To determine how much of a radioactive substance remains after a certain period, we can use the concept of half-life. The half-life is the time it takes for half of a sample of a radioactive substance to decay. In this case, the half-life of the substance 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 as follows: \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{30 \text{ years}}{10 \text{ years}} = 3 \] - This means that 30 years is equivalent to 3 half-lives. 3. **Calculating Remaining Mass**: - We start with a 100-gram sample. After each half-life, the remaining mass can be calculated as follows: - After 1 half-life (10 years): \[ \text{Remaining mass} = \frac{100 \text{ grams}}{2} = 50 \text{ grams} \] - After 2 half-lives (20 years): \[ \text{Remaining mass} = \frac{50 \text{ grams}}{2} = 25 \text{ grams} \] - After 3 half-lives (30 years): \[ \text{Remaining mass} = \frac{25 \text{ grams}}{2} = 12.5 \text{ grams} \] 4. **Final Result**: - After 30 years, the remaining mass of the radioactive substance is **12.5 grams**. ### Why the Other Options Are Incorrect - **Option B (25 grams)**: This is the amount remaining after 20 years (2 half-lives), not 30 years. - **Option C (50 grams)**: This is the amount remaining after 10 years (1 half-life), not 30 years. - **Option 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 just one half-life, which is incorrect. ### Summary of Key Points - 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 100-gram sample with a half-life of 10 years, after 30 years (3 half-lives), 12.5 grams remain. This thorough understanding of half-lives and decay processes is crucial for solving problems related to radioactive substances in physics.