A saample of radioactive elements contains `4xx10^(16)` active nuclei. If half-life of element is `10` days, then the number of decayed nuclei after `30` days is

Options

A. `0.5xx10^(10)`
B. `2xx10^(10)`
C. `3.5xx10^(10)`
D. `1xx10^(10)`

Answer

To solve the problem, we need to determine the number of decayed nuclei after 30 days given the initial number of active nuclei and the half-life of the radioactive element.

### Step-by-Step Solution:

1. **Identify the Initial Number of Nuclei**:
The initial number of active nuclei, \( N_0 \), is given as:
\[ N_0 = 4 \times 10^{16} \]

2. **Determine the Half-Life**:
The half-life of the element, \( t_{1/2} \), is given as:
\[ t_{1/2} = 10 \text{ days} \]

3. **Calculate the Total Time**:
We need to find the number of decayed nuclei after \( t = 30 \) days.

4. **Calculate the Number of Half-Lives**:
The number of half-lives that have passed in 30 days can be calculated as:
\[ n = \frac{t}{t_{1/2}} = \frac{30 \text{ days}}{10 \text{ days}} = 3 \]

5. **Calculate the Remaining Nuclei After 30 Days**:
The remaining number of nuclei after \( n \) half-lives can be calculated using the formula:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
Substituting the values:
\[ N = 4 \times 10^{16} \left( \frac{1}{2} \right)^3 = 4 \times 10^{16} \times \frac{1}{8} = \frac{4 \times 10^{16}}{8} = 0.5 \times 10^{16} \]

6. **Convert to Standard Form**:
Converting \( 0.5 \times 10^{16} \) to standard form gives:
\[ N = 5 \times 10^{15} \]

7. **Calculate the Number of Decayed Nuclei**:
The number of decayed nuclei, \( N_d \), can be calculated as:
\[ N_d = N_0 - N \]
Substituting the values:
\[ N_d = 4 \times 10^{16} - 0.5 \times 10^{16} = 3.5 \times 10^{16} \]

### Final Answer:
The number of decayed nuclei after 30 days is:
\[ \boxed{3.5 \times 10^{16}} \]