Final answer:
The differential equation is dT/dt = -k(T - TA), and temperatures need to be converted to Kelvin using the formula K = °C + 273 to ensure the relationship holds.
Step-by-step explanation:
The differential equation that describes the time evolution of the temperature T of the object, taking into account that the constant of proportionality k equals 0.4 and the ambient temperature TA(t) is a constant 84 degrees Celsius, would typically take the form of Newton's Law of Cooling: dT/dt = -k(T - TA). However, it's important to note that this relationship is valid when temperatures are measured in Kelvin, as the direct relationship between temperature change and time will hold in Kelvin but not in degrees Celsius. To convert degrees Celsius to Kelvin, you can use the relationship K = °C + 273. Therefore, the equation for the time evolution of temperature, using the given ambient temperature in Kelvin, would be: dT/dt = -k(T - (84 + 273)).
Since the size of a kelvin and a degree Celsius are the same, both coefficient of linear expansion (a) and temperature change (ΔT) can be expressed in units of kelvins or degrees Celsius for small changes in temperature, and the calculation will be accurate.