If `10%` of a radioactive substance decays in every 5 year, then the percentage of the substance that will have decaed in `20 years` will be .

Options

A. `40%`
B. `50%`
C. `65.6%`
D. `34.4%`

Answer

To solve the problem of how much of a radioactive substance decays over 20 years when 10% decays every 5 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Decay Rate**:
- We are given that 10% of the substance decays in 5 years. This means that after 5 years, 90% of the substance remains.

2. **Determine Remaining Substance After Each Interval**:
- If 10% decays, then 90% remains after 5 years. We can express this as:
\[ n = 0.9 \cdot n_0 \]
- Where \( n_0 \) is the initial amount of the substance.

3. **Calculate Remaining Substance After 20 Years**:
- Since 20 years is 4 intervals of 5 years, we can calculate the remaining substance after 20 years by applying the decay formula repeatedly:
\[ n = n_0 \cdot (0.9)^4 \]
- Calculate \( (0.9)^4 \):
\[ (0.9)^4 = 0.6561 \]
- Therefore, after 20 years, the remaining amount is:
\[ n = n_0 \cdot 0.6561 \]

4. **Calculate the Decayed Amount**:
- The amount that has decayed after 20 years can be found by subtracting the remaining amount from the initial amount:
\[ \text{Decayed amount} = n_0 - n = n_0 - n_0 \cdot 0.6561 = n_0 (1 - 0.6561) = n_0 \cdot 0.3439 \]

5. **Convert Decayed Amount to Percentage**:
- To find the percentage of the substance that has decayed, we use the formula:
\[ \text{Percentage decayed} = \left( \frac{\text{Decayed amount}}{n_0} \right) \times 100 = 0.3439 \times 100 = 34.39\% \]

### Final Answer:
The percentage of the substance that will have decayed in 20 years is approximately **34.39%**.

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