a piece of wood from an acient burial site has an activity of 3.0 x 10 Bq. the original activity is estimated to be 2.4 x 10$$ ^2 $$Bq. what is the estimated age of the wood?where R= 100($$ \frac12 $$)$$ ^{\frac{t}{5730}} $$

To find the estimated age of the wood, we need to use the given formula and solve for $ t $, the time. Here's the step-by-step process:

1. **Given Data:**
   - Current activity, $ R $: $ 3.0 \times 10 $ Bq
   - Original activity, $ R_0 $: $ 2.4 \times 10^2 $ Bq
   - Half-life of Carbon-14: $ 5730 $ years

2. **Formula:**
   \[
   R = R_0 \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]

3. **Substitute the values:**
   \[
   3.0 \times 10 = 2.4 \times 10^2 \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]

4. **Simplify the equation:**
   \[
   \frac{3.0 \times 10}{2.4 \times 10^2} = \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]
   \[
   \frac{3.0}{2.4 \times 10} = \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]
   \[
   \frac{3.0}{24} = \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]
   \[
   0.125 = \left(\frac{1}{2}\right)^{\frac{t}{5730}}
   \]

5. **Take the logarithm of both sides:**
   \[
   \log(0.125) = \log\left[\left(\frac{1}{2}\right)^{\frac{t}{5730}}\right]
   \]

6. **Use the power rule of logarithms:**
   \[
   \log(0.125) = \frac{t}{5730} \log\left(\frac{1}{2}\right)
   \]

7. **Calculate the logarithms (base 10 or natural logarithm can be used):**
   \[
   \log(0.125) \approx -0.903
   \]
   \[
   \log\left(\frac{1}{2}\right) \approx -0.301
   \]

8. **Solve for $ t $:**
   \[
   -0.903 = \frac{t}{5730} (-0.301)
   \]
   \[
   t = \frac{-0.903 \times 5730}{-0.301}
   \]
   \[
   t \approx \frac{5169.69}{0.301}
   \]
   \[
   t \approx 17178 \text{ years}
   \]

### Conclusion
The estimated age of the wood is approximately $ 17178 $ years.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the significance of the half-life in radiocarbon dating?
2. How does the activity of a radioactive sample change over time?
3. What are some common uses of radiocarbon dating besides determining the age of ancient wood?
4. How accurate is radiocarbon dating and what factors can affect its accuracy?
5. What other isotopes are used for radiometric dating, and what are their applications?
6. How do scientists determine the original activity of a sample?
7. Can radiocarbon dating be used for materials other than wood? If so, which ones?
8. What are the limitations of radiocarbon dating for very old samples?

### Tip
Always remember to check the units and ensure consistency when solving physics or chemistry problems involving formulas and calculations.

Learn how to estimate the age of wood using radioactive decay formula with a detailed step-by-step solution. This problem involves understanding radioactive decay, exponential functions, and logarithms. Suitable for high school students.

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