A wooden artifact from an ancient tomb contains 35 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.)
To solve this problem, we can use the concept of half-life.
Since the artifact contains 35 percent of the carbon-14 present in living trees, it means that 65 percent of the carbon-14 has decayed.
The half-life of carbon-14 is 5730 years, which means that after 5730 years, half of the carbon-14 will decay.
So, if 65 percent of the carbon-14 has decayed, it means that the artifact is around 2 half-lives old.
5730 years * 2 = 11460 years
Therefore, the artifact was made approximately 11460 years ago to the nearest year.
To find out how long ago the artifact was made, we can use the concept of half-life and the given information.
The half-life of carbon-14 is 5730 years, meaning that after 5730 years, half of the carbon-14 in a sample decays.
Since the wooden artifact contains 35 percent of the carbon-14 found in living trees, this means that 65 percent of the carbon-14 has decayed (100% - 35% = 65%).
Now, we need to determine the number of half-lives it took for 65% of the carbon-14 to decay. We can use the formula:
Number of half-lives = (log(initial amount / final amount)) / (log(1/2))
Let's substitute the values:
Number of half-lives = (log(100% / 35%)) / (log(1/2))
Number of half-lives = (log(1 / 0.35)) / (log(1/2))
Using a calculator:
Number of half-lives ≈ 0.155 / -0.301
Number of half-lives ≈ -0.514
We get a negative value because the logarithms result in a negative number.
To determine the time in years, we multiply the number of half-lives by the half-life of carbon-14:
Time = Number of half-lives * Half-life
Time ≈ -0.514 * 5730
Time ≈ -2946.942
The negative sign suggests that the time is in the past, so we drop it to determine years ago:
Time ≈ 2946.942 years ago
Therefore, the artifact was made approximately 2946 years ago (to the nearest year).