A paleobotanist estimates the age of a fossil layer using carbon-14 decay, knowing the half-life is 5730 years. If a sample has 12.5% of its original carbon-14 remaining, how old is the sample?
["Title: How Paleobotanists Use Carbon-14 Half-Life to Estimate Fossil Age — A Step-by-Step Guide", "When studying fossilized plant remains, paleobotanists often face the challenge of determining how long ago a specimen died. One of the most reliable methods for dating organic material up to about 50,000 years old is carbon-14 dating—an analytical technique based on the predictable decay of the carbon-14 isotope. Understanding how this process works helps unlock the timeline hidden in ancient plant fossils.", "### The Science Behind Carbon-14 Dating", "Carbon-14 is a radioactive isotope of carbon naturally formed in the upper atmosphere when cosmic rays interact with nitrogen. Living organisms continuously exchange carbon with their environment, maintaining a stable ratio of carbon-14 to stable carbon-12. Once an organism dies, it stops absorbing new carbon, and the carbon-14 within its remains begins to decay at a known rate—this is known as its half-life, which is approximately 5,730 years.", "By measuring the remaining amount of carbon-14 in a fossil sample and comparing it to the initial expected level, scientists calculate how many half-lives have passed, thus estimating the fossil's age.", "### Determining Age from Remaining Carbon-14", "According to the problem, a fossil sample retains only 12.5% of its original carbon-14. Because each half-life reduces the carbon-14 by half, we can calculate how many halvies have passed:", "- After 1 half-life: 50% remains\n- After 2 half-lives: 25% remains\n- After 3 half-lives: 12.5% remains", "Since 12.5% corresponds to exactly three half-lives, the fossil has aged through three full decay cycles.", "### Calculating the Total Age", "With each half-life equal to 5,730 years, the total age is:", "[ 3 \ imes 5,730 = 17,190 \ ext{ years} ]", "### Why This Method Matters in Paleobotany", "Accurately dating fossil plant layers helps paleobotanists reconstruct ancient environments, track climate changes, and understand evolutionary timelines. Carbon-14 dating is especially valuable for relatively recent fossils, giving insight into ecosystems from the last Ice Age and beyond—provided the sample is within its effective dating range.", "In summary, when a fossil retains 12.5% of its original carbon-14, paleobotanists conclude it is about 17,190 years old—a window into Earth’s past captured in plant fossils.", "Keywords: carbon-14 dating, paleobotany, fossil age estimation, half-life, radioactive decay, ancient plant fossils, 14C dating method, radiocarbon dating, decay process, 5730-year half-life, fossil timeline."]
