Half life of a ratio-active element is 8 years, how much amount will be present after 32 years ?

Options

A. `(1)/(4)`
B. `(1)/(8)`
C. `(1)/(16)`
D. `(1)/(32)`

Answer

To solve the problem of how much of a radioactive element will remain after 32 years given its half-life of 8 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the concept of half-life**: The half-life of a radioactive element is the time taken for half of the radioactive atoms to decay. In this case, the half-life is given as 8 years.

2. **Determine the number of half-lives in 32 years**:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{32 \text{ years}}{8 \text{ years}} = 4 \]
This means that 32 years is equivalent to 4 half-lives.

3. **Calculate the remaining amount after each half-life**:
- Start with an initial amount, which we can assume to be 1 unit (this can represent any quantity).
- After the first half-life (8 years), the amount remaining is:
\[ \text{Amount after 1st half-life} = \frac{1}{2} \]
- After the second half-life (16 years), the amount remaining is:
\[ \text{Amount after 2nd half-life} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
- After the third half-life (24 years), the amount remaining is:
\[ \text{Amount after 3rd half-life} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]
- After the fourth half-life (32 years), the amount remaining is:
\[ \text{Amount after 4th half-life} = \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \]

4. **Conclusion**: After 32 years, the amount of the radioactive element remaining is \( \frac{1}{16} \) of the initial amount.

### Final Answer:
The amount of the radioactive element present after 32 years is \( \frac{1}{16} \) of the initial amount.