Problem

Allyson carries an 80 degree cup of coffee into a room which has been heated to 20 degrees. According to Newton's Law of Cooling, the temperature of the coffee at time t will be

where t is the amount of time the cup of coffee is sitting out in the room. Find

the temperature the coffee will reach if it is left in the room indefinitely.

Solution

A graph of the temperature function looks like this.

Both this graph and our common sense tell us the answer should be 20 degrees, the temperature of the room. Let's solve this mathematically to confirm our guess.

Using the properties of limits, we can write

The last term can be written

As t approaches infinity, the exponent in the expression approaches negative infinity, and the expression approaches zero.

Therefore

as we expected.