To determine how old the skeletons were in 1909, we will use the exponential decay model for carbon-14, which is given by:
[tex]\[ A = A_0 e^{-0.000424 t} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of carbon-14 remaining after time [tex]\( t \)[/tex],
- [tex]\( A_0 \)[/tex] is the initial amount of carbon-14,
- [tex]\( 0.000424 \)[/tex] is the decay constant,
- [tex]\( t \)[/tex] is the time in years.
Given data:
- The skeletons contain 86% of the expected amount of carbon-14 found in a living person. Thus, [tex]\( \frac{A}{A_0} = 0.86 \)[/tex].
We need to find the time [tex]\( t \)[/tex] that has passed since the skeletons contained the initial amount of carbon-14.
Firstly, let's set up the equation with the given information:
[tex]\[ 0.86 = e^{-0.000424 t} \][/tex]
To solve for [tex]\( t \)[/tex], we will take the natural logarithm (ln) of both sides:
[tex]\[ \ln(0.86) = \ln(e^{-0.000424 t}) \][/tex]
Using the property of logarithms, [tex]\(\ln(e^x) = x\)[/tex], this simplifies to:
[tex]\[ \ln(0.86) = -0.000424 t \][/tex]
Now, solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.86)}{-0.000424} \][/tex]
After calculating the above expression, we find:
[tex]\[ t \approx 356 \text{ years} \][/tex]
This means the skeletons were approximately 356 years old in 1989.
To find out how old the skeletons were in 1909, we calculate the difference in years between 1989 and 1909, and then add this difference to the age in 1989:
[tex]\[
\text{Difference in years} = 1989 - 1909 = 80 \text{ years}
\][/tex]
Thus, the age of the skeletons in 1909:
[tex]\[
\text{Age in 1909} = 356 + 80 = 436 \text{ years}
\][/tex]
Therefore, our answers are:
In 1989, the skeletons were 356 years old (rounded to the nearest integer).
In 1909, the skeletons were 436 years old (rounded to the nearest integer).
