Carbon`-14` used to determine the age of organic material. The procedure is absed on the formation of `C^(14)` by neutron capture iin the upper atmosphere.
`._(7)N^(14)+._(0)n^(1) rarr ._(6)C^(14)+._(1)H^(1)`
`C^(14)` is absorbed by living organisms during photosynthesis. The `C^(14)` content is constant in living organism. Once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of `C^(14)` in the dead being falls due to the decay, which `C^(14)` undergoes.
`._(6)C^(14)rarr ._(7)N^(14)+beta^(c-)`
The half`-` life period of `C^(14)` is 5770 year. The decay constant `(lambda)` can be calculated by using the following formuls `:`
`lambda=(0.693)/(t_(1//2))`
The comparison of the `beta^(c-)` activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method, however, ceases to be accurate over periods longer than 30000 years. The proportion of `C^(14)` to `C^(12)` in living matter is `1:10^(12)`.
What should be the age of fossil for meaningful determination of its age ?

Options

A. 6 years
B. 6000 years
C. 60000years
D. It can be used to calculate any age

Explanation

To determine the age of a fossil using Carbon-14 dating, we need to consider the following steps:

### Step 1: Understand the Half-Life of Carbon-14
The half-life of Carbon-14 (C-14) is given as 5770 years. This means that after 5770 years, half of the original amount of C-14 will have decayed.

### Step 2: Calculate the Decay Constant (λ)
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} \]
Substituting the half-life into the formula:
\[ \lambda = \frac{0.693}{5770 \text{ years}} \approx 1.20 \times 10^{-4} \text{ year}^{-1} \]

### Step 3: Understand the Proportion of C-14 to C-12
The proportion of C-14 to C-12 in living organisms is given as 1:10^12. This ratio is crucial for determining the age of fossils because it indicates how much C-14 is present in comparison to C-12.

### Step 4: Determine the Effective Age Range for C-14 Dating
The question states that the method ceases to be accurate over periods longer than 30,000 years. Therefore, for a meaningful determination of the fossil's age, it should be less than or equal to 30,000 years.

### Step 5: Evaluate the Options
Given the options, we need to select an age that is suitable for C-14 dating. The options include various ages, and we need to find the one that is less than or equal to 30,000 years.

### Conclusion
From the analysis, the age of the fossil for a meaningful determination using C-14 dating should be less than or equal to 30,000 years. Therefore, the most suitable option from the provided choices would be 6000 years, as it is well within the effective range of C-14 dating.

### Final Answer
The age of the fossil for meaningful determination of its age is **6000 years**.

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