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 radioactive atoms in a sample to decay. In this case, the half-life is given as 5 years. ### Step-by-Step Explanation 1. **Understanding Half-Life**: - The half-life tells us that after each period of 5 years, the amount of the substance will reduce to half of its previous amount. 2. **Calculating the Number of Half-Lives**: - We need to find out how many half-lives fit into the total time of 15 years. - Total time = 15 years - Half-life = 5 years - Number of half-lives = Total time / Half-life = 15 years / 5 years = 3 half-lives. 3. **Calculating Remaining Amount**: - We start with an initial amount of 100 g. - After each half-life, the amount remaining is halved: - After 1st half-life (5 years): \[ \text{Remaining} = \frac{100 \text{ g}}{2} = 50 \text{ g} \] - After 2nd half-life (10 years): \[ \text{Remaining} = \frac{50 \text{ g}}{2} = 25 \text{ g} \] - After 3rd half-life (15 years): \[ \text{Remaining} = \frac{25 \text{ g}}{2} = 12.5 \text{ g} \] 4. **Final Calculation**: - After 15 years, the remaining amount of the radioactive substance is 12.5 g. ### Conclusion The correct answer is **A. 12.5 g**. ### Explanation of Other Options - **B. 25 g**: This amount represents the remaining quantity after 10 years (2 half-lives), not 15 years. - **C. 50 g**: This amount is what remains after 5 years (1 half-life), not after 15 years. - **D. 75 g**: 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 half-life concept and think that the amount remaining after each half-life is cumulative rather than halving the previous amount. - **Incorrect Calculation of Time**: Ensure that the total time is correctly divided by the half-life to find the number of half-lives. ### Revision Summary - The half-life is the time required for half of a radioactive sample to decay. - To find the remaining amount after a certain time, divide the total time by the half-life to determine the number of half-lives. - Each half-life reduces the remaining amount by half. - For a 100 g sample with a half-life of 5 years, after 15 years (3 half-lives), 12.5 g remains.