An artifact was found and tested for its carbon-14 content. If 74% 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.
1 yr
the fraction left after t years is
(1/2)^(t/5730)
so, you need to solve for t:
.74 = (1/2)^(t/5730)
ln .74 = t/5730 * ln .5
t = 5730 * ln .74/ln .5
t = 2489, or 2500 years
To find the age of the artifact based on the remaining carbon-14 content, we can use the concept of half-life.
The half-life of carbon-14 is 5,730 years, which means that after every 5,730 years, the amount of carbon-14 in a sample will decrease by half.
Since the artifact has 74% of the original carbon-14 remaining, it means that 26% of the carbon-14 has decayed. We can assume that the 26% decay occurred in one or more complete half-lives.
To determine the number of half-lives the decayed carbon-14 represents, we can use the formula:
Decay Factor = (1/2)^n
Where "n" is the number of half-lives.
In this case, the decay factor is 0.26 (26% decayed). Substituting the values into the equation, we get:
0.26 = (1/2)^n
To solve for "n", we can take the logarithm of both sides of the equation:
log(0.26) = log((1/2)^n)
log(0.26) = n * log(1/2)
Using the logarithm base 10 logarithm in this case, we can solve for "n":
n = log(0.26) / log(1/2)
n ≈ 0.585 / (-0.301)
n ≈ -1.942
Since the number of half-lives cannot be negative, we take the absolute value of "n" to get the positive number of half-lives:
n ≈ 1.942
Now, we know that each half-life is 5,730 years. Therefore, to find the probable age of the artifact, we multiply the number of half-lives by the half-life time:
Age = n * half-life
Age = 1.942 * 5,730
Age ≈ 11,059
Hence, the probable age of the artifact, to the nearest 100 years, is approximately 11,100 years.