The half-life of redioactive element is 600 yr. The fraction of sample that would remain after 3000 yr is

Options

A. `1//2`
B. `1//16`
C. `1//8`
D. `1//32`

Explanation

To find the fraction of a radioactive sample that remains after a certain period, we can use the concept of half-life. Here's a step-by-step solution to the problem:

### Step 1: Understand the half-life concept
The half-life of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life (T_half) is given as 600 years.

### Step 2: Determine the total time elapsed
We need to find out how much time has passed. The total time (T) given in the question is 3000 years.

### Step 3: Calculate the number of half-lives
To find out how many half-lives fit into the total time, we can use the formula:
\[ n = \frac{T}{T_{half}} \]
Where:
- \( n \) is the number of half-lives,
- \( T \) is the total time (3000 years),
- \( T_{half} \) is the half-life (600 years).

Substituting the values:
\[ n = \frac{3000 \text{ years}}{600 \text{ years}} = 5 \]

### Step 4: Calculate the remaining fraction of the sample
The fraction of the sample remaining after \( n \) half-lives can be calculated using the formula:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^n \]

Substituting the value of \( n \):
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^5 = \frac{1}{32} \]

### Step 5: Conclusion
Thus, the fraction of the sample that would remain after 3000 years is \( \frac{1}{32} \).

### Final Answer
The fraction of the sample that would remain after 3000 years is \( \frac{1}{32} \).

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