The Fertilizer Runoff Model
A lake receives nitrogen from two sources: a farm field and a wastewater plant. The farm contributes N_f(t) = 40 + 3t kilograms per month, where t is months since monitoring began. The plant contributes N_p(t) = 90 - 2t kg/month under a scheduled upgrade. A biologist samples the lake and finds algal bloom risk rises sharply once total monthly nitrogen exceeds 120 kg. The farm's contribution is measured with ±10% error; the plant's with ±5%. A stakeholder claims: 'Total input is roughly flat, so the lake is stable - no action needed.' The monitoring budget only allows quarterly sampling, so decisions are made on 3-month averages.
1. Compute when total monthly nitrogen first exceeds 120 kg, and assess the stakeholder's 'roughly flat' claim.
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Total: N(t) = (40 + 3t) + (90 - 2t) = 130 + t.
The total is not flat - it rises by 1 kg/month. The farm rises at 3 and the plant falls at 2, so the net is +1, not 0. The stakeholder likely subtracted incorrectly or assumed the rates canceled.
Threshold: 130 + t = 120 gives t = -10, which is before monitoring began. So the total is already above 120 at t = 0 (it is 130). The threshold was crossed 10 months before t = 0, assuming the linear models held then - a caveat worth flagging, since extrapolating backward may be invalid.
Stakeholder verdict: Wrong on two counts. The trend is upward, and the level already exceeds the risk threshold. 'Roughly flat' would require the slopes to sum to zero, which they do not (3 + (-2) = 1). The correct statement is: total nitrogen is 130 kg/month and climbing at 1 kg/month.
2. Using the stated measurement errors, compute the uncertainty in total nitrogen at t = 6 months and determine whether the exceedance of 120 kg is statistically robust.
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At t = 6: Farm = 40 + 18 = 58 kg; plant = 90 - 12 = 78 kg; total = 136 kg.
Uncertainties: Farm ±10% of 58 = ±5.8 kg; plant ±5% of 78 = ±3.9 kg.
Combined (independent, root-sum-square): sqrt(5.8² + 3.9²) = sqrt(33.64 + 15.21) = sqrt(48.85) ≈ 6.99 kg. So total ≈ 136 ± 7 kg.
Robustness: The lower bound is 129 kg, still above 120 by 9 kg. The exceedance is robust - the uncertainty band does not reach the threshold.
Quarterly-sampling caveat: Decisions use 3-month averages. The average of N(t) = 130 + t over a quarter starting at t is 130 + t + 1.5 (average of a linear function over an interval is its midpoint value). So quarterly averages run about 1.5 kg above the instantaneous start value, making exceedance even more certain. Averaging reduces noise but does not remove the systematic upward trend.