To determine how much of a radioactive substance remains after a certain period, we can use the concept of half-life. The half-life of a substance is the time it takes for half of the original amount of the substance to decay. ### Step-by-Step Explanation 1. **Understanding Half-Life**: - The half-life of the radioactive substance in this question is given as 10 years. This means that every 10 years, half of the remaining substance will decay. 2. **Calculating the Number of Half-Lives**: - We need to find out how many half-lives fit into the 30 years we are considering. - To do this, we divide the total time (30 years) by the half-life (10 years): \[ \text{Number of half-lives} = \frac{30 \text{ years}}{10 \text{ years/half-life}} = 3 \] - This means that 30 years is equivalent to 3 half-lives. 3. **Applying the Half-Life Concept**: - Starting with 100% of the original substance, we can calculate how much remains after each half-life: - After the **first half-life (10 years)**: \[ \text{Remaining} = 100\% \times \frac{1}{2} = 50\% \] - After the **second half-life (20 years)**: \[ \text{Remaining} = 50\% \times \frac{1}{2} = 25\% \] - After the **third half-life (30 years)**: \[ \text{Remaining} = 25\% \times \frac{1}{2} = 12.5\% \] 4. **Final Calculation**: - After 30 years, which is 3 half-lives, only **12.5%** of the original radioactive substance remains. ### Conclusion The correct answer is **A. 12.5%**. ### Explanation of Other Options - **B. 25%**: This option represents the amount remaining after 20 years (2 half-lives), not 30 years. - **C. 50%**: This option represents the amount remaining after 10 years (1 half-life), not 30 years. - **D. 75%**: This option is incorrect because it suggests that only a quarter of the substance has decayed, which is not the case after 30 years. ### Common Pitfalls - **Misunderstanding Half-Life**: Students often confuse the concept of half-life with the total decay time. Remember, each half-life reduces the remaining amount by half. - **Forgetting to Count Half-Lives**: Ensure you correctly calculate the number of half-lives that fit into the total time period. - **Assuming Linear Decay**: Radioactive decay is exponential, not linear. Each half-life reduces the remaining amount by half, not by a fixed amount. ### Revision Summary - The half-life is the time required for half of a substance to decay. - To find the remaining amount after a certain time, calculate how many half-lives fit into that time. - After each half-life, the remaining amount is halved. - For a half-life of 10 years, after 30 years (3 half-lives), 12.5% of the original substance remains.