To solve this problem, we need to estimate the age of prehistoric cave paintings, given that the paint still contains 8% of the original carbon-14. We will use the exponential decay model for carbon-14:
[tex]\[ A = A_0 e^{kt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the remaining amount of carbon-14 (8% in this case).
- [tex]\( A_0 \)[/tex] is the original amount of carbon-14 (100% in this case).
- [tex]\( k \)[/tex] is the decay constant ([tex]\(0.000121\)[/tex] per year).
- [tex]\( t \)[/tex] is the time that has passed, which is what we want to find.
First, we start by rearranging the decay formula to solve for [tex]\( t \)[/tex]:
[tex]\[ A = A_0 e^{kt} \][/tex]
[tex]\[ \frac{A}{A_0} = e^{kt} \][/tex]
[tex]\[ \ln\left(\frac{A}{A_0}\right) = kt \][/tex]
[tex]\[ t = \frac{1}{k} \ln\left(\frac{A_0}{A}\right) \][/tex]
Plugging the given values into the equation:
- [tex]\( A_0 = 100 \)[/tex] (original amount of carbon-14)
- [tex]\( A = 8 \)[/tex] (remaining amount of carbon-14)
- [tex]\( k = 0.000121 \)[/tex]
The formula now becomes:
[tex]\[ t = \frac{1}{0.000121} \ln\left(\frac{100}{8}\right) \][/tex]
Now, let's calculate this step-by-step:
1. Calculate the ratio [tex]\( \frac{100}{8} \)[/tex]:
[tex]\[ \frac{100}{8} = 12.5 \][/tex]
2. Calculate the natural logarithm of 12.5:
[tex]\[ \ln(12.5) \approx 2.5257 \][/tex]
3. Divide the result by the decay constant:
[tex]\[ t = \frac{2.5257}{0.000121} \approx 20873.79 \][/tex]
4. Round the result to the nearest integer to get the estimated age:
[tex]\[ t \approx 20874 \][/tex]
So, the prehistoric cave paintings are approximately 20,874 years old.
