium-40 (K-40) decays into Argon-40 (Ar-40). The half-life of this decay process is 1.3 billion years.
2. Identify the Given Amounts:
- We have 25 grams of Potassium-40.
- We have 75 grams of Argon-40.
3. Determine the Initial Amount of Potassium-40:
- The Potassium-40 that has decayed turned into Argon-40. Therefore, the initial amount of Potassium-40 must have been the current amount of Potassium-40 plus the amount that has decayed into Argon-40.
- Thus, the initial amount of Potassium-40 was [tex]\( 25 \, \text{grams} + 75 \, \text{grams} = 100 \, \text{grams} \)[/tex].
4. Calculate the Number of Half-Lives Passed:
- With the initial amount of Potassium-40 being 100 grams and the current amount being 25 grams, we need to find out how many half-lives have passed.
- The fraction of the original Potassium-40 remaining is [tex]\( \frac{25}{100} = 0.25 \)[/tex].
- To find the number of half-lives that have passed, we use the formula:
[tex]\[
\left(\frac{1}{2}\right)^n = 0.25
\][/tex]
where [tex]\( n \)[/tex] is the number of half-lives.
- [tex]\( 0.25 \)[/tex] is actually [tex]\( \left(\frac{1}{2}\right)^2 \)[/tex], so [tex]\( n = 2 \)[/tex].
5. Calculate the Age of the Rock:
- Each half-life is 1.3 billion years. Since two half-lives have passed, the age of the rock is:
[tex]\[
2 \times 1.3 \, \text{billion years} = 2.6 \, \text{billion years}
\][/tex]
Therefore, the age of the rock is approximately 2.6 billion years.
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