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 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 Mass**: - We start with a 200 gram sample. After each half-life, the remaining mass can be calculated as follows: - After 1 half-life (10 years): \[ \text{Remaining mass} = \frac{200 \text{ grams}}{2} = 100 \text{ grams} \] - After 2 half-lives (20 years): \[ \text{Remaining mass} = \frac{100 \text{ grams}}{2} = 50 \text{ grams} \] - After 3 half-lives (30 years): \[ \text{Remaining mass} = \frac{50 \text{ grams}}{2} = 25 \text{ grams} \] 4. **Final Calculation**: - After 30 years, which corresponds to 3 half-lives, the remaining mass of the radioactive substance is 25 grams. ### Conclusion The correct answer is **A. 25 grams**. ### Explanation of Other Options - **B. 50 grams**: This would be the amount remaining after 20 years (2 half-lives), not 30 years. - **C. 100 grams**: This is the amount remaining after 10 years (1 half-life), not 30 years. - **D. 12.5 grams**: This would be the amount remaining after 40 years (4 half-lives), not 30 years. ### Revision Summary - The half-life is the time it takes for half of a radioactive substance to decay. - To find the remaining mass after a certain time, calculate how many half-lives fit into that time. - After each half-life, the remaining mass is halved. - For a 200 gram sample with a 10-year half-life, after 30 years (3 half-lives), 25 grams remain.