A radioactive decay graph plots remaining mass (g) on the y-axis against time (days) on the x-axis, showing a smooth curve that starts at 200. g and drops to 25.0 g at the 30.0-day mark. Using this single data point (not assuming any half-life value in advance), determine the isotope's half-life.

Fraction remaining = 25.0 ÷ 200. = 0.125 = 1/8, corresponding to exactly 3 half-lives, since (1/2)³ = 1/8. With 3 half-lives spanning the full 30.0 days shown, one half-life is 30.0 ÷ 3 = 10.0 days. Dividing 30.0 days by 8 instead of 3 (mistaking the denominator of the 1/8 remaining fraction for a direct half-life count) gives an incorrect 3.75-day half-life. Assuming the curve's shape means exactly 2 half-lives have passed rather than checking the actual 1/8 fraction against (1/2)ⁿ (reasoning informally that 'the curve looks like it's dropped a lot, so maybe 2 half-lives') gives an incorrect 15.0-day half-life without doing the actual fraction-matching calculation. Treating the decay as linear and reasoning that since 87.5% of the mass was lost over 30.0 days, the half-life must be roughly half of that elapsed time (≈15.0 days) misapplies linear reasoning to an exponential decay curve.