Problem

A cup of coffee is initially 170 degrees Fahrenheit and is left in a room with ambient temperature 70 degrees Fahrenheit. Suppose that when the coffee is first placed in the room, it is cooling at a rate of 20 degrees per minute.

Assuming Newton's law of cooling applies, how long does it take for the coffee to cool to 110 degrees?

Solution

Newton's law of cooling states that this differential equation holds for the temperature of a cooling object:

where

T = temperature of object
t = time (in minutes for this problem)
A = ambient temperature
k = constant

We know all these terms except for k, so we set up the equation and solve for k.

At time t = 0, we are told that

so now we know

Next we solve the differential equation for T. We use the method of separation of variables introduced in section 3.1.

Again, the initial conditions let us solve for C.

Here's a graph of T(t).

It looks from the graph as if the temperature reaches 110 degrees around 5 minutes. Substitute 110 for the temperature and solve for t.

So the coffee cools to the desired temperature in about 4.58 minutes.