We are designing a building for aretic conditions. For safety in case the heater mal$-$

functions, the insulation must be rated to keep the indoor temperature above the

ambient outdoor temperature for $24$ hours. The dynamics of heat loss are modeled

by the differential equation

$mc\frac{dT}{dt}=h(T-T_{\text{a}\text{m}\text{b}})$

where $T[K]$ is the inside temperature, $T_{\text{a}\text{m}\text{b}}[K]$ is the outdoor temperature. $m[kg]$

is the mass of the air in the building, $c[\frac{J}{kgK}]$ is the specific heat of air, $A[m^2]$ is the

wall surface area, and $h{|}_{{?}_{?}}\frac{1}{s}\frac{1}{(m^2(K){)}_{{?}_{?}}}|$ is the heat transfer coefficient of the walls.

$($a$)$ Find the transfer function from the outdoor temperature $T_{\text{a}\text{m}\text{b}}$ to the indoor

temperature $T.$

$($b$)$ What are the poles and zeros of the transfer function? What is the order of the

transfer function e$.$g$.$ first or second?

$($c$)$ Compute the heat transfer coefficient $h$ so that the indoor temperature $T$ requires

$24$ hours to reach the outdoor temperature $T_{\text{a}\text{r}\text{m}\text{b}}$ when $m=300[kg],c={10}^3$

${|}_{{?}_{?}}\frac{J}{(kgK{)}_{{?}_{?}}}|,A=220[m^2].$

HINT: Make sure you convert time to seconds.

$($d$)$ Plot the step$-$response for your value of $h.$