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 years old. (round to the nearest integer.)

Explanation:

Step1: Set up the equation

We know that $A = A_0e^{-0.000121t}$, and $A = 0.12A_0$. Substitute $A$ into the formula: $0.12A_0=A_0e^{-0.000121t}$. Since $A_0
eq0$, we can divide both sides by $A_0$ to get $0.12 = e^{-0.000121t}$.

Step2: Take the natural - logarithm of both sides

$\ln(0.12)=\ln(e^{-0.000121t})$. According to the property of logarithms $\ln(e^x)=x$, the right - hand side simplifies to $- 0.000121t$. So, $\ln(0.12)=-0.000121t$.

Step3: Solve for $t$

$t=\frac{\ln(0.12)}{-0.000121}$. We know that $\ln(0.12)\approx - 2.12026$. Then $t=\frac{-2.12026}{-0.000121}\approx17523$.

Answer:

$17523$