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

Options

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

Explanation

To find the fraction of a radioactive material that remains after a certain period, we can use the concept of half-life. The half-life is the time taken for half of the radioactive material to decay.

**Given:**
- Half-life (T) = 10 days
- Time elapsed (t) = 30 days

**Step 1: Determine the number of half-lives that have passed.**
To find out how many half-lives fit into the elapsed time, we can use the formula:
\[ \text{Number of half-lives} = \frac{t}{T} \]
Substituting the values:
\[ \text{Number of half-lives} = \frac{30 \text{ days}}{10 \text{ days}} = 3 \]

**Step 2: Calculate the remaining fraction of the material.**
The remaining fraction of the radioactive material after a certain number of half-lives can be calculated using the formula:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^{\text{Number of half-lives}} \]
Substituting the number of half-lives:
\[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^{3} = \frac{1}{8} \]

**Step 3: Convert the fraction to decimal form.**
To express the fraction in decimal form:
\[ \frac{1}{8} = 0.125 \]

**Final Answer:**
The fraction of the radioactive material that remains after 30 days is \( \frac{1}{8} \) or 0.125.

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