1-069-TorricelliHowSure-ModelingScenario

Students ask how certain the fitted value is, and what the model says beyond the data. Students examine residuals, compute a 95% confidence interval for b by linearization, turn it into an interval for the time at which the column empties, and interpret b physically as a discharge coefficient of the exit hole. They then fit only the early data and use that fit to predict the later measurements. Finally, they follow the model past the moment the column is empty, where the closed-form solution gives a physically impossible answer and the uniqueness theorem for differential equations no longer applies. The mathematics involved: separable differential equations, least squares, linearized confidence intervals, the existence and uniqueness theorem, and the difference between a solution formula and a solution of the differential equation. The scenario is written for a first course in ordinary differential equations; no statistics course is assumed.