Carbon-14 has a half-life of 5700 years. The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon contained 59.5% of the carbon-14 found in living matter. Approximate the age of Crater Lake.
.595=1.0 e^(.693t/5700)
ln of each side..
ln (.595)=.693t/5700
solve for t in years.
To approximate the age of Crater Lake, we can use the concept of half-life.
Given that Carbon-14 has a half-life of 5700 years, this means that after 5700 years, half of the Carbon-14 atoms in a sample will decay.
We are told that the charcoal from a tree killed in the volcanic eruption contained 59.5% of the Carbon-14 found in living matter. This means that 40.5% of the Carbon-14 has decayed.
To find the age of Crater Lake, we need to determine how many half-lives it took for 40.5% of Carbon-14 to decay.
Using the formula:
Remaining amount = Initial amount x (1/2)^(number of half-lives),
we can solve for the number of half-lives:
40.5% = 100% x (1/2)^(number of half-lives)
Divide both sides by 100%:
0.405 = (1/2)^(number of half-lives)
To solve for the number of half-lives, we can take the logarithm of both sides:
log base (1/2) (0.405) = number of half-lives
Using a calculator, we find:
number of half-lives = 2.68
Since each half-life is 5700 years, we can find the age of Crater Lake by multiplying the number of half-lives by the half-life:
Age of Crater Lake = 2.68 x 5700 years
Age of Crater Lake is approximately 15276 years.
To approximate the age of Crater Lake, we can use the concept of half-life and the given information about the percentage of carbon-14 left in the charcoal from the tree killed in the volcanic eruption.
The half-life of carbon-14 is 5700 years, which means that after 5700 years, half of the carbon-14 in a sample will have decayed.
If the charcoal from the tree killed in the volcanic eruption contains 59.5% of the carbon-14 found in living matter, it means that 40.5% of the carbon-14 has decayed. This is because 100% - 59.5% = 40.5%.
So, after one half-life (5700 years), 40.5% of carbon-14 has decayed. Therefore, the remaining carbon-14 would be 59.5% after 5700 years.
To find the number of half-lives it took for 40.5% of carbon-14 to decay to 59.5%, we can use the formula:
n = (log(R) / log(0.5))
Where:
- n is the number of half-lives
- R is the remaining fraction of carbon-14 (0.595 in this case)
- log is the logarithm function
Substituting the values into the formula:
n = (log(0.595) / log(0.5))
By evaluating this expression, we find that n is approximately 0.455.
Since one half-life is equal to 5700 years, we can calculate the age of Crater Lake by multiplying the number of half-lives (0.455) by the half-life duration (5700 years):
Age ≈ 0.455 x 5700 ≈ 2593.5 years
Therefore, the approximate age of Crater Lake is around 2593.5 years.