$(30$ pts$)[$Simscape$]$ The figure below depicts a thermal system consisting of two capacitive rooms

separated from each other and the outside temperature by resistive walls with effective resistances

$R_1$;$R_2$; and $R_3.$

a$.(10$ pts$)$ Derive the governing equations of motion for this system.

b$.(5$ pts$)$ Condense the governing equations to obtain a single ODE describing the dynamics of

the temperature $T_1$ in response to the outside temperature $T_o$ : Show that sign agreement

holds for this equation.

c$.(10$ pts$)$ Assuming the effective resistances $R_1,R_2,$ and $R_3$ comprise convection that occurs at

each side of both walls and conduction that occurs through each wall, develop a Simscape

model of this system.

d$.(5$ pts$)$ Add sensors to the Simscape model to measure the heat flow from room $1$ to room $2$

and the temperature $T_m(t)$ in the middle of the outer wall.