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.