An artifact was found and tested for its carbon-14 content. If 88% 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.
If you start with amount P, then after t years, R, the fraction remaining, is
R = 2^(-t/5730)
See how that works? Every 5730 years, the power of 2 goes down by 1.
2^(-t/5730) = .88
-t/5730 = ln(.88)/ln(2) = −0.321928095
t = 5730 * 0.321928095
t = 1844.64798435
or, to the nearest 100 years, 1800.
To determine the probable age of the artifact, we can use the concept of half-life of carbon-14.
The half-life of carbon-14 is 5,730 years. This means that every 5,730 years, the amount of carbon-14 in a sample is reduced by half.
In this case, we are told that 88% of the original carbon-14 is still present. This means that 12% of the carbon-14 has decayed.
To find the number of half-life intervals that have passed, we can use the formula:
Number of half-life intervals = log(initial amount of carbon-14 / remaining amount of carbon-14) / log(1/2)
Let's substitute the given values:
Number of half-life intervals = log(100% / 12%) / log(1/2)
Simplifying:
Number of half-life intervals = log(8.33) / log(1/2)
Using logarithmic properties, we can solve this equation by changing the base of the logarithm:
Number of half-life intervals = log(8.33) / log(1/2) ≈ log(8.33) / log(2)
Using a calculator, we get:
Number of half-life intervals ≈ 2.092
Since we want the age to the nearest 100 years, we need to multiply the number of half-life intervals by the half-life of carbon-14:
Age ≈ 2.092 x 5,730 years ≈ 11,961 years
Therefore, the probable age of the artifact is approximately 11,961 years.