The `C^(14) " to " C^(12)` ratio in a wooden article is 13% that of the fresh wood. Calculate the age of the wooden article. Given that the half-life of `C^(14)` is 5770 years

Options

A. 16989 years
B. 16858 years
C. 15675 years
D. 17700 years

Explanation

To solve the problem, we need to determine the age of the wooden article based on the given ratio of Carbon-14 to Carbon-12 and the half-life of Carbon-14.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The problem states that the ratio of \( C^{14} \) to \( C^{12} \) in the wooden article is 13% of that in fresh wood. In fresh wood, we can assume the ratio of \( C^{14} \) to \( C^{12} \) is 100:100 (or simply 1:1 for the sake of comparison). Therefore, in the wooden article, the ratio of \( C^{14} \) is:
\[ \text{Amount of } C^{14} = 0.13 \times 100 = 13 \]

2. **Calculating the Remaining Carbon-14**:
The initial amount of \( C^{14} \) in fresh wood is 100. After some time, due to radioactive decay, the remaining amount of \( C^{14} \) is 13. We can express this mathematically using the formula for radioactive decay:
\[ N_t = N_0 \left( \frac{1}{2} \right)^{t/T_{1/2}} \]
Where:
- \( N_t \) = remaining quantity of \( C^{14} \) (which is 13)
- \( N_0 \) = initial quantity of \( C^{14} \) (which is 100)
- \( t \) = time elapsed (what we want to find)
- \( T_{1/2} \) = half-life of \( C^{14} \) (5770 years)

3. **Setting Up the Equation**:
Plugging in the values, we get:
\[ 13 = 100 \left( \frac{1}{2} \right)^{t/5770} \]

4. **Solving for \( t \)**:
Rearranging the equation gives:
\[ \left( \frac{1}{2} \right)^{t/5770} = \frac{13}{100} \]
Taking the logarithm of both sides:
\[ \frac{t}{5770} \log\left( \frac{1}{2} \right) = \log\left( \frac{13}{100} \right) \]
Now, we can solve for \( t \):
\[ t = 5770 \times \frac{\log\left( \frac{13}{100} \right)}{\log\left( \frac{1}{2} \right)} \]

5. **Calculating the Logs**:
Using a calculator:
- \( \log\left( \frac{13}{100} \right) = \log(13) - \log(100) = 1.1139 - 2 = -0.8861 \)
- \( \log\left( \frac{1}{2} \right) \approx -0.3010 \)

6. **Substituting the Values**:
Now substituting the values back into the equation:
\[ t = 5770 \times \frac{-0.8861}{-0.3010} \approx 5770 \times 2.943 = 17080.51 \text{ years} \]

7. **Final Result**:
Rounding it off, the age of the wooden article is approximately:
\[ t \approx 17080 \text{ years} \]

### Conclusion:
The age of the wooden article is approximately **17080 years**.