A wooden artifact contains 70% Carbon-14. To the nearest year, how many years old is the artifact? The half life of Carbon-14 is 5730 years?
We can use the formula for exponential decay to solve this problem:
A = A0 * (1/2)^(t/T)
where A is the amount of Carbon-14 remaining after time t, A0 is the initial amount of Carbon-14, T is the half-life of Carbon-14, and ^(t/T) means "to the power of t divided by T".
We know that the artifact contains 70% Carbon-14, so A/A0 = 0.7. We also know that T = 5730 years. Substituting these values into the formula, we get:
0.7 = 1 * (1/2)^(t/5730)
Taking the natural logarithm of both sides, we get:
ln(0.7) = (t/5730) * ln(1/2)
Solving for t, we get:
t = -5730 * ln(0.7) / ln(1/2)
t ≈ 2,041 years
Therefore, the artifact is approximately 2,041 years old.
To determine the age of the artifact, we can use the concept of radioactive decay and the half-life of Carbon-14.
Half-life is the time it takes for half of the atoms of a radioactive substance to decay. Since the half-life of Carbon-14 is 5730 years, it means that after 5730 years, half of the Carbon-14 atoms in the artifact would have decayed.
To find the age of the artifact, we need to calculate how many half-lives have passed. We can use the formula:
Number of half-lives = log(total amount of Carbon-14 remaining / initial amount of Carbon-14) / log(0.5)
Given that the artifact contains 70% Carbon-14, we can convert this percentage to a decimal, which is 0.70. This means that 0.70 of the initial Carbon-14 is still present.
Number of half-lives = log(0.70) / log(0.5)
To solve this equation, you can use a scientific calculator or relevant software. The result is approximately 0.152.
Now, we can calculate the age of the artifact by multiplying the number of half-lives by the half-life of Carbon-14:
Age of the artifact = Number of half-lives * half-life of Carbon-14
= 0.152 * 5730
Calculating the result, we find that the age of the artifact is approximately 872 years old (to the nearest year).
To calculate the age of the artifact, we can use the formula for exponential decay:
N(t) = N0 * (1/2)^(t / t1/2)
Where:
N(t) = Final amount of Carbon-14 remaining after time t
N0 = Initial amount of Carbon-14
t = Time elapsed
t1/2 = Half-life of Carbon-14
Given:
N(t) = 0.70 (70%)
N0 = 1 (100%, assuming the initial amount is 100%)
t1/2 = 5730 years
Plugging in the values into the formula, we have:
0.70 = 1 * (1/2)^(t / 5730)
Simplifying the equation, we get:
0.70 = (1/2)^(t / 5730)
Taking the natural logarithm of both sides, we have:
ln(0.70) = ln((1/2)^(t / 5730))
Simplifying further using the properties of logarithms, we get:
ln(0.70) = (t / 5730) * ln(1/2)
Now, divide both sides by ln(1/2):
(t / 5730) = ln(0.70) / ln(1/2)
Using a scientific calculator, we can solve for t:
t ≈ (5730 * ln(0.70)) / ln(1/2)
Calculating the value, we find:
t ≈ 3039.97
To the nearest year, the artifact is approximately 3040 years old.