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
The fraction remaining of a radioactive substance is the ratio of the final amount (N) to the initial amount (N_0). It can be calculated using the formula: Fraction remaining = (N)/(N_0) = ((1)/(2))^n where: • N is the final amount of the substance remaining. • N_0 is the initial amount of the substance. • n is the number of half-lives that have passed, calculated as n = (t)/(T_1/2). • t is the total time elapsed. • T_1/2 is the half-life of the substance. Example: A radioactive sample has a half-life of 5 years. What fraction of the original sample will remain after 15 years? Step 1: Determine the number of half-lives (n) that have passed. n = (t)/(T_1/2) n = 15 years5 years/half-life = 3 half-lives Step 2: Use the formula for the fraction remaining. Fraction remaining = ((1)/(2))^n Fraction remaining = ((1)/(2))^3 Fraction remaining = (1)/(2) × (1)/(2) × (1)/(2) Fraction remaining = (1)/(8) After 15 years, (1)/(8) of the original sample will remain.