Question

prehistoric cave - paintings were discovered in a cave in france. the paint contained 7% 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 decay - model equation

We know that the exponential decay model is $A = A_0e^{-0.000121t}$, where $A$ is the amount of carbon - 14 at time $t$, $A_0$ is the initial amount of carbon - 14. Given that $A = 0.07A_0$. Substitute $A = 0.07A_0$ into the equation:
$0.07A_0=A_0e^{- 0.000121t}$
Since $A_0
eq0$, we can divide both sides of the equation by $A_0$ to get $0.07 = e^{-0.000121t}$.

Step2: Take the natural logarithm of both sides

$\ln(0.07)=\ln(e^{-0.000121t})$
Using the property $\ln(e^x)=x$, the right - hand side simplifies to $-0.000121t$. So we have $\ln(0.07)=-0.000121t$.

Step3: Solve for $t$

We know that $\ln(0.07)\approx - 2.659257$. Then $t=\frac{\ln(0.07)}{-0.000121}$.
$t=\frac{-2.659257}{-0.000121}\approx22002$.

Answer:

$22002$