An anthropologist finds there is so little remaining Carbon-14 in a prehistoric bone that instruments cannot measure it. This means that there is less than 0.5% of the amount of Carbon-14 the bones would have contained when the person was alive. How long ago did the person die? Round your answer to the nearest thousand. (22,000, etc)
Find the half-life of C14. Call it n years
2^(-t/n) < .005
-t/n < ln(.005)/ln(2) = -7.64
t > 7.64n
check: 2^-7.64 = .005013, so after 7.64 half-lives, we are down to .5% of the original.
To find out how long ago the person died, we can use the concept of half-life. Carbon-14 has a half-life of approximately 5730 years, which means that after 5730 years, half of the amount of Carbon-14 in a sample will have decayed.
Given that there is less than 0.5% of the original amount of Carbon-14 remaining, we know that more than 99.5% of the Carbon-14 has decayed. Let's set up an equation to solve for the number of half-lives:
(1/2)^(n) ≤ 0.005
Here, 'n' represents the number of half-lives that have occurred. We can solve for 'n' by taking the logarithm of both sides:
log((1/2)^(n)) ≤ log(0.005)
Using the logarithmic property that log(a^b) = b(log(a)), we can simplify the equation further:
n * log(1/2) ≤ log(0.005)
Since log(1/2) is a negative number, we can divide both sides of the inequality by log(1/2) without flipping the sign:
n ≤ log(0.005) / log(1/2)
Using a calculator, we find that log(0.005) ≈ -2.3 and log(1/2) ≈ -0.3:
n ≤ -2.3 / -0.3
n ≤ 7.67
Since we cannot have a fractional number of half-lives, we can round down to the nearest whole number. Therefore, 'n' must be less than or equal to 7.
Each half-life represents 5730 years, so the person must have died at least 5730 * 7 = 40,110 years ago.
Rounding this value to the nearest thousand gives us an estimated answer of 40,000 years ago.