You are at a archeological dig an discover a strange artifact. It is a utinsil tha appears to be carved out of an elephant's tusk You analyze it and discover that it has 0.625 the amount of carbon -14 you woul find in a living specimen. How many half lives has the specimen been through? How old is the specimen? How much of the present amount of carbon - 14 will be in the artifact in another 11,460 years?
To answer these questions, we need to understand the concept of half-life and apply it to carbon-14 dating.
1. How many half-lives has the specimen been through?
The ratio of the carbon-14 present in the artifact to that in a living specimen tells us how many half-lives have passed. In this case, the artifact has 0.625 times the amount of carbon-14 compared to a living specimen. To determine the number of half-lives, we can use the formula:
Number of half-lives = (log (present ratio) / log (0.5))
In this case, the present ratio is 0.625. Plugging this value into the formula:
Number of half-lives = (log (0.625) / log (0.5))
Number of half-lives ≈ 1.678
So, the artifact has been through approximately 1.678 half-lives.
2. How old is the specimen?
Each half-life of carbon-14 is approximately 5730 years. To find the age of the specimen, we need to multiply the number of half-lives (from the previous step) by the length of one half-life:
Age ≈ (Number of half-lives) × (Half-life period)
Age ≈ 1.678 × 5730 ≈ 9604.74 years
Therefore, the specimen is approximately 9604.74 years old.
3. How much of the present amount of carbon-14 will be in the artifact in another 11,460 years?
Since each half-life is approximately 5730 years, we can calculate the number of half-lives in 11,460 years:
Number of half-lives = (Time elapsed) / (Half-life period)
Number of half-lives = 11460 / 5730 ≈ 2
So, in another 11,460 years, there will be an additional 2 half-lives. To calculate the remaining amount of carbon-14 in the artifact, we use the formula:
Remaining amount = (Initial amount) × (1/2)^(Number of half-lives)
Given that the initial amount is 0.625, we can plug in the values:
Remaining amount = 0.625 × (1/2)^2
Remaining amount = 0.625 × 0.25 ≈ 0.15625
Hence, in another 11,460 years, approximately 15.625% of the present amount of carbon-14 will remain in the artifact.