b. Suppose that an artifact contains \( 20 \% \) of the amount of carbon-14 that it contained originally. How old is the artifact? Show your work.

To determine the age of the artifact based on the remaining amount of carbon-14, we can use the formula for radioactive decay. The amount of carbon-14 remaining can be expressed using the following equation: \[ N(t) = N_0 \cdot e^{-\lambda t} \] Where: - \( N(t) \) is the remaining amount of carbon-14 at time \( t \). - \( N_0 \) is the original amount of carbon-14. - \( \lambda \) is the decay constant. - \( t \) is the time that has passed. Carbon-14 has a half-life of approximately 5730 years. The decay constant \( \lambda \) can be calculated using the half-life formula: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \] Substituting the half-life of carbon-14: \[ \lambda = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \text{ years}^{-1} \] Given that the artifact contains \( 20\% \) of its original carbon-14, we can express this as: \[ N(t) = 0.2 N_0 \] Substituting this into the decay equation: \[ 0.2 N_0 = N_0 \cdot e^{-\lambda t} \] We can divide both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)): \[ 0.2 = e^{-\lambda t} \] Next, we take the natural logarithm of both sides: \[ \ln(0.2) = -\lambda t \] Now, substituting \( \lambda \): \[ \ln(0.2) = -\left(1.2097 \times 10^{-4}\right) t \] Calculating \( \ln(0.2) \): \[ \ln(0.2) \approx -1.6094 \] Now we can solve for \( t \): \[ -1.6094 = -\left(1.2097 \times 10^{-4}\right) t \] \[ t = \frac{1.6094}{1.2097 \times 10^{-4}} \approx 13311.5 \text{ years} \] Thus, the age of the artifact is approximately: \[ \boxed{13312} \text{ years} \]