Question

prehistoric cave paintings were discovered in a cave in france. the paint contained 25% 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 11450 years old. (round to the nearest integer.)

Explanation:

Step1: Set up the equation

Since the paint contains 25% (or 0.25) of the original carbon - 14, we set $A = 0.25A_0$ in the decay model $A = A_0e^{-0.000121t}$. So, $0.25A_0=A_0e^{-0.000121t}$. Divide both sides by $A_0$ (since $A_0
eq0$), getting $0.25 = e^{-0.000121t}$.

Step2: Take the natural logarithm of both sides

$\ln(0.25)=\ln(e^{-0.000121t})$. By the property of logarithms $\ln(e^x)=x$, the right - hand side simplifies to $- 0.000121t$. So, $\ln(0.25)=-0.000121t$.

Step3: Solve for t

We know that $\ln(0.25)\approx - 1.3863$. Then $t=\frac{\ln(0.25)}{-0.000121}$. Substitute $\ln(0.25)\approx - 1.3863$ into the formula: $t=\frac{-1.3863}{-0.000121}\approx11457$. Rounding to the nearest integer gives $t = 11450$.

Answer:

11450