The half life period of a radioactive material is 15 minutes. What percent will remain after 45 minutes ?

Options

A. `10%`
B. `12.5%`
C. `15%`
D. `17.5%`

Explanation

To solve the problem of how much of a radioactive material remains after 45 minutes given its half-life of 15 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Half-Life Period**:
The half-life of the radioactive material is given as 15 minutes.

2. **Determine the Total Time**:
We need to find out how much of the material remains after 45 minutes.

3. **Calculate the Number of Half-Lives**:
To find the number of half-lives that fit into 45 minutes, we divide the total time by the half-life period:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{45 \text{ minutes}}{15 \text{ minutes}} = 3 \]
So, there are 3 half-lives in 45 minutes.

4. **Calculate Remaining Material**:
After each half-life, the amount of radioactive material remaining is halved. The formula for the remaining quantity after \( n \) half-lives is:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^n \]
Substituting \( n = 3 \):
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

5. **Convert to Percentage**:
To find the percentage of the material remaining, we multiply the remaining fraction by 100:
\[ \text{Percentage remaining} = \frac{1}{8} \times 100 = 12.5\% \]

### Final Answer:
After 45 minutes, 12.5% of the radioactive material will remain.

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