Interpreting a Blood-Alcohol Elimination Estimate
A community health pamphlet states that blood-alcohol concentration (BAC) falls at roughly 0.015 per hour once drinking stops. Jordan, mass 70 kg, drank steadily and stopped at 11:00 pm. A police reading at 11:30 pm was 0.085. Jordan plans to drive at 7:00 am, and the legal limit is 0.05. The pamphlet also notes that estimates vary by about ±20% between individuals, and that the elimination rate is not perfectly linear at low concentrations. Jordan wants a safety margin, not a borderline calculation.
1. Using the pamphlet's rate, estimate Jordan's BAC at 7:00 am and decide whether driving is acceptable.
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Time from 11:30 pm to 7:00 am = 7.5 hours. Estimated fall = 7.5 × 0.015 = 0.1125. Estimated BAC at 7:00 am = 0.085 − 0.1125 = −0.0275, i.e. effectively zero (BAC cannot be negative; the linear model breaks down near zero).
On the central estimate Jordan would be well under 0.05 by 7:00 am. However, the model over-predicts elimination at low BAC, so the honest statement is 'predicted to be at or near zero, comfortably below 0.05'. The judgement call is not to report a negative BAC as a real value but to state the model's limit.
2. Repeat the estimate using the slowest plausible elimination rate and the highest plausible starting reading, and state the safe earliest driving time.
Read the worked answer
Slowest rate = 0.015 × 0.8 = 0.012 per hour. Highest plausible start = 0.085 × 1.2 = 0.102. Time to fall to 0.05: need a drop of 0.102 − 0.05 = 0.052. At 0.012/h, time = 0.052 / 0.012 ≈ 4.33 hours after 11:30 pm, i.e. about 3:50 am. But Jordan wants a margin below the limit; targeting 0.03 instead: drop = 0.102 − 0.03 = 0.072; time = 0.072 / 0.012 = 6 hours, i.e. about 5:30 am.
So even the cautious case clears well before 7:00 am. The safe earliest time with margin is around 5:30 am; 7:00 am leaves roughly 1.5 hours of buffer. The key judgement is using the unfavourable end of both ranges, not the central estimate.
3. Explain why a single-number answer ('you will be 0.00 at 7 am') is misleading, and what the pamphlet should say instead.
Read the worked answer
A single number hides two problems. First, the ±20% variation means the true elimination rate could be 0.012/h, so the central estimate is optimistic. Second, the linear model cannot go below zero: once BAC is low, elimination slows and the straight-line prediction is unrealistic, so '−0.0275' is not a real BAC.
A better statement gives a range and a margin: 'On average elimination, BAC would be near zero by 7 am; allowing for slower elimination, it would still be below 0.03 by about 5:30 am. Do not drive on a calculated figure alone.' The judgement call is recognising that a model's arithmetic output is not the same as a safe decision, and that uncertainty should be carried through to the recommendation rather than discarded.