An artifact was found and tested for its carbon-14 content. If 76% 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.
.76 = 1(1/2)^(t/5730)
ln both sides
ln .76 = ln (.5)^(t/5730)
t/5730 ln.5 = ln.76
t/5730 = ln.76/ln.5 = .3959..
t =2268.67
age is appr. 2269 years
To determine the probable age of the artifact, we can use the concept of half-life and the given information. The half-life of carbon-14 is 5,730 years.
Since 76% of the original carbon-14 content is still present, it means that the remaining 24% has decayed.
We can set up the following equation to solve for the number of half-lives that have passed:
0.24 = (1/2)^(n)
Where 'n' represents the number of half-lives.
Taking the logarithm of both sides (base 2):
log(0.24) = log((1/2)^(n))
log(0.24) = n * log(1/2)
Using the properties of logarithms (log(x^(y)) = y * log(x)):
n * log(1/2) = log(0.24)
Simplifying:
n = log(0.24) / log(1/2)
Using a calculator:
n ≈ 1.933
Since we want the probable age to the nearest 100 years, we need to round this number to the nearest whole number:
n ≈ 2
Therefore, approximately 2 half-lives have passed.
To find the probable age, we multiply the number of half-lives by the half-life of carbon-14:
Probable age = 2 * 5,730 years
Probable age ≈ 11,460 years
Therefore, the probable age of the artifact is approximately 11,460 years.
To determine the probable age of the artifact, we can use the concept of half-life and the remaining carbon-14 content.
The half-life of carbon-14 is 5,730 years, which means that over this period, half of the carbon-14 in a sample will decay. Given that 76% of the original carbon-14 is still present, it means that 24% has decayed (100% - 76%).
To calculate the number of half-life periods elapsed, we can use the formula:
(number of half-life periods) = (log(initial % of carbon-14 remaining) / log(1/2))
Using this formula, we can calculate:
(number of half-life periods) = (log(0.24) / log(1/2))
Now, let's solve this equation:
(number of half-life periods) ≈ (-1.415 / -0.301)
(number of half-life periods) ≈ 4.7
Since the number of half-life periods can't be a fraction, we'll round it to the nearest whole number. In this case, the artifact has gone through approximately 5 half-life periods.
To find the age of the artifact, we'll multiply the half-life by the number of half-life periods:
(age of the artifact) = (half-life) * (number of half-life periods)
(age of the artifact) = (5,730 years) * (5)
(age of the artifact) ≈ 28,650 years
Therefore, the probable age of the artifact is approximately 28,650 years (to the nearest 100 years).