If a radioactive material remains 25% after 16 days, then its half life will be

Options

A. 32 days
B. 8 days
C. 64 days
D. 28 days

Answer

To find the half-life of a radioactive material that remains at 25% of its original amount after 16 days, we can follow these steps:

### Step 1: Understand the relationship between remaining material and half-lives
If a radioactive material remains at 25% of its original amount, it means that it has gone through a certain number of half-lives. Specifically, 25% is equivalent to \( \frac{N_0}{4} \), where \( N_0 \) is the initial quantity of the material.

### Step 2: Determine how many half-lives have passed
Each half-life reduces the quantity of the material by half. Therefore:
- After 1 half-life, the remaining amount is \( \frac{N_0}{2} \).
- After 2 half-lives, the remaining amount is \( \frac{N_0}{4} \).

This indicates that to reach 25% of the original amount, the material must have undergone 2 half-lives.

### Step 3: Relate the total time to half-lives
We know from the problem that the total time taken to reach 25% of the original amount is 16 days. Since this corresponds to 2 half-lives, we can express this relationship mathematically:
\[ 2 \times t_{1/2} = 16 \text{ days} \]

### Step 4: Solve for the half-life
To find the half-life (\( t_{1/2} \)), we can rearrange the equation:
\[ t_{1/2} = \frac{16 \text{ days}}{2} = 8 \text{ days} \]

### Conclusion
Thus, the half-life of the radioactive material is 8 days.

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