Question

{"given":"Half-life $T_{1/2} = 100\\ \\mu s$, final activity $R = \\dfrac{R_0}{16}$","key_observation":"Activity decays as $R = R_0 \\left(\\dfrac{1}{2}\\right)^n$ where $n = \\dfrac{t}{T_{1/2}}$ is the number of half-lives elapsed.","option_analysis":[{"label":"(A)","text":"$400\\ \\mu s$","verdict":"correct","explanation":"Setting $\\left(\\frac{1}{2}\\right)^n = \\frac{1}{16} = \\left(\\frac{1}{2}\\right)^4$ gives $n = 4$, so $t = 4 \\times 100\\ \\mu s = 400\\ \\mu s$."},{"label":"(B)","text":"$63\\ \\mu s$","verdict":"incorrect","explanation":"This value does not correspond to any integer number of half-lives and does not satisfy the decay equation for $R = R_0/16$."},{"label":"(C)","text":"$40\\ \\mu s$","verdict":"incorrect","explanation":"This corresponds to $n = 0.4$ half-lives, giving $R/R_0 \\approx 0.76$, not $1/16$."},{"label":"(D)","text":"$300\\ \\mu s$","verdict":"incorrect","explanation":"This corresponds to $n = 3$ half-lives, giving $R = R_0/8$, not $R_0/16$."}],"answer":"(A)","formula_steps":[]}