To determine what percentage of Carbon-14 remains after 34,400 years, you'll need to follow these steps:
1. Understand the Definition of Half-Life:
The half-life is the time required for half of the radioactive substance to decay. In this case, the half-life of Carbon-14 is 5,730 years.
2. Calculate the Number of Half-Lives that Have Passed:
To find the number of half-lives that have passed, divide the total time elapsed by the half-life of the substance.
[tex]\[
\text{Number of half-lives} = \frac{\text{Time elapsed}}{\text{Half-life}} = \frac{34,400 \text{ years}}{5,730 \text{ years}} \approx 6.0035
\][/tex]
3. Understand the Exponential Decay Model:
With each half-life that passes, the amount of the substance reduces to half of its previous amount. Thus, after [tex]\( n \)[/tex] half-lives, the remaining percentage of the substance can be calculated using the formula:
[tex]\[
\text{Remaining percentage} = \left(0.5\right)^n \times 100
\][/tex]
where [tex]\( n \)[/tex] is the number of half-lives.
4. Calculate the Remaining Percentage:
Using the number of half-lives, you can find the remaining percentage of Carbon-14.
[tex]\[
\text{Remaining percentage} = \left(0.5\right)^{6.0035} \times 100 \approx 1.5587\%
\][/tex]
So, after approximately 34,400 years, about 1.5587% of the Carbon-14 remains.
