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

Options

A. `31%`
B. `3.125%`
C. `0.3%`
D. `1%`

Explanation

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 required for half of the material to decay. After each half-life, the quantity of the material is halved.

### Step 2: Calculate the remaining quantity after n half-lives
The formula to calculate the remaining quantity after n half-lives is:
\[ \text{Remaining quantity} = \left(\frac{1}{2}\right)^n \times \text{Initial quantity} \]
where \( n \) is the number of half-lives.

### Step 3: Substitute n with 5
In this case, we want to find the remaining quantity after 5 half-lives, so we substitute \( n = 5 \):
\[ \text{Remaining quantity} = \left(\frac{1}{2}\right)^5 \times \text{Initial quantity} \]

### Step 4: Calculate \(\left(\frac{1}{2}\right)^5\)
Calculating \(\left(\frac{1}{2}\right)^5\):
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

### Step 5: Express the remaining quantity as a percentage
To find the percentage of the initial quantity that remains, we multiply by 100%:
\[ \text{Percentage remaining} = \frac{1}{32} \times 100\% \]

### Step 6: Calculate the final percentage
Calculating \(\frac{1}{32} \times 100\%\):
\[ \text{Percentage remaining} = \frac{100}{32} = 3.125\% \]

### Final Answer
Thus, the percentage of the quantity of radioactive material that remains after 5 half-lives is **3.125%**.

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