An artifact was found and tested for its carbon-14 content. If 75% of the original carbon-14 was still present, what is its probable age (to the nearest 100 years)? (Carbon-14 has a half-life of 5,730 years.)
thank you
amount = initial (.5)^(t/5730)
.75 = 1 (.5)^(t/5730)
log .75 = (t/5730) log .5
t/5730 = log .75/log .5 = .415...
t = 2378.16
age is about 2400 years to the nearest century
To determine the probable age of the artifact, we can use the concept of radioactive decay and the half-life of carbon-14.
The half-life of carbon-14 is 5,730 years, which means that after each 5,730-year period, half of the carbon-14 atoms in a sample will decay.
In this case, if only 75% of the original carbon-14 is still present, we need to determine how many half-lives have passed.
Let's denote the number of half-lives as "n."
Since each half-life represents a 50% decrease in the amount of carbon-14, we can calculate the amount remaining after n half-lives using the formula:
Remaining amount = original amount × (1/2)^n
Given that 75% of the original carbon-14 is still present, we can rewrite this equation as:
0.75 = 1 × (1/2)^n
To solve for n, we can take the logarithm of both sides of the equation (base 2, because half-lives are involved):
logbase2(0.75) = n
Using a calculator, we find that logbase2(0.75) is approximately -0.415.
Now we need to convert this to the number of half-lives. Since one half-life corresponds to a decrease of 50%, we need to divide -0.415 by -0.5 to get the number of half-lives:
n = (-0.415) / (-0.5) = 0.83
So, approximately 0.83 half-lives have passed.
To find the age of the artifact, we multiply the number of half-lives by the half-life duration of carbon-14:
Age = n × half-life duration = 0.83 × 5730 ≈ 4,759.1 years
Rounding that to the nearest 100 years, the probable age of the artifact is approximately 4,800 years.