A wooden artifact from an ancient tomb contains 45% of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made
what is the half-life? If it is N years, then you need to solve for t in
(1/2)^(t/N) = 0.25
Expect the answer to be just a bit over one half-life...
To determine the age of the wooden artifact, we can utilize the concept of carbon dating. Carbon dating is a method that measures the ratio of carbon-14 to carbon-12 in a sample to determine its age.
The half-life of carbon-14 is approximately 5730 years, which means that every 5730 years, half of the carbon-14 in a sample decays. Given that the artifact contains 45% of the carbon-14 found in living trees, we can use this information to estimate the age.
The general equation for carbon dating is:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
- N(t) represents the remaining amount of carbon-14 after time t
- N₀ is the initial amount of carbon-14 (which is 100% for living trees)
- t is the time that has passed
- t₁/₂ is the half-life of carbon-14
We can rearrange the equation to solve for t:
t = t₁/₂ * (log(N₀/N(t)) / log(1/2))
Since we know that the artifact contains 45% of the carbon-14, we can substitute N(t) = 0.45N₀ into the equation:
t = t₁/₂ * (log(N₀ / 0.45N₀) / log(1/2))
Simplifying further, we have:
t = t₁/₂ * (log(1 / 0.45) / log(1/2))
Using t₁/₂ = 5730 years, we can insert this value into the equation:
t = 5730 * (log(1 / 0.45) / log(1/2))
Calculating this expression, we find that t is approximately 3722 years.
Therefore, the wooden artifact was made roughly 3722 years ago, to the nearest year.