To determine how much of a 100-gram sample of a radioactive substance remains after 30 years, given that its half-life is 10 years, we can follow these steps: ### Step 1: Understand Half-Life The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life is 10 years, meaning that every 10 years, half of the remaining substance will decay. ### Step 2: Calculate the Number of Half-Lives To find out how many half-lives fit into 30 years, we divide the total time by the half-life: \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{30 \text{ years}}{10 \text{ years}} = 3 \] This means that 30 years encompasses 3 complete half-lives. ### Step 3: Calculate Remaining Mass After Each Half-Life Starting with the initial mass of 100 grams, we can calculate the remaining mass after each half-life: 1. **After 1st half-life (10 years)**: \[ \text{Remaining mass} = \frac{100 \text{ grams}}{2} = 50 \text{ grams} \] 2. **After 2nd half-life (20 years)**: \[ \text{Remaining mass} = \frac{50 \text{ grams}}{2} = 25 \text{ grams} \] 3. **After 3rd half-life (30 years)**: \[ \text{Remaining mass} = \frac{25 \text{ grams}}{2} = 12.5 \text{ grams} \] ### Final Answer After 30 years, the remaining mass of the radioactive substance is **12.5 grams**. Therefore, the correct option is **A**. ### Explanation of Other Options - **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 and is incorrect. ### Common Pitfalls - **Misunderstanding Half-Life**: Some students may confuse the concept of half-life with total decay time. Remember, each half-life reduces the remaining amount by half. - **Incorrect Calculation of Half-Lives**: Ensure you divide the total time by the half-life correctly to find the number of half-lives. - **Forgetting to Apply the Half-Life Process Sequentially**: It’s important to apply the half-life reduction step-by-step rather than trying to calculate the final amount in one step. ### Revision Summary - The half-life is the time required 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. - Apply the half-life reduction sequentially to find the remaining mass. - After 30 years (3 half-lives), 12.5 grams of the original 100-gram sample remains.