A radioactive element has a half-life of 20 minutes. How much time should elapse before the element is reduced to `(1)/(8)th` of the original mass

Options

A. 40 minutes
B. 60 minutes
C. 80 minutes
D. 160 minutes

Explanation

To solve the problem of how much time should elapse before a radioactive element is reduced to \( \frac{1}{8} \) of its original mass, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Half-Life**: The half-life of a radioactive element is the time it takes for half of the substance to decay. In this case, the half-life is given as 20 minutes.

2. **Determine the Number of Half-Lives**: We need to find out how many half-lives it takes for the substance to reduce to \( \frac{1}{8} \) of its original mass.
- After 1 half-life (20 minutes), the mass will be \( \frac{1}{2} \) of the original mass.
- After 2 half-lives (40 minutes), the mass will be \( \frac{1}{4} \) of the original mass.
- After 3 half-lives (60 minutes), the mass will be \( \frac{1}{8} \) of the original mass.

3. **Calculate the Total Time**: Since it takes 3 half-lives to reach \( \frac{1}{8} \) of the original mass, we multiply the number of half-lives by the duration of each half-life:
\[ \text{Total Time} = 3 \times \text{Half-life} = 3 \times 20 \text{ minutes} = 60 \text{ minutes} \]

4. **Conclusion**: Therefore, the time that should elapse before the element is reduced to \( \frac{1}{8} \) of its original mass is 60 minutes.

### Final Answer:
The time that should elapse is **60 minutes**.

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