Question

prehistoric cave paintings were discovered in a cave in france. the paint contained 12% of the original carbon - 14. use the exponential decay model for carbon - 14, ( a = a_0e^{-0.000121t} ), to estimate the age of the paintings. the paintings are approximately ( square ) years old. (round to the nearest integer.)

Explanation:

Step1: Set up the equation

Given the formula \(A = A_0e^{- 0.000121t}\), and since the paint contains \(12\%\) of the original carbon - 14, we have \(A=0.12A_0\). Substitute \(A = 0.12A_0\) into the formula:
\(0.12A_0=A_0e^{-0.000121t}\)
Divide both sides by \(A_0\) (since \(A_0
eq0\)), we get \(0.12 = e^{-0.000121t}\)

Step2: Take the natural logarithm of both sides

Using the property \(\ln(e^{x})=x\), take the natural logarithm of both sides of the equation \(0.12 = e^{-0.000121t}\).
\(\ln(0.12)=\ln(e^{-0.000121t})\)
Since \(\ln(e^{-0.000121t})=- 0.000121t\), our equation becomes \(\ln(0.12)=-0.000121t\)

Step3: Solve for \(t\)

We know that \(\ln(0.12)\approx - 2.1203\). Then, \(t=\frac{\ln(0.12)}{-0.000121}\)
Substitute \(\ln(0.12)\approx - 2.1203\) into the formula: \(t=\frac{-2.1203}{-0.000121}\)
\(t\approx17523.14\)

Answer:

\(17523\)