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 isotope is the time it takes for half of the radioactive atoms in a sample to decay. ### Step-by-Step Explanation 1. **Understanding Half-Life**: - The half-life of the isotope in this question is given as 5 years. This means that every 5 years, half of the remaining radioactive material will decay. 2. **Calculating the Number of Half-Lives**: - We need to find out how many half-lives fit into the 15 years we are considering. - To do this, we divide the total time (15 years) by the half-life (5 years): \[ \text{Number of half-lives} = \frac{15 \text{ years}}{5 \text{ years}} = 3 \] - This means that 3 half-lives will occur in 15 years. 3. **Calculating Remaining Mass**: - We start with an initial mass of 100 grams. After each half-life, the mass of the remaining radioactive material is halved. - After the first half-life (5 years): \[ \text{Remaining mass} = \frac{100 \text{ grams}}{2} = 50 \text{ grams} \] - After the second half-life (10 years): \[ \text{Remaining mass} = \frac{50 \text{ grams}}{2} = 25 \text{ grams} \] - After the third half-life (15 years): \[ \text{Remaining mass} = \frac{25 \text{ grams}}{2} = 12.5 \text{ grams} \] 4. **Final Answer**: - After 15 years, the remaining mass of the radioactive isotope is **12.5 grams**. ### Why the Other Options Are Incorrect - **Option B (25 grams)**: This is the amount remaining after 10 years (2 half-lives), not 15 years. It does not account for the third half-life. - **Option C (50 grams)**: This is the amount remaining after 5 years (1 half-life). It does not consider the decay that occurs in the subsequent 10 years. - **Option D (75 grams)**: This option suggests that only a quarter of the sample has decayed, which is incorrect. After 15 years, three half-lives have passed, meaning more than half of the sample has decayed. ### Summary of Key Points - The half-life is the time required for half of a radioactive sample 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 5 years, after 15 years (3 half-lives), 12.5 grams remain. This method of calculating remaining mass is crucial in understanding radioactive decay and is applicable in various fields, including nuclear physics, medicine, and environmental science.