The first "observation" we make is that these three problems appear to be "the same." It is natural to model them all by the system of differential equations.
where
The first and third settings can be described in this way, The middle setting could be more complicated. This description is based on Newton's Model of Cooling and it may or may not be appropriate. If the metal bar is large then its temperature is not uniform and a partial differential equation is a better model. See Newton's Model of Cooling, Revisited for a series of modules that discuss this question.
By changing the temperature scale so that A is zero, we can describe all three settings by the system of differential equations
where the constants R1, R2, and R3 are all positive.
We will only consider the first question -- is the water level/temperature/charge on the left always below that on the right? The other question is here to make this a more open-ended exercise.
We can approach these questions in at least four different ways.
Copyright c 1998 by
Frank Wattenberg, Department of Mathematics, Montana State University,
Bozeman, MT 59717
