Explanation

After each half-life, the sample is left with half of the isotopes it started with in the previous period. After one half-life, the sample goes from 16,000 isotopes to 8,000. After two half-lives, it goes from 8,000 to 4,000.

Continue this pattern to determine the total number of half-lives that have passed: 4,000 becomes 2,000 after 3 half-lives, 2,000 becomes 1,000 after 4 half-lives, 1,000 becomes 500 after 5 half-lives. The sample will have 500 isotopes after 5 half-lives. Thus, multiply 5 times the half-life, or 5 × 5,730 = 28,650 years.

Note that the answer choices are very spread apart. After determining that 5 half-lives have passed, estimate: \(5 \times 5,000 = 25,000\) years; answer (D) is the only possible answer.
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