A radioactive material has half-life of `10` days. What fraction of the material would remain after `30` days ?

Options

A. 0.5
B. 0.25
C. 0.125
D. 0.33

Explanation

To solve the problem of how much fraction of a radioactive material remains after 30 days given its half-life of 10 days, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a radioactive material is the time taken for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 10 days.

### Step 2: Calculate the number of half-lives in 30 days
To find out how many half-lives fit into 30 days, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{30 \text{ days}}{10 \text{ days}} = 3 \]

### Step 3: Determine the fraction remaining after each half-life
After each half-life, the amount of radioactive material remaining is halved. Therefore, after \( n \) half-lives, the fraction of material remaining can be calculated using the formula:
\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives.

### Step 4: Apply the number of half-lives to the formula
Since we have determined that there are 3 half-lives in 30 days, we substitute \( n = 3 \) into the formula:
\[ \text{Fraction remaining} = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]

### Step 5: Convert the fraction to decimal form
To express the fraction \( \frac{1}{8} \) in decimal form:
\[ \frac{1}{8} = 0.125 \]

### Conclusion
Thus, the fraction of the radioactive material that remains after 30 days is \( 0.125 \) or \( \frac{1}{8} \).