What is the age of an ancient wooden piece if it is known that the specific activity of `C^(14)` nuclide in it amouts to `3//5` of that in fresh trees? Given: the half of `C` nuclide is `5570 years` and `log_(e)(5//3)=0.5` .

Options

A. `1000 years`
B. `2000 years`
C. `3000 years`
D. `4000 years`

Explanation

To solve the problem of determining the age of an ancient wooden piece based on the specific activity of the `C^(14)` nuclide, we can follow these steps:

### Step 1: Understand the relationship between the activities
Let \( n_0 \) be the number of `C^(14)` atoms in fresh trees. The number of `C^(14)` atoms in the ancient wooden piece is given as \( \frac{3}{5} n_0 \).

### Step 2: Relate the number of atoms to time
The number of radioactive atoms decreases over time according to the equation:
\[ N(t) = N_0 e^{-\alpha t} \]
where:
- \( N(t) \) is the number of atoms at time \( t \),
- \( N_0 \) is the initial number of atoms,
- \( \alpha \) is the decay constant,
- \( t \) is the time elapsed.

Substituting the values, we have:
\[ \frac{3}{5} n_0 = n_0 e^{-\alpha t} \]

### Step 3: Simplify the equation
Dividing both sides by \( n_0 \) (assuming \( n_0 \neq 0 \)):
\[ \frac{3}{5} = e^{-\alpha t} \]

### Step 4: Take the natural logarithm
Taking the natural logarithm of both sides gives:
\[ \ln\left(\frac{3}{5}\right) = -\alpha t \]

### Step 5: Express \( \alpha \) in terms of half-life
The decay constant \( \alpha \) is related to the half-life \( T_{1/2} \) by the formula:
\[ \alpha = \frac{\ln(2)}{T_{1/2}} \]
Given that the half-life of `C^(14)` is \( 5570 \) years, we have:
\[ \alpha = \frac{\ln(2)}{5570} \]

### Step 6: Substitute \( \alpha \) into the equation
Substituting \( \alpha \) into the equation:
\[ \ln\left(\frac{3}{5}\right) = -\frac{\ln(2)}{5570} t \]

### Step 7: Solve for \( t \)
Rearranging gives:
\[ t = -\frac{5570 \ln\left(\frac{3}{5}\right)}{\ln(2)} \]

### Step 8: Calculate \( \ln\left(\frac{3}{5}\right) \)
Using the property of logarithms:
\[ \ln\left(\frac{3}{5}\right) = \ln(3) - \ln(5) \]
We can use the given value \( \ln\left(\frac{5}{3}\right) = 0.5 \), thus:
\[ \ln\left(\frac{3}{5}\right) = -0.5 \]

### Step 9: Substitute and calculate \( t \)
Now substituting this back into the equation for \( t \):
\[ t = -\frac{5570 \cdot (-0.5)}{\ln(2)} \]

### Step 10: Calculate \( \ln(2) \)
Using the approximate value \( \ln(2) \approx 0.693 \):
\[ t = \frac{5570 \cdot 0.5}{0.693} \]

### Step 11: Final calculation
Calculating this gives:
\[ t \approx \frac{2785}{0.693} \approx 4018.7 \text{ years} \]

### Conclusion
Thus, the age of the ancient wooden piece is approximately \( 4000 \) years.