To determine how much of a 100 g sample of carbon-14 will be left after 1,000 years, we use the concept of exponential decay, which is typically modelled with the formula:
[tex]\[ \text{remaining\_mass} = \text{initial\_mass} \times \left(\frac{1}{2}\right)^{\frac{\text{elapsed\_time}}{\text{half\_life}}} \][/tex]
### Step-by-Step Solution:
1. Identify the parameters given:
- Initial mass ([tex]\(\text{initial\_mass}\)[/tex]): 100 grams
- Half-life of carbon-14 ([tex]\(\text{half\_life}\)[/tex]): 5730 years
- Elapsed time ([tex]\(\text{elapsed\_time}\)[/tex]): 1000 years
2. Apply the decay formula:
Substitute the values into the formula:
[tex]\[
\text{remaining\_mass} = 100 \times \left(\frac{1}{2}\right)^{\frac{1000}{5730}}
\][/tex]
3. Calculate the exponent:
Divide the elapsed time by the half-life:
[tex]\[
\frac{1000}{5730} \approx 0.1745
\][/tex]
4. Evaluate the power of 1/2:
Raise [tex]\( \frac{1}{2} \)[/tex] to the power of the result from step 3:
[tex]\[
\left(\frac{1}{2}\right)^{0.1745} \approx 0.88606
\][/tex]
5. Multiply by the initial mass:
Multiply this result by the initial mass of 100 grams:
[tex]\[
\text{remaining\_mass} = 100 \times 0.88606 \approx 88.606
\][/tex]
6. Round the result:
Round the result to the nearest hundredth:
[tex]\[
\text{remaining\_mass\_rounded} = 88.61 \text{ grams}
\][/tex]
### Conclusion
After 1,000 years, approximately [tex]\( 88.61 \)[/tex] grams of the original 100-gram sample of carbon-14 will remain.
