The age of the wood if only `1//16` part of original `C^(14)` is present in its piece is (in years) ( `T` of `C^(14)` is `5,580` years)

Options

A. 5580 years
B. 11160 years
C. 22320 years
D. 16740 years

Answer

To find the age of the wood given that only \( \frac{1}{16} \) of the original \( C^{14} \) is present, we can follow these steps:

### Step 1: Understand the relationship between the remaining quantity and the original quantity
Let:
- \( N_0 \) = initial amount of \( C^{14} \)
- \( N \) = remaining amount of \( C^{14} \)

Given that \( N = \frac{1}{16} N_0 \).

### Step 2: Use the radioactive decay formula
The decay of a radioactive substance can be described by the equation:
\[ N = N_0 e^{-\lambda t} \]
Where:
- \( \lambda \) = decay constant
- \( t \) = time elapsed

### Step 3: Set up the equation with the given information
Substituting \( N \) in the decay equation:
\[ \frac{1}{16} N_0 = N_0 e^{-\lambda t} \]

### Step 4: Simplify the equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{16} = e^{-\lambda t} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ \ln\left(\frac{1}{16}\right) = -\lambda t \]

### Step 6: Express \( \frac{1}{16} \) in terms of powers of 2
We know that:
\[ \frac{1}{16} = 2^{-4} \implies \ln\left(\frac{1}{16}\right) = -4 \ln(2) \]
So we can rewrite the equation as:
\[ -4 \ln(2) = -\lambda t \]

### Step 7: Solve for \( t \)
Rearranging gives:
\[ t = \frac{4 \ln(2)}{\lambda} \]

### Step 8: Calculate the decay constant \( \lambda \)
The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
Given \( T_{1/2} = 5580 \) years, we have:
\[ \lambda = \frac{\ln(2)}{5580} \]

### Step 9: Substitute \( \lambda \) back into the equation for \( t \)
Substituting \( \lambda \) into the equation for \( t \):
\[ t = \frac{4 \ln(2)}{\frac{\ln(2)}{5580}} = 4 \times 5580 \]

### Step 10: Calculate \( t \)
Calculating gives:
\[ t = 22320 \text{ years} \]

### Conclusion
The age of the wood is \( 22320 \) years.

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