$1.$ The thermometer readings follows a first order behavior to any changes in the surrounding temperature. The thermometer is immersed in a liquid bath measured as the input of the system and the temperature reading was determined as output of the system. a$)$ Show that the thermometer reading follows a first order behavior to any changes in the surrounding temperature. b$)$ A thermometer having a time constant of $5,$ determine the temperature reading when a step input to the system of $40$