How can rates of change model the cooling of an object?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
The cooling of an object is modeled using Newton’s law of cooling, which states that the rate of heat loss is proportional to the temperature difference between the object and its surroundings.1
In mathematical analysis, this process is described by a differential equation where the rate of change of the temperature is proportional to the current value of the temperature difference. Because the only functions whose derivatives are proportional to themselves are exponential functions, the temperature of a cooling object follows an exponential decay model.1
Essentially, as the object cools and its temperature approaches that of the environment, the temperature difference decreases, causing the rate of cooling to slow down over time. This principle is a fundamental application of exponential decay, where the same proportion of the temperature difference is lost in any fixed period.1