For a radioactive material, half-life is `10` minutes. If initially there are `600` number of nuclei, the time taken (in minutes) for the disintegration of `450` nuclei is.

Options

A. 15
B. 20
C. 30
D. 10

Explanation

To solve the problem, we need to determine the time taken for the disintegration of 450 nuclei from an initial amount of 600 nuclei, given that the half-life of the radioactive material is 10 minutes.

### Step-by-Step Solution:

1. **Identify Initial and Final Nuclei:**
- Initial number of nuclei, \( N_0 = 600 \)
- Number of disintegrated nuclei, \( N' = 450 \)
- Remaining nuclei, \( N = N_0 - N' = 600 - 450 = 150 \)

2. **Use the Half-Life Formula:**
The relationship between the number of remaining nuclei and the initial number of nuclei can be expressed using the half-life formula:
\[ N = N_0 \left(\frac{1}{2}\right)^{\frac{T}{T_{1/2}}} \]
where \( T_{1/2} \) is the half-life of the material.

3. **Substitute Known Values:**
Here, \( T_{1/2} = 10 \) minutes, so we can substitute the values into the equation:
\[ 150 = 600 \left(\frac{1}{2}\right)^{\frac{T}{10}} \]

4. **Simplify the Equation:**
Divide both sides by 600:
\[ \frac{150}{600} = \left(\frac{1}{2}\right)^{\frac{T}{10}} \]
This simplifies to:
\[ \frac{1}{4} = \left(\frac{1}{2}\right)^{\frac{T}{10}} \]

5. **Express \(\frac{1}{4}\) as a Power of \(\frac{1}{2}\):**
We know that:
\[ \frac{1}{4} = \left(\frac{1}{2}\right)^{2} \]
Therefore, we can equate the powers:
\[ \left(\frac{1}{2}\right)^{2} = \left(\frac{1}{2}\right)^{\frac{T}{10}} \]

6. **Set the Exponents Equal:**
Since the bases are the same, we can set the exponents equal to each other:
\[ 2 = \frac{T}{10} \]

7. **Solve for \( T \):**
Multiply both sides by 10 to solve for \( T \):
\[ T = 20 \text{ minutes} \]

### Final Answer:
The time taken for the disintegration of 450 nuclei is **20 minutes**.