To determine the percentage of the remaining sample of Carbon-14 ([tex]\(_6^{14}C\)[/tex]) that has a half-life of 5730 years and is estimated to be 16,230 years old, you can follow these steps:
1. Understanding the Concept of Half-Life:
- The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay.
- For Carbon-14, the half-life is 5730 years, meaning every 5730 years, half of the remaining Carbon-14 decays.
2. Formula for Remaining Percentage:
- The general formula to calculate the remaining percentage of a substance after a certain amount of time is:
[tex]\[
\text{Remaining Percentage} = \left( \frac{1}{2} \right)^{\frac{\text{time}}{\text{half-life}}} \times 100
\][/tex]
- In this formula, "time" is the age of the sample and "half-life" is the half-life of the substance.
3. Substitute the Given Values:
- Age of the sample (time): 16,230 years.
- Half-life of Carbon-14: 5730 years.
Substitute these values into the formula:
[tex]\[
\text{Remaining Percentage} = \left( \frac{1}{2} \right)^{\frac{16230}{5730}} \times 100
\][/tex]
4. Simplifying the Exponent:
- Calculate the exponent:
[tex]\[
\frac{16230}{5730} = 2.833 (approximately)
\][/tex]
So the formula becomes:
[tex]\[
\text{Remaining Percentage} = \left( \frac{1}{2} \right)^{2.833} \times 100
\][/tex]
5. Evaluate the Power:
- Calculate [tex]\(\left( \frac{1}{2} \right)^{2.833}\)[/tex]. This value is approximately 0.14039.
6. Convert to Percentage:
- Multiply by 100 to convert the decimal to a percentage:
[tex]\[
0.14039 \times 100 = 14.039
\][/tex]
Therefore, the percentage of remaining Carbon-14 in the sample estimated to be 16,230 years old is approximately 14.039%.
