The half life of a radioactive substance is 30 days. What is the time taken to disintegrate to `3//4^(th)` of its original mass ?

Options

A. 30 days
B. 15 days
C. 60 days
D. 90 days

Explanation

To solve the problem of determining the time taken for a radioactive substance to disintegrate to three-fourths of its original mass, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life is given as 30 days.

2. **Determine the Mass Remaining**:
If the original mass of the substance is \( m_0 \), then after disintegrating to three-fourths of its original mass, the remaining mass will be:
\[ m_t = m_0 - \frac{3}{4} m_0 = \frac{1}{4} m_0 \]
This means that after some time, only one-fourth of the original mass remains.

3. **Relate Remaining Mass to Half-Lives**:
We need to find out how many half-lives it takes for the original mass \( m_0 \) to reduce to \( \frac{1}{4} m_0 \).
- After 1 half-life (30 days), the remaining mass will be:
\[ m_0 \times \frac{1}{2} = \frac{1}{2} m_0 \]
- After 2 half-lives (60 days), the remaining mass will be:
\[ \frac{1}{2} m_0 \times \frac{1}{2} = \frac{1}{4} m_0 \]

4. **Calculate the Total Time**:
Since it takes 2 half-lives for the mass to reduce from \( m_0 \) to \( \frac{1}{4} m_0 \), and each half-life is 30 days, the total time taken is:
\[ \text{Total Time} = 2 \times 30 \text{ days} = 60 \text{ days} \]

5. **Conclusion**:
Therefore, the time taken for the radioactive substance to disintegrate to three-fourths of its original mass is **60 days**.

### Final Answer:
The time taken to disintegrate to three-fourths of its original mass is **60 days**.

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