To determine the amount of carbon-14 remaining in an artifact after 7186 years using the decay model [tex]\( A = 16 e^{-0.000121 t} \)[/tex], we can follow these steps:
1. Identify the given values:
- Initial amount of carbon-14, [tex]\( A_0 = 16 \)[/tex] grams
- Decay constant, [tex]\( k = -0.000121 \)[/tex]
- Time period, [tex]\( t = 7186 \)[/tex] years
2. Substitute the given values into the decay model:
The decay model is [tex]\( A = 16 e^{-0.000121 \cdot 7186} \)[/tex]
3. Calculate the exponent:
Compute the value of the exponent [tex]\( -0.000121 \cdot 7186 \)[/tex]:
[tex]\[
-0.000121 \cdot 7186 = -0.869306
\][/tex]
4. Evaluate the exponential function:
Calculate [tex]\( e^{-0.869306} \)[/tex]:
The value of [tex]\( e^{-0.869306} \)[/tex] approximately equals 0.419158.
5. Multiply by the initial amount:
Multiply the initial amount of carbon-14 by the value of the exponential function:
[tex]\[
16 \cdot 0.419158 \approx 6.7065
\][/tex]
6. Round to the nearest whole number:
Since the value [tex]\( 6.7065 \)[/tex] is closer to 7, we round it to the nearest whole number.
Therefore, the amount of carbon-14 present in 7186 years will be approximately 7 grams.
