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.
just solve for t in
(1/2)^(t/5730) = 0.75
To determine the probable age of the artifact, we need to use the concept of half-life and calculate the number of half-lives that have passed based on the remaining carbon-14.
Given that carbon-14 has a half-life of 5,730 years, which means that after each half-life, half of the carbon-14 decays, we can set up the following equation:
Remaining Carbon-14 = Original Carbon-14 × (1/2)^(Number of Half-Lives)
Let's denote the remaining carbon-14 as R (since 75% of the original is still present), and the original carbon-14 as C. We can rewrite the equation as:
R = C × (1/2)^(Number of Half-Lives)
Since we are given that 75% of the original carbon-14 is still present, we know that R = 0.75C. Substituting this into the equation, we get:
0.75C = C × (1/2)^(Number of Half-Lives)
To solve for the number of half-lives (Number of Half-Lives), we need to isolate it on one side of the equation. Let's divide both sides by C:
0.75 = (1/2)^(Number of Half-Lives)
To get rid of the exponent, we can take the logarithm of both sides of the equation. The base of the logarithm is not crucial, but let's compute it using base 2, resulting in:
log₂(0.75) = Number of Half-Lives
Using a calculator, we find that log₂(0.75) is approximately -0.415.
Now, to find the actual number of half-lives, we need to solve for Number of Half-Lives. Rearranging the equation:
Number of Half-Lives = log₂(0.75) / log₂(1/2)
Dividing -0.415 by -1 (since log₂(1/2) is -1) gives us:
Number of Half-Lives ≈ 0.415
The probable age (in years) can be calculated by multiplying the number of half-lives by the half-life of carbon-14:
Probable Age = Number of Half-Lives × Half-Life of Carbon-14
Probable Age = 0.415 × 5,730 years
Probable Age ≈ 2,377 years
Therefore, the probable age of the artifact, rounded to the nearest 100 years, is approximately 2,400 years.