An artifact was found and tested for its carbon-14 content. If 70% 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.
Use
amount = c (1/2)^(t/k), where k is the half-life period
so ....
.7 = 1(1/2)^(t/5730)
.7 = (2)^(-t/5730)
ln .7 = ln ( 2^(-t/5730) )
ln .7 = -t/5730 (ln2)
-t/5730 = ln .7/ln 2 = -.514573...
t = 2948.5 years
or 2900 to the nearest 100 years
To determine the probable age of the artifact, we need to use the concept of carbon-14 dating and its half-life.
Carbon-14 is a radioactive isotope of carbon that is present in the atmosphere and absorbed by plants through photosynthesis. When an organism dies, it stops absorbing carbon-14, and the remaining carbon-14 in the organism starts to decay.
The half-life of carbon-14 is the time it takes for half of the carbon-14 in a sample to decay. In this case, the half-life of carbon-14 is given as 5,730 years. This means that after 5,730 years, only half of the original carbon-14 will remain.
In this problem, we are told that 70% of the original carbon-14 is still present in the artifact. This means that the remaining 30% has decayed. Since the half-life is 5,730 years, we know that 50% of the carbon-14 decays every 5,730 years. Therefore, the 30% decayed in some fraction of the half-life.
To find this fraction, we can set up the following equation:
0.30 = (0.50)^n
where n represents the number of half-lives.
To solve for n, we can take the logarithm of both sides of the equation:
log(0.30) = n * log(0.50)
n = log(0.30) / log(0.50)
Using a calculator, we find that n is approximately 0.5229.
Since each half-life is 5,730 years, we can multiply the number of half-lives by the length of each half-life to get the probable age of the artifact:
Age = n * 5,730 years
Plugging in the value of n, we get:
Age ≈ 0.5229 * 5,730 years
Age ≈ 2,997.16 years
Therefore, the probable age of the artifact is approximately 2,997 years (to the nearest 100 years).