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 substance to decay. In this case, the half-life is given as 5 years. ### Step-by-Step Explanation 1. **Understanding Half-Life**: - The half-life of a substance is the time required for half of the radioactive atoms in a sample to decay. After one half-life, 50% of the original amount remains; after two half-lives, 25% remains; after three half-lives, 12.5% remains, and so on. 2. **Calculating the Number of Half-Lives**: - We need to find out how many half-lives fit into the 15 years we are considering. - Since the half-life is 5 years, we can calculate the number of half-lives in 15 years: \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{15 \text{ years}}{5 \text{ years}} = 3 \] - This means that 15 years is equivalent to 3 half-lives. 3. **Calculating Remaining Amount**: - We start with a 100-gram sample. After each half-life, the amount remaining is halved: - After 1 half-life (5 years): \[ 100 \text{ grams} \times \frac{1}{2} = 50 \text{ grams} \] - After 2 half-lives (10 years): \[ 50 \text{ grams} \times \frac{1}{2} = 25 \text{ grams} \] - After 3 half-lives (15 years): \[ 25 \text{ grams} \times \frac{1}{2} = 12.5 \text{ grams} \] 4. **Final Calculation**: - After 15 years, the remaining amount of the radioactive substance is 12.5 grams. ### Conclusion The correct answer is **A. 12.5 grams**. ### Explanation of Other Options - **B. 25 grams**: This would be the amount remaining after 10 years (2 half-lives), not 15 years. - **C. 50 grams**: This is the amount remaining after 5 years (1 half-life), not 15 years. - **D. 75 grams**: This option does not correspond to any point in the decay process and is incorrect. ### Common Pitfalls - **Misunderstanding Half-Life**: Some students may confuse the concept of half-life with total decay time. Remember, each half-life reduces the remaining amount by half. - **Incorrect Calculation of Half-Lives**: Ensure you divide the total time by the half-life correctly to find the number of half-lives. - **Forgetting to Apply the Half-Life Process**: It’s essential to apply the halving process for each half-life sequentially. ### Revision Summary - The half-life is the time taken for half of a radioactive substance to decay. - To find the remaining amount after a certain time, calculate how many half-lives fit into that time. - Apply the halving process for each half-life to find the final amount. - After 15 years (3 half-lives), a 100-gram sample reduces to 12.5 grams.