A radioactive sample remains undecayed `9/16` after time t.How much sample remains undecayed after time `t/2`

Options

A. `3/4`
B. `9/16`
C. `4/3`
D. `16/9`

Answer

To solve the problem of how much of a radioactive sample remains undecayed after time \( t/2 \), we can follow these steps:

### Step 1: Understand the initial conditions
Let the initial number of undecayed atoms in the sample be \( N_0 \). After time \( t \), the number of undecayed atoms is given as \( \frac{9}{16} N_0 \).

### Step 2: Use the radioactive decay formula
The radioactive decay of a sample can be described by the equation:
\[ N_t = N_0 e^{-\lambda t} \]
where:
- \( N_t \) is the number of undecayed atoms at time \( t \),
- \( N_0 \) is the initial number of undecayed atoms,
- \( \lambda \) is the decay constant,
- \( t \) is the time elapsed.

### Step 3: Set up the equation for time \( t \)
From the information given, we can write:
\[ \frac{N_t}{N_0} = e^{-\lambda t} = \frac{9}{16} \]

### Step 4: Find the expression for \( N_{t/2} \)
Now, we want to find the number of undecayed atoms after time \( t/2 \). Using the same decay formula:
\[ N_{t/2} = N_0 e^{-\lambda (t/2)} \]
We can express this in terms of \( N_0 \):
\[ \frac{N_{t/2}}{N_0} = e^{-\lambda (t/2)} \]

### Step 5: Relate \( e^{-\lambda (t/2)} \) to \( e^{-\lambda t} \)
We know that:
\[ e^{-\lambda t} = \frac{9}{16} \]
Now, we can express \( e^{-\lambda (t/2)} \) as:
\[ e^{-\lambda (t/2)} = \left(e^{-\lambda t}\right)^{1/2} = \left(\frac{9}{16}\right)^{1/2} = \frac{3}{4} \]

### Step 6: Calculate \( N_{t/2} \)
Thus, we have:
\[ \frac{N_{t/2}}{N_0} = \frac{3}{4} \]
This means that after time \( t/2 \), the number of undecayed atoms is:
\[ N_{t/2} = \frac{3}{4} N_0 \]

### Conclusion
The amount of the radioactive sample that remains undecayed after time \( t/2 \) is \( \frac{3}{4} N_0 \).

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