Final answer:
To determine the half-life of an unknown isotope, we started with 8500 atoms and ended with 2125 atoms after 3.8 years. Since the remaining atoms are a quarter of the original number, two half-lives have passed, indicating that the half-life of the isotope is 1.9 years.
Step-by-step explanation:
To calculate the half-life of an unknown isotope when given the starting amount of atoms and the remaining amount of atoms after a certain period, we can use the half-life decay formula. The original question states that an isotope starts with 8500 atoms, and after 3.8 years, only 2125 atoms remain. To find the half-life, we use the relationship that after one half-life, half the original number of isotopes will remain. By applying this concept repeatedly, we can determine the number of half-lives that have passed in 3.8 years.
Since the remaining number of atoms is a quarter of the original amount (8500 to 2125), we can conclude that two half-lives have passed (8500 to 4250 after the first half-life, and then 4250 to 2125 after the second half-life). Thus, two half-lives equal 3.8 years, meaning that one half-life is 3.8 years divided by 2, which equals 1.9 years.