Heating$/$Cooling System Description: One of the most common mechanical feedback systems is the heating system found in most homes and buildings. Once a temperature has been set on the thermostat, the system will try to keep the temperature in the house as close to that level as possible. If the temperature drops below that level, the thermostat responds by turning the furnace on$.$ The furnace produces heat, which warms the air back up$.$

Eventually the temperature rises above the desired level and the thermostat shuts the furnace off. Then, if it is colder outside than inside, the house will cool down again until the thermostat turns the furnace on$,$ and the whole cycle is repeated. You should include the effects of both the environment and the furnace on the room temperature. While the furnace is either on or off, the effect of the environment is somewhat more complicated. Hints: Both the furnace and the outside environment affect the state of the house by adding and removing energy in the form of heat. Changes in the house$$s temperature are proportional to the change in the amount of heat in the house. The rate at which heat is transferred from the environment to the house is directly proportional to the difference in temperature between the house and the environment. 

Based on the above description: 

$1)$ Write at least two different scenarios and report the results. $$Defining the different values for variables is up to you.