Question

Prehistoric cave paintings were discovered in a cave in France. The paint contained 31 % of the original​ carbon-14. Use the exponential decay model for​ carbon-14, Upper A equals Upper A 0 e Superscript negative 0.000121 t​, to estimate the age of the paintings.
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Part 1
The paintings are approximately
  
enter your response here years old. ​(Round to the nearest​ integer.)

Answer 1

To estimate the age of the prehistoric cave paintings using the exponential decay model for carbon-14, we start with the formula given:

\[
A = A_0 e^{-0.000121t}
\]

Where:
- \( A \) is the amount of carbon-14 remaining (31% of the original amount).
- \( A_0 \) is the original amount of carbon-14.
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm.

Since the paintings contain 31% of the original carbon-14, we can express:

\[
A = 0.31 A_0
\]

Substituting this into the model gives:

\[
0.31 A_0 = A_0 e^{-0.000121t}
\]

We can cancel \( A_0 \) from both sides (assuming \( A_0 \) is not zero):

\[
0.31 = e^{-0.000121t}
\]

Next, we take the natural logarithm of both sides:

\[
\ln(0.31) = -0.000121t
\]

Now we solve for \( t \):

\[
t = \frac{\ln(0.31)}{-0.000121}
\]

Calculating \( \ln(0.31) \):

\[
\ln(0.31) \approx -1.171
\]

Now substituting this into the equation for \( t \):

\[
t \approx \frac{-1.171}{-0.000121} \approx 9685.12
\]

Rounding to the nearest integer gives:

\[
t \approx 9685 \text{ years}
\]

Therefore, the paintings are approximately **9685** years old.