Activity 7.4.3.
Suppose that a cup of coffee is initially at a temperature of \(105^\circ\) F and is placed in a \(75^\circ\) F room. If \(T\) is the temperature of the coffee in degrees Fahrenheit at time \(t\) in minutes, Newton’s law of cooling says that
\begin{equation*}
\frac{dT}{dt} = -k(T-75)\text{,}
\end{equation*}
where \(k\) is a constant of proportionality.
(a)
Suppose you measure that the coffee is cooling at one degree per minute at the time the coffee is brought into the room. Use the differential equation to determine the value of the constant \(k\text{.}\)
(b)
Find all the solutions of this differential equation.
(c)
What happens to all the solutions as \(t\to\infty\text{?}\) Explain how this agrees with your intuition.
(d)
What is the temperature of the cup of coffee after 20 minutes?
(e)
How long does it take for the coffee to cool to \(80^\circ\text{?}\)
