This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Determine the number of half-lives (n) that have passed. The fraction of radioactive atoms remaining after time t is given by the formula: (N)/(N_0) = ((1)/(2))^n where N is the number of atoms remaining, N_0 is the initial number of atoms, and n is the number of half-lives. Given that the fraction of atoms left after 120 years is (1)/(64): (1)/(64) = ((1)/(2))^n We know that 2^6 = 64, so (1)/(64) = ((1)/(2))^6. Therefore, n = 6. This means 6 half-lives have passed in 120 years. Step 2: Calculate the half-life (T_1/2) of the material. The number of half-lives (n) is also related to the total time (t) and the half-life (T_1/2) by the formula: n = (t)/(T_1/2) We need to find T_1/2, so we can rearrange the formula: T_1/2 = (t)/(n) Substitute the given total time t = 120 years and the calculated number of half-lives n = 6: T_1/2 = 120 \, years6 T_1/2 = 20 \, years Step 3: Compare the result with the given options. The calculated half-life is 20 years, which corresponds to option C. The half-life of the material is 20 years.