Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference between the temperature of the object and the ambient temperature. (a) Let u(t) be the temperature (in °C, degrees Celsius) of an object as a function of time t minutes, and let a be the ambient room temperature. Show that the differential equation governing the cooling is du/dt = -k(u - a) for some positive constant k. (b) Solve this differential equation. (c) To make tea, David puts a tea bag and boiling water (at exactly 100°C) into a cup and lets it sit for 5 minutes to brew in a room with ambient temperature 20°C. After 5 minutes, the tea has cooled to 60°C. Find the value of k. (d) David likes to have milk in his tea. When he adds milk to the tea, it decreases the temperature of the tea by exactly 5°C. He wants his tea as hot as possible. Should David add the milk before brewing the tea for 5 minutes, or after? Justify your answer mathematically.