If a radioactive element having half-life of `30min` is undergoing beta decay,the fraction of radioactive element remains undecayed after `90min`.will be

Options

A. `(1)/(8)`
B. `(1)/(2)`
C. `(1)/(4)`
D. `(1)/(16)`

Answer

To solve the problem of determining the fraction of a radioactive element that remains undecayed after 90 minutes, given its half-life of 30 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**: The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, the half-life is given as 30 minutes.

2. **Calculate the Number of Half-Lives in 90 Minutes**:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{90 \text{ minutes}}{30 \text{ minutes}} = 3 \]
This means that 90 minutes corresponds to 3 half-lives.

3. **Determine the Remaining Fraction After Each Half-Life**:
- After the first half-life (30 minutes), the fraction remaining is:
\[ \frac{1}{2} \text{ of } N_0 \]
- After the second half-life (60 minutes), the fraction remaining is:
\[ \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \text{ of } N_0 \]
- After the third half-life (90 minutes), the fraction remaining is:
\[ \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \text{ of } N_0 \]

4. **Calculate the Fraction of Undecayed Nuclei**:
The fraction of the radioactive element that remains undecayed after 90 minutes is:
\[ \text{Fraction remaining} = \frac{N}{N_0} = \frac{1}{8} \]

### Final Answer:
The fraction of the radioactive element that remains undecayed after 90 minutes is \(\frac{1}{8}\).