A paleontologist finds an unidentified bone. In the laboratory, she finds that the Carbon-14 found in the bone is ⅓ of that found in living bone tissue. How old is the bone? Round to the nearest whole number.

To determine the age of the bone, we can use the concept of the half-life of Carbon-14. The half-life of Carbon-14 is approximately 5730 years, which means that after 5730 years, half of the Carbon-14 in an organism will have decayed.

Given that the Carbon-14 found in the unidentified bone is ⅓ of that found in living bone tissue, we can set up the following equation:

⅓ = (1/2)^(n/5730)

Where n is the number of years since the bone was living tissue.

To find the value of n, we can solve for it using logarithms. Taking the logarithm base 2 of both sides of the equation:

log₂(⅓) = log₂((1/2)^(n/5730))

log₂(⅓) = (n/5730) * log₂(1/2)

Using log₂(1/2) ≈ -0.693 (the logarithm base 2 of 1/2):

log₂(⅓) = (n/5730) * (-0.693)

Solving for n:

n ≈ (log₂(⅓) / -0.693) * 5730

Using a calculator:

n ≈ ( -0.58496 / -0.693) * 5730

n ≈ 4877

Therefore, the age of the bone is approximately 4877 years. Rounding to the nearest whole number gives us a final answer of 4877 years old.

To determine the age of the bone, we can use the concept of radioactive decay. Carbon-14 is a radioactive isotope that is used for dating organic materials, like bone. It has a half-life of approximately 5730 years, which means that over time, half of the Carbon-14 in a sample will decay.

In this case, since the Carbon-14 in the unidentified bone is 1/3 of that found in living bone tissue, it means that there has been a loss of 2/3 (1 - 1/3) of the Carbon-14. Since each half-life corresponds to a loss of half of the remaining Carbon-14, we need to determine how many half-lives it takes for 2/3 to decay.

The formula to solve for the number of half-lives is:
(Total decay) = (Initial amount) * (1/2)^(number of half-lives)

In this case, the total decay is 2/3 (since 2/3 of the Carbon-14 has decayed), and the initial amount is 1 (since the Carbon-14 in living bone tissue is the starting point).

We can rewrite the equation as:
2/3 = 1 * (1/2)^(number of half-lives)

Now we can solve for the number of half-lives:
2/3 = (1/2)^(number of half-lives)

Taking the logarithm of both sides, we get:
log(2/3) = log((1/2)^(number of half-lives))

Using logarithm properties, we can bring the exponent down:
log(2/3) = (number of half-lives) * log(1/2)

Rearranging the equation to solve for the number of half-lives:
(number of half-lives) = log(2/3) / log(1/2)

Using a calculator, we find that (number of half-lives) is approximately 0.6309.

Since each half-life is approximately 5730 years, we multiply the number of half-lives by the length of a half-life:
Age of the bone ≈ 0.6309 * 5730 ≈ 3609.24 years.

Therefore, the bone is approximately 3609 years old when rounded to the nearest whole number.

If the half-life of carbon is n years, then

(1/2)^(t/n) = 1/3
t/n log(1/2) = log(1/3)
t/n = log3/log2 ≈ 1.58

so, about 1.58 half-lives