To determine how much of a radioactive isotope remains after a certain period, we can use the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life is given as 10 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 30 years we are considering. Since the half-life is 10 years, we can calculate the number of half-lives in 30 years: \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{30 \text{ years}}{10 \text{ years}} = 3 \] - This means that 30 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 (10 years): \[ 100 \text{ grams} \times \frac{1}{2} = 50 \text{ grams} \] - After 2 half-lives (20 years): \[ 50 \text{ grams} \times \frac{1}{2} = 25 \text{ grams} \] - After 3 half-lives (30 years): \[ 25 \text{ grams} \times \frac{1}{2} = 12.5 \text{ grams} \] 4. **Final Calculation**: - After 30 years, the remaining amount of the radioactive isotope 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 20 years (2 half-lives), not 30 years. - **C. 50 grams**: This is the amount remaining after 10 years (1 half-life), not after 30 years. - **D. 75 grams**: This option does not correspond to any point in the decay process; it suggests that less than half of the original sample remains after 10 years, which is incorrect. ### Common Pitfalls: - **Misunderstanding Half-Life**: Some students may confuse the concept of half-life with the 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 Sequentially**: It’s important to apply the halving process for each half-life step rather than trying to calculate the final amount in one step. ### Revision Summary: - The half-life is the time taken for half of a radioactive sample to decay. - To find the remaining amount after a certain time, calculate how many half-lives fit into that time. - Apply the halving process sequentially for each half-life. - After 30 years with a half-life of 10 years, 12.5 grams of a 100 gram sample remains.