If the half-life of a radioisotope is 20,000 years, then a sample in which three-quarters of that radioisotope has decayed is _____ years old.a. 15,000b. 26,667c. 30,000d. 40,000

Key Concepts: Radioisotopes, Half-Life, Exponential Decay Explanation: The half-life of a radioisotope is the time it takes for half of the original sample to decay. We can use the information given to find the number of half-lives that have passed, and then use that information to find the age of the sample. Step by Step Solution: Step 1. If a radioisotope has a half-life of 20,000 years, then after one half-life, half of the original sample will be decayed. After two half-lives, only one-fourth (half of half) of the original sample will remain. After three half-lives, only one-eighth (half of half of half) of the original sample will remain, and so on. Step 2. We are told that three-quarters of the original sample has decayed. This means that one-quarter of the original sample remains. We can use logarithms to find the number of half-lives that have passed: log(1/4) = -0.602. This means that 0.602 of the original sample has decayed. Since each half-life represents half of the sample decaying, we can then divide -0.602 by the natural logarithm of 2: -0.602 / ln(2) ≈ 1.636. Therefore, approximately 1.636 half-lives have passed. Step 3. To find the age of the sample, we can multiply the number of half-lives by the length of one half-life: 1.636 x 20,000 years = 32,720 years. Therefore, the sample is approximately 32,720 years old. Final Answer: b. 26,667