after some ancient bones were excavated, carbon was taken from one of the bones to find that there was 69% as much carbon as from current bones. the decay constant of carbon is k=0.0001216, for time in years. how many years old is the bone?

To determine the age of the bone, we can use the exponential decay formula:

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

Where:
- \(N(t)\) is the amount of carbon remaining at time \(t\).
- \(N_0\) is the initial amount of carbon (when the bone was new).
- \(k\) is the decay constant.
- \(t\) is the time in years.

Given:
- \(N(t) = 0.69N_0\) (since 69% of the carbon remains)
- \(k = 0.0001216\)

We need to solve for \(t\). Start by substituting the given values into the formula:

\[
0.69N_0 = N_0 \cdot e^{-0.0001216t}
\]

Next, divide both sides by \(N_0\):

\[
0.69 = e^{-0.0001216t}
\]

To solve for \(t\), take the natural logarithm (ln) of both sides:

\[
\ln(0.69) = -0.0001216t
\]

Now, solve for \(t\):

\[
t = \frac{\ln(0.69)}{-0.0001216}
\]

I'll calculate that for you.The bone is approximately 3,052 years old.

Would you like more details on how this was calculated, or do you have any other questions? Here are five related topics we could explore:
1. Exponential decay and its applications.
2. The half-life concept in radioactive decay.
3. How carbon dating is used in archaeology.
4. Calculating decay constants from half-life values.
5. Understanding logarithmic functions in decay problems.

**Tip:** When working with exponential functions, remember that taking the logarithm is a powerful tool to solve for time or other variables.

Learn how to calculate the age of ancient bones using the exponential decay formula. This problem involves determining the age based on the decay constant of carbon and the remaining carbon content. Suitable for high school students studying exponential functions and decay processes.

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