Final answer:
True, Carbon-14 has a half-life of 5,730 years. Using the exponential decay formula, it takes approximately 1903.7 years for 10.0% of a sample of Carbon-14 to decay, which showcases the utility of Carbon-14 in radiometric dating.
Step-by-step explanation:
True, Carbon-14 does indeed have a half-life of 5,730 years. To determine how long it takes for 10.0% of a sample of carbon-14 to decay, we would use the exponential decay formula involving the half-life. Since 10.0% decay means that 90.0% is left, we use the equation N(t) = N_0(1/2)t/t_1/2, where N(t) is the remaining amount after time t, N_0 is the original amount, and t_1/2 is the half-life. In this case, we're solving for t when N(t)/N_0 = 0.9.
First, we find the number of half-lives that pass for the sample to decay to 90% of its original amount by rearranging the formula to t = (log(N(t)/N_0)/log(1/2)) * t_1/2. Substituting N(t)/N_0 = 0.9 and t_1/2 = 5730 years leads to t = (log(0.9)/log(0.5)) * 5730 years. When this calculation is carried out, we find that it takes about 1903.7 years for 10.0% of carbon-14 to decay.
Carbon-14 Dating Procedures
These procedures are integral to radiometric dating, a technique that has had immense impact on fields like archaeology and paleontology. It allows scientists to determine the age of organic artifacts with a high degree of accuracy. Carbon-14 dating is perhaps the most famous application of this principle.