If `T` is the half-life of a radioactive material, then the fraction that would remain after a time `(T)/(2)` is

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. `(1)/(sqrt(2))`
D. `(sqrt(2) -1)/(sqrt(2))`

Explanation

To find the fraction of a radioactive material that remains after a time of \( \frac{T}{2} \), where \( T \) is the half-life of the material, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**: The half-life \( T \) is the time required for half of the radioactive material to decay. After one half-life, the fraction of the material remaining is \( \frac{1}{2} \).

2. **Using the Decay Formula**: The general formula for the remaining quantity of a radioactive substance is given by:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
where:
- \( N \) is the remaining quantity after time \( t \),
- \( N_0 \) is the initial quantity,
- \( T \) is the half-life,
- \( t \) is the elapsed time.

3. **Substituting \( t = \frac{T}{2} \)**: We need to find the fraction remaining after a time of \( \frac{T}{2} \):
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{\frac{T}{2}}{T}} \]

4. **Simplifying the Exponent**: The exponent simplifies as follows:
\[ \frac{\frac{T}{2}}{T} = \frac{1}{2} \]
Therefore, the equation becomes:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{1}{2}} \]

5. **Calculating the Fraction**: The fraction \( \frac{N}{N_0} \) is:
\[ \frac{N}{N_0} = \left( \frac{1}{2} \right)^{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

6. **Final Result**: Thus, the fraction of the radioactive material that remains after a time of \( \frac{T}{2} \) is:
\[ \frac{1}{\sqrt{2}} \]

### Conclusion:
The fraction that would remain after a time \( \frac{T}{2} \) is \( \frac{1}{\sqrt{2}} \).