A piece of wood from a recently cut tree shows 20 decays per minute. A wooden piece of same size placed in a museum (obtained from a tree cut many years back) shows 2 decays per minute. If half-life of `C^(14)` is 5,730 years, then age of the wooden piece placed in the museum is approximately

Options

A. 10,439 years
B. 13,094 years
C. 19,039 years
D. 39,049 years

Explanation

To find the age of the wooden piece placed in the museum, we can use the concept of radioactive decay and the half-life of Carbon-14 (\(C^{14}\)). Here’s a step-by-step solution:

### Step 1: Understand the decay rates
We know the activity (decay rate) of the two pieces of wood:
- Recently cut wood: \(A_0 = 20\) decays per minute
- Museum wood: \(A = 2\) decays per minute

### Step 2: Use the decay formula
The relationship between the initial activity (\(A_0\)), the current activity (\(A\)), and time (\(t\)) can be expressed using the decay formula:
\[ A = A_0 e^{-\lambda t} \]
Where \(\lambda\) is the decay constant.

### Step 3: Relate the decay constant to half-life
The decay constant \(\lambda\) can be related to the half-life (\(t_{1/2}\)) using the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Given that the half-life of \(C^{14}\) is \(5730\) years, we can calculate \(\lambda\):
\[ \lambda = \frac{0.693}{5730} \approx 1.21 \times 10^{-4} \text{ years}^{-1} \]

### Step 4: Substitute values into the decay formula
Now we can substitute the values into the decay formula:
\[ 2 = 20 e^{-\lambda t} \]
Dividing both sides by \(20\):
\[ \frac{2}{20} = e^{-\lambda t} \]
\[ 0.1 = e^{-\lambda t} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ \ln(0.1) = -\lambda t \]
Calculating \(\ln(0.1)\):
\[ \ln(0.1) \approx -2.302 \]
Thus, we have:
\[ -2.302 = -\lambda t \]
Substituting \(\lambda\):
\[ -2.302 = -\left(\frac{0.693}{5730}\right) t \]

### Step 6: Solve for \(t\)
Rearranging gives:
\[ t = \frac{2.302 \times 5730}{0.693} \]
Calculating this:
\[ t \approx \frac{13163.46}{0.693} \approx 19038.7 \text{ years} \]

### Conclusion
The age of the wooden piece placed in the museum is approximately **19039 years**.