Wooden artifact and freshly cut tree give 7.7 and `"15.4 min"^(-1)g^(-1)` of carbon `(t_((1)/(2))="5770 years")` respectively. The age of the artifact is

Options

A. 5770 years
B. `5770xx(15.4)/(7.7)" years"`
C. `5770xx(7.7)/(15.4)" years"`
D. None of these

Answer

To find the age of the wooden artifact, we will use the information provided about the decay rates of carbon in the artifact and a freshly cut tree, as well as the half-life of carbon-14.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Decay rate of the wooden artifact, \( k_1 = 7.7 \, \text{min}^{-1} \, \text{g}^{-1} \)
- Decay rate of a freshly cut tree, \( k_0 = 15.4 \, \text{min}^{-1} \, \text{g}^{-1} \)
- Half-life of carbon-14, \( t_{1/2} = 5770 \, \text{years} \)

2. **Set Up the Equations:**
- For the wooden artifact, we can express the decay in terms of the number of carbon atoms:
\[ N_t = N_0 \cdot e^{-k_1 t} \]
- For the freshly cut tree:
\[ N_0 = N_0 \cdot e^{-k_0 \cdot 0} = N_0 \]
- Since we are comparing the two, we can express the ratio of the two decay rates:
\[ \frac{N_t}{N_0} = \frac{k_1}{k_0} \]

3. **Calculate the Ratio:**
- Substitute the values of \( k_1 \) and \( k_0 \):
\[ \frac{N_t}{N_0} = \frac{7.7}{15.4} = \frac{1}{2} \]
- This indicates that the amount of carbon-14 in the artifact is half of that in the freshly cut tree.

4. **Relate to Half-Lives:**
- The relationship between the remaining quantity of a radioactive substance and its half-lives is given by:
\[ N_t = N_0 \left( \frac{1}{2} \right)^n \]
- From the previous step, we found that \( \frac{N_t}{N_0} = \frac{1}{2} \), which implies:
\[ n = 1 \]
- This means the artifact is one half-life old.

5. **Calculate the Age of the Artifact:**
- Since \( n = 1 \) and the half-life \( t_{1/2} = 5770 \, \text{years} \):
\[ \text{Age of the artifact} = n \cdot t_{1/2} = 1 \cdot 5770 \, \text{years} = 5770 \, \text{years} \]

### Final Answer:
The age of the wooden artifact is **5770 years**.