The percentage of quantity of a radioactive material that remains after 5 half-lives will be .

Options

A. `1%`
B. `0.3%`
C. `3.125%`
D. `0.2%`

Answer

To find the percentage of a radioactive material that remains after 5 half-lives, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a radioactive material is the time taken for half of the material to decay. After each half-life, the amount of material remaining is halved.

### Step 2: Determine the remaining quantity after multiple half-lives
If we start with an initial quantity \( n_0 \), after one half-life, the remaining quantity \( n_1 \) will be:
\[ n_1 = \frac{n_0}{2} \]
After two half-lives:
\[ n_2 = \frac{n_1}{2} = \frac{n_0}{2^2} \]
After three half-lives:
\[ n_3 = \frac{n_2}{2} = \frac{n_0}{2^3} \]
Continuing this pattern, after \( t \) half-lives, the remaining quantity \( n_t \) can be expressed as:
\[ n_t = \frac{n_0}{2^t} \]

### Step 3: Calculate the remaining quantity after 5 half-lives
For 5 half-lives, we substitute \( t = 5 \):
\[ n_5 = \frac{n_0}{2^5} = \frac{n_0}{32} \]

### Step 4: Calculate the percentage of the remaining quantity
To find the percentage of the remaining quantity, we use the formula:
\[ \text{Percentage remaining} = \left( \frac{n_5}{n_0} \right) \times 100 \]
Substituting \( n_5 \):
\[ \text{Percentage remaining} = \left( \frac{n_0/32}{n_0} \right) \times 100 = \left( \frac{1}{32} \right) \times 100 \]

### Step 5: Simplify the calculation
Calculating \( \frac{100}{32} \):
\[ \frac{100}{32} = 3.125 \]

### Final Answer
Thus, the percentage of the quantity of a radioactive material that remains after 5 half-lives is:
\[ \text{Percentage remaining} = 3.125\% \]