An artifact was found and tested for its carbon-14 content. If 83% of the original carbon-14 was still present, what is its probable age (to the nearest 100 years)? Use that carbon-14 has a half-life of 5,730 years.
.83 = 1(1/2)^(t/5730
.83 = .5^(t/5730)
ln .83 = (t/5730) ln .5
t/5730 = ln .83/ln .5
t = 5730(ln .83/ln .5) = appr 1540.3 years
To determine the probable age of the artifact based on its carbon-14 content, you can use the concept of half-life and exponential decay.
The half-life of carbon-14 is known to be 5,730 years, which means that after every 5,730 years, the amount of carbon-14 in a sample is reduced by half.
In this case, we know that 83% of the original carbon-14 is still present. This indicates that 17% of the carbon-14 has decayed. Therefore, we can set up the following equation:
0.17 = (1/2)^(n/5730)
In this equation, 'n' represents the number of half-lives that have passed.
To solve for 'n', we can apply logarithms. Taking the logarithm (base 2) on both sides of the equation gives us:
log2(0.17) = (n/5730) * log2(1/2)
By rearranging the equation, we can solve for 'n':
n = (log2(0.17) / log2(1/2)) * 5730
Now, we can calculate 'n':
n ≈ (−2.796 / −1) * 5730
n ≈ 16,412
So, approximately 16,412 years have passed since the artifact's carbon-14 content decayed to 83% of its original amount.
To find the probable age of the artifact, we add this value to the present year. Let's assume the present year is 2022:
Probable age = 2022 + 16,412 ≈ 18,434 years
Therefore, the probable age of the artifact is approximately 18,434 years.