Answer: To determine the age of the bones when they were discovered, we can use the concept of radioactive decay and the given information about carbon-14's half-life.
The half-life of carbon-14 is 5750 years, which means that after every 5750 years, the amount of carbon-14 in the bones is reduced by half. We are given that the bones have lost 63.8% of their carbon-14, which means that the remaining amount of carbon-14 is 100% - 63.8% = 36.2% of the original amount.
Let N be the original amount of carbon-14 in the bones (100%). After one half-life (5750 years), the remaining amount is 50% (N * 0.5), after two half-lives (11500 years), the remaining amount is 25% (N * 0.25), and so on.
We need to find how many half-lives it takes for the remaining amount to be 36.2% of the original amount. Let's set up an equation to find the number of half-lives (n):
(1/2)^n = 36.2% = 0.362.
Now, let's solve for n:
n * log(1/2) = log(0.362)
n = log(0.362) / log(1/2)
Using a calculator:
n ≈ 2.5042.
Since we cannot have a fraction of a half-life, we'll consider 2 half-lives. Each half-life is 5750 years, so the age of the bones when they were discovered is:
Age = 2 * 5750 ≈ 11500 years.
Therefore, the bones were approximately 11500 years old when they were discovered.