Abstract
We begin with a paradoxical three-point problem where the standard parameter estimation formula fails because temperature data must satisfy a concavity condition reflecting Newton's Law of Cooling's exponential structure. Although a closed-form solution for A exists for equally-spaced measurements, high sensitivity to error motivates the use of overdetermined systems with many measurements. The "profiling over A" technique transforms this three-parameter nonlinear problem into a sequence of simple linear regressions, providing computational efficiency and conceptual transparency. This approach can be generalized to many parameter estimation problems in science and engineering, making it a valuable tool for undergraduates interested in applied mathematics. The problem exemplifies how mathematical modeling combines theoretical analysis, model reduction, numerical implementation, and statistical validation in a real experimental context. The exercises provide further opportunities for critical analysis and promote the discovery that algorithms can produce estimates with incompatible data, R^2 values may mislead, and that one's own insight about the physical world plays an essential role in validating mathematical models.