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 solve the problem of determining the age of the wooden artifact, we can follow these steps:

### Step 1: Understand the decay rates
We are given the decay rates for the wooden artifact and the freshly cut tree:
- Wooden artifact: \( R_a = 7.7 \, \text{min}^{-1} \, \text{g}^{-1} \)
- Freshly cut tree: \( R_f = 15.4 \, \text{min}^{-1} \, \text{g}^{-1} \)

### Step 2: Set up the relationship between the decay rates
Using the relationship between the decay rates and the amount of carbon remaining, we can express this as:
\[ \frac{R_a}{R_f} = \frac{n_t}{n_0} \]
Where:
- \( n_t \) is the amount of carbon in the wooden artifact
- \( n_0 \) is the amount of carbon in the freshly cut tree

### Step 3: Substitute the values
Substituting the values we have:
\[ \frac{7.7}{15.4} = \frac{n_t}{n_0} \]
This simplifies to:
\[ \frac{1}{2} = \frac{n_t}{n_0} \]
From this, we can deduce:
\[ n_0 = 2n_t \]

### Step 4: Relate the number of half-lives
The relationship between the remaining amount of carbon and the initial amount can also be expressed as:
\[ n_t = n_0 \left( \frac{1}{2} \right)^n \]
Where \( n \) is the number of half-lives that have passed.

### Step 5: Equate the two expressions
From the previous steps, we have:
\[ n_0 = 2n_t \implies n_t = n_0 \left( \frac{1}{2} \right)^n \]
Substituting \( n_0 = 2n_t \) into this equation gives:
\[ n_t = 2n_t \left( \frac{1}{2} \right)^n \]
Dividing both sides by \( n_t \) (assuming \( n_t \neq 0 \)):
\[ 1 = 2 \left( \frac{1}{2} \right)^n \]
This simplifies to:
\[ \left( \frac{1}{2} \right)^n = \frac{1}{2} \implies n = 1 \]

### Step 6: Calculate the age of the artifact
The age of the artifact can be calculated using the formula:
\[ \text{Age} = n \times t_{1/2} \]
Where \( t_{1/2} = 5770 \, \text{years} \):
\[ \text{Age} = 1 \times 5770 = 5770 \, \text{years} \]

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

---