The half-life for decay of `.^(14)C " by " beta-`emission is 5730 years. The fraction of `.^(14)C` decays, in a sample that is 22,920 years old, would be

Options

A. `1//8`
B. `1//16`
C. `7//8`
D. `15//16`

Explanation

To solve the problem of determining the fraction of Carbon-14 that has decayed in a sample that is 22,920 years old, we can follow these steps:

### Step 1: Understand the half-life concept
The half-life of a radioactive substance is the time required for half of the substance to decay. For Carbon-14, the half-life is given as 5730 years.

### Step 2: Calculate the number of half-lives that have passed
To find out how many half-lives have passed in 22,920 years, we can divide the total time by the half-life:

\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{22,920 \text{ years}}{5730 \text{ years}} = 4 \]

### Step 3: Determine the remaining fraction of Carbon-14
After each half-life, the remaining amount of Carbon-14 can be calculated as follows:

- After 1 half-life: \( \frac{1}{2} \)
- After 2 half-lives: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
- After 3 half-lives: \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \)
- After 4 half-lives: \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \)

Thus, after 4 half-lives, the remaining fraction of Carbon-14 is:

\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^4 = \frac{1}{16} \]

### Step 4: Calculate the fraction that has decayed
To find the fraction that has decayed, we subtract the remaining fraction from 1:

\[ \text{Fraction decayed} = 1 - \text{Remaining fraction} = 1 - \frac{1}{16} = \frac{16}{16} - \frac{1}{16} = \frac{15}{16} \]

### Conclusion
The fraction of Carbon-14 that has decayed in a sample that is 22,920 years old is:

\[ \frac{15}{16} \]