at an archaeological site remains of plant material will found to have 28.9% of the amount of carbon-14 in a living sample. What was the approximate age of the plant material?
look up the half-life of C-14. If it is k years, then the amount left after t years is
(1/2)^(t/k)
So, you want to solve for t in
(1/2)^(t/k) = 0.289
To determine the approximate age of the plant material based on the amount of carbon-14, we need to use a process called carbon dating. Carbon-14 dating relies on the fact that carbon-14, a radioactive isotope of carbon, undergoes radioactive decay at a constant rate.
The half-life of carbon-14 is approximately 5730 years, which means that after this period, half of the carbon-14 atoms in a sample will have decayed into stable nitrogen-14. By comparing the ratio of carbon-14 to carbon-12 (a stable isotope) in a sample, we can estimate its age.
In this case, the plant material is found to have 28.9% of the amount of carbon-14 compared to a living sample. We can use this information to set up an equation:
carbon-14 remaining (as a decimal) = initial carbon-14 (as a decimal) * (1/2)^(number of half-lives)
Now, let's solve for the number of half-lives:
0.289 = 1 * (1/2)^n
Here, "n" represents the number of half-lives.
To solve for "n," we can take the logarithm (base 2) of both sides of the equation:
log(0.289) = log[(1/2)^n]
log(0.289) = n * log(1/2)
n = log(0.289) / log(1/2)
Using a calculator, we can find that n is approximately 2.945.
Since each half-life is approximately 5730 years, we can multiply 2.945 by 5730 to find the approximate age of the plant material:
Age ≈ 2.945 * 5730
The approximate age of the plant material is around 16,877 years.