(Solved): 5. Consider a cooling process of a hot cup of coffee in a room with a constant temperature. The rat ...

5. Consider a cooling process of a hot cup of coffee in a room with a constant temperature. The rate of change of the coffee's temperature with respect to time is directly proportional to the difference between the temperature of the coffee and the room temperature, where the proportionality constant is $k>0$. Initially, the temperature of the coffee is $80^{?}\mathrm{C}$, and the room temperature is $25^{?}\mathrm{C}$. After 10 minutes, the temperature of the coffee decreases to $50^{?}\mathrm{C}$. Let $T(t)$ represent the temperature of the coffee at time $t$ (in min), where $t?0$. (a) Formulate a differential equation for $T(t)$ and solve it. (b) What is the temperature of the cup of coffee after 15 minutes? (c) What is the temperature of the cup of coffee in the long run?