The half life for radioactive decay of `.^(14)C` is 5730 years. An archaeological artifact containing wood had only `80%` of the `.^(14)C` found in a living tree. Estimat the age of the sample.

Options

A. 1485 years
B. 1845 years
C. 530 years
D. 4767 years

Explanation

To estimate the age of the archaeological artifact containing wood with only 80% of the carbon-14 found in a living tree, we can follow these steps:

### Step 1: Understand the Problem
We know that the half-life of carbon-14 (\(^{14}C\)) is 5730 years. The artifact has 80% of the carbon-14 compared to a living tree.

### Step 2: Determine the Decay Constant
The decay constant (\(k\)) can be calculated using the formula:
\[ k = \frac{0.693}{T_{1/2}} \]
where \(T_{1/2}\) is the half-life of the substance.

Substituting the values:
\[ k = \frac{0.693}{5730 \text{ years}} \approx 1.209 \times 10^{-4} \text{ years}^{-1} \]

### Step 3: Set Up the Age Calculation Formula
The age of the sample can be calculated using the formula:
\[ t = \frac{2.303}{k} \log \left(\frac{N_0}{N}\right) \]
where:
- \(t\) is the age of the sample,
- \(N_0\) is the initial amount of carbon-14 (100%),
- \(N\) is the remaining amount of carbon-14 (80%).

### Step 4: Substitute the Values
Here, \(N_0 = 100\) and \(N = 80\). Now we can substitute these values into the equation:
\[ t = \frac{2.303}{1.209 \times 10^{-4}} \log \left(\frac{100}{80}\right) \]

### Step 5: Calculate the Logarithm
First, calculate the logarithm:
\[ \log \left(\frac{100}{80}\right) = \log (1.25) \approx 0.09691 \]

### Step 6: Calculate the Age
Now substitute this back into the equation for \(t\):
\[ t = \frac{2.303}{1.209 \times 10^{-4}} \times 0.09691 \]
Calculating this gives:
\[ t \approx \frac{2.303 \times 0.09691}{1.209 \times 10^{-4}} \approx 1845 \text{ years} \]

### Conclusion
The estimated age of the archaeological artifact is approximately **1845 years**.

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