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 initial amount after 16 days, we can follow these steps:

### Step 1: Understand the Remaining Amount
We know that after 16 days, 25% of the initial amount of the radioactive material remains. This can be expressed as:
\[ \text{Remaining Amount} = \frac{25}{100} \times I = \frac{1}{4} I \]
where \( I \) is the initial amount.

### Step 2: Relate Remaining Amount to Half-Lives
The remaining amount of \( \frac{1}{4} I \) indicates that the material has undergone decay through multiple half-lives. Specifically, we can express \( \frac{1}{4} \) in terms of half-lives:
\[ \frac{1}{4} = \left(\frac{1}{2}\right)^2 \]
This means that the material has gone through 2 half-lives to reach \( \frac{1}{4} I \).

### Step 3: Set Up the Equation
Let \( T_{1/2} \) be the half-life of the material. Since it takes 2 half-lives to decay to \( \frac{1}{4} I \), we can write:
\[ 2 \times T_{1/2} = 16 \text{ days} \]

### Step 4: Solve for Half-Life
Now, we can solve for \( T_{1/2} \):
\[ T_{1/2} = \frac{16 \text{ days}}{2} = 8 \text{ days} \]

### Conclusion
The half-life of the radioactive material is \( 8 \) days.

---