Cooling the Perfect Cup with Laplace

After waiting in a long line for your favourite cup of coffee, you finally sit down with your mug and find that the coffee is still scalding hot! How long do you need to wait before you can enjoy it? Once it cools enough, how much time do you have to enjoy it? Are there ways to speed up the process? These questions motivate the presented modelling scenario about tracking the temperature of a cup of coffee as it cools. Students are put in the role of an inquisitive coffee enthusiast who does their due diligence in preventing burns and carefully experimenting on their coffee so that they, and others to come, can enjoy that perfect cup. They identify their assumptions and interventions before developing model differential equations for each case, which force discontinuities on the derivative and even on the solution itself. Being familiar with basic Laplace transforms and learning key properties of the unit step and unit impulse functions, they solve these differential equations and compare the interval of time when the coffee will be at its peak level of enjoyment. This paper includes an implementation guide, grading rubric, example solutions, and example assessment questions.