A cup of hot coffee will, over time, cool down to room temperature. The principle of physics governing the process is Newton's Law of Cooling. Experiments with a covered cup of coffee show that the temperature (in degrees Fahrenheit) of the coffee can be modelled by the following equation $$ f(t) = 110e^{-0.08t}+75. $$ Here the time \(t\) is measured in minutes after the coffee was poured into the cup.
The same reasoning gives, as an expression for the time when the coffee has cooled to \(100\) degrees, $$ \frac{\ln{\left(\frac{25}{110}\right)}}{-0.08}. $$ This is about \(18.5\) minutes.
Commentary
The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.