To determine the ages of the meteorites, follow these steps:
1. Gather Information:
The given half-life of potassium-40 is 1.3 billion years.
2. Formula:
The age of the meteorite can be calculated using the formula:
[tex]\[
t = \frac{\text{half-life} \times \log \left(\frac{\text{initial amount}}{\text{final amount}}\right)}{\log(2)}
\][/tex]
where:
- [tex]\( t \)[/tex] is the age of the meteorite,
- [tex]\(\text{half-life}\)[/tex] is the half-life of potassium-40,
- [tex]\(\text{initial amount}\)[/tex] is the initial amount of potassium-40,
- [tex]\(\text{final amount}\)[/tex] is the final amount of potassium-40.
3. Find the Age of Meteorite 1:
- Initial Amount: 30 grams
- Final Amount: 7.5 grams
Substituting into the formula:
[tex]\[
t = \frac{1.3 \times \log \left(\frac{30}{7.5}\right)}{\log(2)}
\][/tex]
After calculating, we can conclude that the age of meteorite 1 is [tex]\( t \approx 2.6 \)[/tex] billion years.
4. Find the Age of Meteorite 2:
- Initial Amount: 80 grams
- Final Amount: 5 grams
Substituting into the formula:
[tex]\[
t = \frac{1.3 \times \log \left(\frac{80}{5}\right)}{\log(2)}
\][/tex]
After calculating, we can conclude that the age of meteorite 2 is [tex]\( t \approx 5.2 \)[/tex] billion years.
5. Find the Age of Meteorite 3:
- Initial Amount: 100 grams
- Final Amount: 12.5 grams
Substituting into the formula:
[tex]\[
t = \frac{1.3 \times \log \left(\frac{100}{12.5}\right)}{\log(2)}
\][/tex]
After calculating, we can conclude that the age of meteorite 3 is [tex]\( t \approx 3.9 \)[/tex] billion years.
Therefore, the ages of the meteorites are:
- Meteorite 1: [tex]\( \boxed{2.6} \)[/tex] billion years
- Meteorite 2: [tex]\( \boxed{5.2} \)[/tex] billion years
- Meteorite 3: [tex]\( \boxed{3.9} \)[/tex] billion years
