Question

prehistoric cave paintings were discovered in a cave in france. the paint contained 18% 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 (square) years old. (round to the nearest integer.)

Explanation:

Step1: Identify the formula and values

The exponential decay model is \( A = A_0 e^{-0.000121t} \), where \( A \) is the remaining amount, \( A_0 \) is the initial amount, and \( t \) is time (in years). We know \( \frac{A}{A_0} = 0.18 \) (since 18% remains).

Step2: Substitute into the formula

Substitute \( \frac{A}{A_0} = 0.18 \) into the equation: \( 0.18 = e^{-0.000121t} \)

Step3: Take natural logarithm of both sides

Take \( \ln \) of both sides: \( \ln(0.18) = \ln(e^{-0.000121t}) \)
Using the property \( \ln(e^x) = x \), we get: \( \ln(0.18) = -0.000121t \)

Step4: Solve for \( t \)

First, calculate \( \ln(0.18) \approx -1.7147 \)
Then, \( t = \frac{\ln(0.18)}{-0.000121} \approx \frac{-1.7147}{-0.000121} \approx 14171.07 \)

Answer:

14171