After some ancient bones were​ excavated, carbon was taken from one of the bones to find that there was 87​% as much Superscript 14 Baseline Upper C as from current bones. The decay constant of Superscript 14 Baseline Upper C is k almost equals 0.0001216​, for time in years. How many years old is the​ bone?

We can solve this problem using the exponential decay formula for radioactive decay, which is:

\[
N(t) = N_0 e^{-kt}
\]

Where:
- \( N(t) \) is the amount of carbon-14 remaining at time \( t \),
- \( N_0 \) is the initial amount of carbon-14 (from current bones),
- \( k \) is the decay constant,
- \( t \) is the time (age of the bone, which we are solving for).

In this case:
- \( N(t) / N_0 = 87\% = 0.87 \),
- \( k = 0.0001216 \).

We need to solve for \( t \). Rearranging the decay formula:

\[
\frac{N(t)}{N_0} = e^{-kt}
\]

Taking the natural logarithm (ln) of both sides:

\[
\ln \left( \frac{N(t)}{N_0} \right) = -kt
\]

Substituting values:

\[
\ln(0.87) = -0.0001216 \cdot t
\]

Now, solving for \( t \):

\[
t = \frac{\ln(0.87)}{-0.0001216}
\]

Let's calculate this.The bone is approximately 1,145 years old.

Do you have any further questions or would you like additional details?

Here are 5 related questions to explore further:
1. How does carbon dating accuracy change over longer periods of time?
2. What other isotopes are used for dating archaeological finds?
3. What is the half-life of carbon-14, and how does it relate to the decay constant?
4. How does the environment affect carbon-14 levels in organisms?
5. What are some limitations of carbon-14 dating?

**Tip**: Always check the validity of assumptions (like initial amounts of isotopes) when using radiometric dating methods.

After some ancient bones were excavated, carbon was taken from one of the bones to find that there was 87% as much 14C as from current bones. The decay constant of 14C is k ≈ 0.0001216. How many years old is the bone?

Learn how to calculate the age of a bone using the exponential decay formula for carbon-14. This problem covers key concepts such as radioactive decay, exponential decay, and logarithms. The decay constant is k ≈ 0.0001216, and the percentage of 14C remaining is 87%. Suitable for students in Grades 10-12.

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