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Part 1
Prehistoric cave paintings were discovered in a cave in France. The paint contained 22 % 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.)

To estimate the age of the prehistoric cave paintings, we will use the exponential decay model given:

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

where:
- \( A \) is the remaining amount of carbon-14,
- \( A_0 \) is the original amount of carbon-14,
- \( t \) is the time in years,
- \( 0.000121 \) is the decay constant for carbon-14.

We know that 22% of the original carbon-14 remains, so \( \frac{A}{A_0} = 0.22 \).

Plugging this into the formula, we get:

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

To solve for \( t \), we take the natural logarithm (ln) of both sides:

\[
\ln(0.22) = \ln(e^{-0.000121 t})
\]

Simplifying using the property \( \ln(e^x) = x \), we get:

\[
\ln(0.22) = -0.000121 t
\]

Solving for \( t \):

\[
t = \frac{\ln(0.22)}{-0.000121}
\]

Now, I'll calculate the value of \( t \).It seems there is a technical issue preventing the calculation. Let's manually go through the steps:

1. Calculate \(\ln(0.22)\):
   \[
   \ln(0.22) \approx -1.5141
   \]

2. Plug this value into the formula for \( t \):
   \[
   t = \frac{-1.5141}{-0.000121} \approx 12516.5
   \]

Rounding this to the nearest integer, the age of the paintings is approximately **12,517 years old**.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the half-life of carbon-14 and how is it related to its decay constant?
2. How accurate is carbon dating, and what factors could affect its precision?
3. What other radioactive elements are used in dating ancient artifacts?
4. Can you explain the process of radioactive decay in simple terms?
5. What are some other methods used to date prehistoric paintings or artifacts?

**Tip:** For exponential decay problems, remember that taking the natural logarithm (ln) helps isolate the variable in the exponent, making the problem easier to solve.

Prehistoric cave paintings were discovered in a cave in France. The paint contained 22% of the original carbon-14. Use the exponential decay model for carbon-14, A = A_0 e^(-0.000121t), to estimate the age of the paintings. The paintings are approximately ___ years old. (Round to the nearest integer.)

Use the carbon-14 exponential decay model to estimate the age of prehistoric cave paintings found in France. The paint contains 22% of the original carbon-14, and by applying logarithms, the age is approximately 12,517 years. Explore key concepts like exponential decay, radioactive decay, and logarithmic calculations. Suitable for high school students (Grades 10-12).

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