2 g of a radioactive sample having half-life of 15 days was synthesised on 1st Jan. 2009. What is the amount of the sample left behind on 1st March, 2009 (including both the days) in g?

Explanation

To determine the amount of a radioactive sample left after a certain period, we can use the concept of half-life. Here’s how we can solve the problem step by step:

### Step 1: Understand the Problem
We have a radioactive sample of 2 g with a half-life of 15 days. We need to find out how much of the sample remains on 1st March 2009, including both 1st January and 1st March.

### Step 2: Calculate the Time Elapsed
From 1st January to 1st March, there are 59 days (31 days in January and 28 days in February). Therefore, the total time elapsed is:
\[ \text{Total Time} = 31 \text{ (January)} + 28 \text{ (February)} = 59 \text{ days} \]

### Step 3: Calculate the Number of Half-Lives
Next, we need to determine how many half-lives fit into the elapsed time. The half-life of the sample is 15 days. Thus, we calculate the number of half-lives:
\[ \text{Number of Half-Lives} = \frac{\text{Total Time}}{\text{Half-Life}} = \frac{59 \text{ days}}{15 \text{ days}} \approx 3.93 \]
This means that approximately 3.93 half-lives have passed.

### Step 4: Calculate the Remaining Amount of Sample
The remaining amount of a radioactive sample can be calculated using the formula:
\[ A = A_0 \left( \frac{1}{2} \right)^n \]
where:
- \( A_0 \) = initial amount of the sample (2 g)
- \( n \) = number of half-lives (3.93)

Substituting the values:
\[ A = 2 \left( \frac{1}{2} \right)^{3.93} \]
Calculating \( \left( \frac{1}{2} \right)^{3.93} \):
\[ \left( \frac{1}{2} \right)^{3.93} \approx 0.125 \]
Now, substituting back into the equation:
\[ A \approx 2 \times 0.125 \approx 0.25 \text{ g} \]

### Step 5: Conclusion
Thus, the amount of the radioactive sample left on 1st March 2009 is approximately:
\[ \boxed{0.25 \text{ g}} \]

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