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

Options

A. a.`0.5xx10^10
B. b. `2xx10^10
C. c.`3.5xx10^10`
D. `d. 1xx10^10`

Explanation

To solve the problem step by step, we will use the formula related to radioactive decay and the concept of half-life.

### Step 1: Identify the given values
- Initial number of active nuclei, \( N_0 = 4 \times 10^{10} \)
- Half-life, \( t_{1/2} = 10 \) days
- Total time, \( t = 30 \) days

### Step 2: Calculate the number of half-lives
To find the number of half-lives that have passed in 30 days, we can use the formula:
\[ n = \frac{t}{t_{1/2}} \]
Substituting the values:
\[ n = \frac{30 \text{ days}}{10 \text{ days}} = 3 \]
So, 3 half-lives have passed.

### Step 3: Calculate the number of undecayed nuclei
We can use the formula for the remaining number of undecayed nuclei after \( n \) half-lives:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
Substituting the values:
\[ N = 4 \times 10^{10} \left(\frac{1}{2}\right)^3 \]
Calculating \( \left(\frac{1}{2}\right)^3 \):
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Now substituting back:
\[ N = 4 \times 10^{10} \times \frac{1}{8} = \frac{4 \times 10^{10}}{8} = 0.5 \times 10^{10} \]

### Step 4: Calculate the number of decayed nuclei
The number of decayed nuclei can be found by subtracting the number of undecayed nuclei from the initial number of active nuclei:
\[ \text{Decayed nuclei} = N_0 - N \]
Substituting the values:
\[ \text{Decayed nuclei} = 4 \times 10^{10} - 0.5 \times 10^{10} = 3.5 \times 10^{10} \]

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