To determine the half-life of a radioactive material when given the fraction of atoms remaining after a certain period, we can use the concept of half-lives and the formula related to radioactive decay. ### Step-by-Step Explanation 1. **Understanding the Problem**: - We know that after 120 years, the fraction of the radioactive material left is \( \frac{1}{64} \). - We need to find the half-life of this material. 2. **Using the Half-Life Concept**: - The half-life (\( t_{1/2} \)) is the time required for half of the radioactive atoms in a sample to decay. - If we denote the number of half-lives that have passed as \( n \), then the remaining fraction of the material can be expressed as: \[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^n \] - In our case, we have: \[ \left( \frac{1}{2} \right)^n = \frac{1}{64} \] 3. **Finding \( n \)**: - We can rewrite \( \frac{1}{64} \) as a power of 2: \[ \frac{1}{64} = \frac{1}{2^6} \quad \text{(since \( 64 = 2^6 \))} \] - Therefore, we can equate the exponents: \[ n = 6 \] - This means that 6 half-lives have passed in the 120 years. 4. **Calculating the Half-Life**: - Since we know that 6 half-lives correspond to 120 years, we can find the duration of one half-life: \[ 6 \times t_{1/2} = 120 \text{ years} \] - To find \( t_{1/2} \), we divide both sides by 6: \[ t_{1/2} = \frac{120 \text{ years}}{6} = 20 \text{ years} \] ### Conclusion The half-life of the radioactive material is **20 years**. ### Explanation of Other Options - **Option A: 24 years**: This is incorrect because if the half-life were 24 years, then after 6 half-lives (144 years), the fraction remaining would not match \( \frac{1}{64} \). - **Option C: 10 years**: If the half-life were 10 years, then after 6 half-lives (60 years), the fraction remaining would be \( \frac{1}{64} \) after only 60 years, which contradicts the 120 years given. - **Option D: 2 years**: This option is also incorrect because if the half-life were 2 years, then after 6 half-lives (12 years), the fraction remaining would be \( \frac{1}{64} \) after only 12 years, which is far less than 120 years. ### Revision Summary - The fraction of remaining radioactive material can be expressed as \( \left( \frac{1}{2} \right)^n \). - The number of half-lives can be determined by equating the remaining fraction to a power of 2. - The total time elapsed is equal to the number of half-lives multiplied by the half-life duration. - The correct half-life for the material in this problem is **20 years**.