It is a good idea to enjoy a cup of coffee before starting a busy day.

Suppose the coffee fresh out of the pot with temperature is too hot, we can immediately add cream to reduce the temperature by *instantly,* then wait for the coffee to cool down naturally to before sipping it comfortably. We can also wait until the temperature of the coffee drops to first, then add the cream to further reduce it instantly to .

Typically, , and .

If we are in a hurry and want to wait the shortest possible time, should the cream be added right after the coffee is made, or should we wait for a while before adding the cream?

The heat flow from the hot water to the surrounding air obeys Newton’s cooling and heating law, described by the following ordinary differential equation:

where , a function of time , is the temperature of the water, is the temperature of its surroundings, and is a constant depends on the heat transfer mechanism, the contact are with the surroundings, and the thermal properties of the water.

Fig. 1 a place where Newton’s law breaks down

Under normal circumstances, we have

Based on Newton’s law, the mathematical model of coffee cooling is:

Fig. 2

Solving (2), an initial-value problem (see Fig. 2) gives

Therefore,

If cream is added immediately (see Fig. 3),

Fig. 3 : cream *first*

by (3),

Otherwise (see Fig. 4),

Fig. 4: cream* last*

And so,

Fig. 5

Since

implies

from (4) , we see that

i.e.,

Hence,

If we are in a hurry and want to wait the shortest possible time, we should wait for a while before adding the cream!

*Exercise-1* Solve (2) without using a CAS.

*Exercise-2 *Show that