1. The problem: Understanding the formula used in carbon dating. 2. Carbon dating is a method used to determine the age of an object containing organic material by measuring the amount of carbon-14 it contains. 3. The key formula used in carbon dating is based on exponential decay: $$N(t) = N_0 e^{-\lambda t}$$ where: - $N(t)$ is the amount of carbon-14 remaining at time $t$, - $N_0$ is the initial amount of carbon-14, - $\lambda$ is the decay constant, - $t$ is the time elapsed (age of the sample). 4. The decay constant $\lambda$ is related to the half-life $T_{1/2}$ of carbon-14 by: $$\lambda = \frac{\ln(2)}{T_{1/2}}$$ where the half-life of carbon-14 is approximately 5730 years. 5. To find the age $t$ of a sample, rearrange the formula: $$t = \frac{1}{\lambda} \ln\left(\frac{N_0}{N(t)}\right)$$ 6. This means by measuring the ratio $\frac{N(t)}{N_0}$ of carbon-14 remaining, we can calculate the time elapsed since the death of the organism. 7. Important rules: - The initial amount $N_0$ is usually estimated based on the atmospheric carbon-14 at the time. - The method assumes a constant decay rate and no contamination. Final answer: The formula for carbon dating is $$t = \frac{1}{\lambda} \ln\left(\frac{N_0}{N(t)}\right)$$ where $\lambda = \frac{\ln(2)}{5730}$ years$^{-1}$.