The half-life period of a radioactive element is 140 days. After 560 days, `1g` of the element will reduce to
a. `0.5 g` b. `0.25 g` c. `1//8 g` d. `1//16 g`

Options

A. a. `0.5 g`
B. b. `0.25 g`
C. c. `1//8 g`
D. d. `1//16 g`

Explanation

To solve the problem, we need to determine how much of a radioactive element remains after a certain period, given its half-life. Here's a step-by-step solution:

### Step 1: Understand the Half-Life Concept
The half-life of a radioactive element is the time required for half of the radioactive substance to decay. In this case, the half-life is given as 140 days.

### Step 2: Calculate the Total Time
We need to find out how much of the element remains after 560 days. We can express 560 days in terms of the half-life:
\[ 560 \text{ days} = 4 \times 140 \text{ days} \]
This means that 560 days corresponds to 4 half-lives.

### Step 3: Use the Half-Life Formula
The formula to calculate the remaining amount of a substance after \( n \) half-lives is:
\[ \text{Remaining amount} = \frac{\text{Initial amount}}{2^n} \]
Where:
- \( n \) is the number of half-lives
- The initial amount is given as 1 gram.

### Step 4: Substitute the Values
Since we have determined that \( n = 4 \) (because 560 days is 4 half-lives), we can substitute into the formula:
\[ \text{Remaining amount} = \frac{1 \text{ g}}{2^4} = \frac{1 \text{ g}}{16} = 0.0625 \text{ g} \]

### Step 5: Final Answer
Thus, after 560 days, the remaining amount of the radioactive element is:
\[ \frac{1}{16} \text{ g} \]
So the correct answer is **d. \( \frac{1}{16} \text{ g} \)**.

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