To calculate the original mass of a sample of Carbon-14 after a certain time has passed, you can use the formula for exponential decay:
\[N(t) = N_0 \cdot e^{-\lambda t}\]
Where:
- \(N(t)\) is the remaining quantity at time \(t\).
- \(N_0\) is the initial quantity (original mass).
- \(\lambda\) is the decay constant (related to the half-life).
- \(t\) is the time that has passed.
Given that the half-life of Carbon-14 is 2,270 years, you can calculate \(\lambda\) as:
\[\lambda = \frac{\ln(2)}{\text{half-life}} = \frac{\ln(2)}{2,270}\]
Now, you want to find the original mass (\(N_0\)) when \(t = 18,160\) years and \(N(t) = 2.8\) grams:
\[2.8 = N_0 \cdot e^{-\frac{\ln(2)}{2,270} \cdot 18,160}\]
Now, solve for \(N_0\):
\[N_0 = \frac{2.8}{e^{-\frac{\ln(2)}{2,270} \cdot 18,160}}\]
Calculating this value will give you the original mass of the sample.