Teaching differential equations through modelling: hot water heater

We present an example of one of the modelling projects we assign to students in our differential equations classes. Students are asked to determine how to run a cost-efficient hot water heating system. We consider a cylindrical tank filled with water and heated by a heating element immersed in it. Together with students we discuss physical laws governing the process and make assumptions necessary to build a model. One of the goals of the modelling activity is to compute the time needed for water to reach the desired temperature given the power of the heating element and the size of the water tank. Several different scenarios can be discussed. Assuming the cost of heating the water is directly related to the time for which the power is ON, students can choose the most cost-efficient scenario. We provide the description of several modelling activities and obtained results.