If the half life of carbon-14 is 5730 years, and you have a sample that has lost 80% of its carbon-14, how old is the sample?

To solve this problem, we need to understand the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the atoms in a sample to decay. In this case, the half-life of carbon-14 is 5730 years.

1. First, we need to determine how many half-lives have passed for 80% of the carbon-14 to have decayed.

2. If 50% of the carbon-14 decays in one half-life (5730 years), then after two half-lives (11460 years), 75% of the carbon-14 would have decayed.

3. After three half-lives (17190 years), 87.5% of the carbon-14 would have decayed.

4. Therefore, if 80% of the carbon-14 has decayed, the sample is between two and three half-lives old.

5. To find the exact age, we can use the formula for exponential decay, which is N = N0 * (1/2)^(t/T), where N is the final amount, N0 is the initial amount, t is the time, and T is the half-life.

6. If we plug in the values and solve for t, we get t = T * log2(N0/N).

7. If we assume that the initial amount of carbon-14 was 100%, and the final amount is 20% (because 80% has decayed), we get t = 5730 * log2(100/20) = 5730 * log2(5) = 5730 * 2.32 = 13316 years.

So, the sample is approximately 13316 years old.