`C^(14)` has a half life of 5700 yrs. At the end of 11400 years, the actual amount left is

Options

A. (a)0.5 times of original amount
B. (b)0.25 times of original amount
C. (c)0.125 times of original amount
D. (d)0.0625 times of original amount

Answer

To solve the problem of how much Carbon-14 (C-14) remains after 11,400 years, we can follow these steps:

### Step 1: Understand the half-life concept
The half-life of a radioactive substance is the time it takes for half of the substance to decay. For C-14, the half-life is given as 5,700 years.

### Step 2: Determine how many half-lives have passed
To find out how many half-lives fit into 11,400 years, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{11400 \text{ years}}{5700 \text{ years}} = 2 \]

### Step 3: Apply the half-life formula
The remaining amount of a radioactive substance after a certain number of half-lives can be calculated using the formula:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \]
where:
- \(N\) is the remaining amount,
- \(N_0\) is the initial amount,
- \(n\) is the number of half-lives.

From Step 2, we found \(n = 2\).

### Step 4: Calculate the remaining amount
Substituting \(n = 2\) into the formula:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
This means that after 11,400 years, \(\frac{1}{4}\) of the original amount remains.

### Step 5: Conclusion
Thus, the actual amount left after 11,400 years is:
\[ N = 0.25 \times N_0 \]
This indicates that 25% of the original amount of C-14 is left.

### Final Answer
The actual amount of C-14 left after 11,400 years is \(0.25 \times N_0\).

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