A hot cup of coffee is placed on the kitchen table in a room that is 68°F. Newton's Law of Cooling (link forward) says that the temperature *t* of the coffee will decrease at a rate proportional to the difference between the temperature of the surrounding room and the temperature of the coffee. This situation can be modeled by the differential equation

Is the constant *k* positive or negative?

Answer

If coffee is cooling, its temperature is decreasing, so

must be negative. Since the coffee is warmer than 68° the quantity (68 - *T*) is negative. In order for the product

*k*(68 - *T*)

to be negative, the constant *k* must be positive.

Some of the problems we've had so far have been way too simplified to be real. People don't just spend $40 per week - in real life, a bank account has some money going in and some money going out.

A population doesn't just increase. In real life, some people are born and some people die.

For most quantities we model with differential equations, we can think of the rate of change in two separate pieces:

- the rate of increase (birth rate, rate of saving)
- the rate of decrease (death rate, rate of spending)

To get the overall rate of change, we combine the rates of increase and decrease:

(overall rate of change) = (rate of increase) - (rate of decrease).

Remember that the units of the derivative are the units of the dependent variable over the units of the independent variable.