To determine how much of a radioactive isotope 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 isotope is 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, you will have half of the original amount remaining. 2. **Initial Amount**: - We start with 80 grams of the radioactive isotope. 3. **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. 4. **Calculating Remaining Amount**: - After each half-life, the amount of the substance is halved. We can calculate the remaining amount after each half-life: - After 1st half-life (5 years): \[ \text{Remaining} = \frac{80 \text{ grams}}{2} = 40 \text{ grams} \] - After 2nd half-life (10 years): \[ \text{Remaining} = \frac{40 \text{ grams}}{2} = 20 \text{ grams} \] - After 3rd half-life (15 years): \[ \text{Remaining} = \frac{20 \text{ grams}}{2} = 10 \text{ grams} \] 5. **Final Calculation**: - After 15 years, we find that 10 grams of the radioactive isotope remains. ### Conclusion The correct answer is **A. 10 grams**. ### Explanation of Other Options - **B. 20 grams**: This would be the amount remaining after 10 years (2 half-lives), not 15 years. - **C. 40 grams**: This is the amount remaining after 5 years (1 half-life), not 15 years. - **D. 5 grams**: This would imply that there was an additional half-life beyond the 3 calculated, which is incorrect. ### Common Pitfalls - **Misunderstanding Half-Lives**: It's crucial to remember that each half-life reduces the remaining amount by half, not by a fixed amount. - **Incorrect Time Calculation**: 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 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. - After each half-life, the remaining amount is halved. - For 15 years with a half-life of 5 years, starting with 80 grams results in 10 grams remaining after 3 half-lives.