This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Calculate the number of half-lives (n). The formula for radioactive decay is N_t = N_0 ((1)/(2))^n, where N_t is the final mass, N_0 is the initial mass, and n is the number of half-lives. 18.5 g = 200 g ((1)/(2))^n (18.5)/(200) = ((1)/(2))^n 0.0925 = (0.5)^n To solve for n, take the logarithm of both sides: (0.0925) = n (0.5) n = ((0.0925))/((0.5)) n ≈ (-1.0339)/(-0.3010) n ≈ 3.4349 Step 2: Calculate the half-life (t_1/2). The total time (T) is related to the number of half-lives (n) and the half-life (t_1/2) by the formula T = n × t_1/2. 100 years = 3.4349 × t_1/2 t_1/2 = 100 years3.4349 t_1/2 ≈ 29.11 years That's 3 down. 2 left today — send the next one.