The ratio of the abundance of carbon -14 to carbon -12 in a sample of dead wood is one quarter the ratio for living wood.If the half life of carbon -14 is 5730 years, which one of the following expressions determines how many years ago the wood died?

Options

A. `2xx5730`
B. `0.75xx5730`
C. `4xx5730`
D. `0.50xx5730`

Answer

To solve the problem, we need to determine how many years ago the wood died based on the given information about the abundance of carbon-14 and carbon-12, and the half-life of carbon-14.

### Step-by-Step Solution:

1. **Understanding the Ratios:**
- Let the ratio of carbon-14 to carbon-12 in living wood be represented as \( R \).
- For dead wood, the ratio is given as \( \frac{R}{4} \).
- This indicates that the abundance of carbon-14 in dead wood is one quarter of that in living wood.

2. **Using the Decay Formula:**
- The amount of carbon-14 remaining after a certain time can be expressed using the formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{n} \]
where:
- \( N \) is the remaining amount of carbon-14,
- \( N_0 \) is the initial amount of carbon-14,
- \( n \) is the number of half-lives that have passed.

3. **Setting Up the Equation:**
- For living wood, the ratio is \( N_0 \) (initial amount).
- For dead wood, we have:
\[ N = N_0 \left( \frac{1}{2} \right)^{n} \]
- Since the ratio for dead wood is \( \frac{R}{4} \), we can set up the equation:
\[ \frac{N_0 \left( \frac{1}{2} \right)^{n}}{N_0} = \frac{1}{4} \]
- This simplifies to:
\[ \left( \frac{1}{2} \right)^{n} = \frac{1}{4} \]

4. **Solving for \( n \):**
- We know that \( \frac{1}{4} = \left( \frac{1}{2} \right)^{2} \).
- Therefore, we can equate the exponents:
\[ n = 2 \]
- This means that 2 half-lives have passed since the wood died.

5. **Calculating the Time:**
- The half-life of carbon-14 is given as 5730 years.
- To find the total time that has passed, we multiply the number of half-lives by the half-life duration:
\[ \text{Total time} = n \times \text{Half-life} = 2 \times 5730 = 11460 \text{ years} \]

### Final Answer:
The wood died approximately **11460 years ago**.

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