A micropaleontologist uses carbon-14 dating on a foraminifera sample. If the sample has 35% of its original carbon-14 remaining, and the half-life of carbon-14 is 5730 years, how old is the sample? Use $N = N_0 e^{-kt}$, $k = rac{\ln 2}{5730}$.

["Using Carbon-14 Dating to Determine the Age of a Foraminifera Sample", "In the field of micropaleontology, determining the precise age of microscopic fossils like foraminifera is essential for reconstructing past climates and understanding Earth’s geological history. One powerful tool in this analysis is radiocarbon dating, which relies on measuring the decay of carbon-14 ($^{14}C$). With a half-life of 5,730 years, carbon-14 allows scientists to date organic materials up to about 50,000 years old.", "A recent study employed carbon-14 dating on a foraminifera sample collected from sediment layers. The sample retained only 35% of its original carbon-14, prompting researchers to calculate its age using the exponential decay formula:", "[\nN = N_0 e^{-kt}\n]", "where:\n- $N$ is the current amount of carbon-14,\n- $N_0$ is the initial amount,\n- $k$ is the decay constant,\n- $t$ is the time elapsed,\n- and $k = \dfrac{\ln 2}{5730} \approx 0.000121$ per year.", "To find $k$, we start with the known fraction remaining:", "[\n\frac{N}{N_0} = 0.35 = e^{-kt}\n]", "Taking the natural logarithm of both sides:", "[\n\ln(0.35) = -kt\n]", "Solving for $t$:", "[\nt = -\frac{\ln(0.35)}{k} = \frac{-\ln(0.35)}{0.000121}\n]", "Calculate $\ln(0.35) \approx -1.0498$, so:", "[\nt = \frac{1.0498}{0.000121} \approx 8676 \ ext{ years}\n]", "Thus, the foraminifera sample is approximately 8,676 years old. This age helps micropaleontologists place the fossil within a well-dated stratigraphic context, supporting broader paleoenvironmental reconstructions.", "Carbon-14 dating remains invaluable in micropaleontology, offering a reliable method to date organic microfossils and unlock historical insights from sediment cores around the globe."]