To determine how much of a 100-gram sample of a radioactive isotope will remain 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 isotope is the time it takes for half of the radioactive atoms in a sample to decay. In this case, the half-life is 10 years, meaning that every 10 years, half of the remaining sample 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 is equivalent to 3 half-lives. ### Step 3: Apply the Half-Life Decay Formula The amount of substance remaining after a certain number of half-lives can be calculated using the formula: \[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{n} \] where \( n \) is the number of half-lives. ### Step 4: Substitute the Values Now we can substitute the initial amount (100 grams) and the number of half-lives (3) into the formula: \[ \text{Remaining amount} = 100 \text{ grams} \times \left(\frac{1}{2}\right)^{3} \] Calculating \( \left(\frac{1}{2}\right)^{3} \): \[ \left(\frac{1}{2}\right)^{3} = \frac{1}{8} \] Now, substituting this back into the equation: \[ \text{Remaining amount} = 100 \text{ grams} \times \frac{1}{8} = 12.5 \text{ grams} \] ### Conclusion After 30 years, 12.5 grams of the original 100-gram sample will remain. Therefore, the correct answer is: **A. 12.5 grams** ### Explanation of Other Options - **B. 25 grams**: This would imply that only two half-lives have passed (100 grams → 50 grams after 10 years, then 50 grams → 25 grams after 20 years). However, since 30 years corresponds to three half-lives, this option is incorrect. - **C. 50 grams**: This would be the amount remaining after 20 years (two half-lives). Since we are looking for the amount after 30 years, this option is also incorrect. - **D. 75 grams**: This option suggests that only a quarter of the sample has decayed, which is not consistent with the half-life decay process over three half-lives. Thus, this option is incorrect as well. ### 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, calculate the number of half-lives that fit into that time. - Use the formula: Remaining amount = Initial amount × (1/2)ⁿ, where n is the number of half-lives. - After 30 years (3 half-lives), 12.5 grams of a 100-gram sample remains.