Consider automatic heating$-$cooling of a room by heat inflow$-$outflow controlled by a thermostat. The

desired temperature is $25$ degrees and the 'speed' of the heating system can be represented by its

'adjustment time' $($AT$)($time for closing the gap between the thermostat setting and the current room

temperature$)$ that is assumed $5$ minutes. The adjustment is thus proportional to the difference between the

desired and actual room temperatures. The room is not perfectly isolated so there is a heat transfer between

the walls of the room and outside, which amounts to $20\%$ of the temperature difference from the outside

per minute. The outside temperature is $15$ degrees.

a$)$ Build a model of this system, and draw a stock$-$flow diagram that represent the model structure.

Then find the solution, and plot it$,$ assuming the initial room temperature $(T)$ is $T_0.$ Discuss your

model briefly and improve it if there are major weaknesses.

b$)$ Now assume that there is a long sensing delay of the heating system. The heating$/$cooling activity

is driven by the room temperature perceived by the controller of the heating system $(P),$ rather than

the instantaneous actual room temperature $(T).$ The perceived temperature changes as a fraction of

$($k$)$ the difference between the actual room temperature $(T)$ and the perceived temperature $(P).$

Revise your model in $($a$)$ and give the new equations, both in single nth order and in $n$ system of

equations form. Also provide the revised stock$-$flow diagram.

c$)$ The house goes through a major renovation, and isolated. The heat transfer to the outside is now

negligible. However, the tenants still complain that the heater cannot keep the room temperature

constant, but it fluctuates around the desired room temperature. Suggest an improvement in the

heating system such that these fluctuations are eliminated.

Can you solve all differential equations explicitly?