The temperature outside of the building was stable $0^{\circ}$ Celsius. At time $t = 0$ inside the building it was $18^{\circ}$. The time constant 1 for the building equals $4$ hrs. In $1/3$ hr a source of heat was turned on and during 1/2 hr was generated heat of intensity 60000 btu/hr. Find the temperature $T (t)$ in the building at $t = 1$ hr if the heat capacity γ of the building is $1/6$ degree per thousand of btu.
Use equation $dT(t)/dt = K(M(t) − T(t)) + γH(t)$ (where $H(t)$ describes production of heat) and hour as unit of time. Start with representation of $H(t)$ by means of unit step functions.
Answer: ≈ $18.53^{\circ}C$
The answer was given to me, however I have no idea how to fill in the blanks for H(t), and M(t)
My guess for H(t) is u(t-1), but I'm not sure.
So it appears the heat is turned on at $t=1/3$ and then off again at $t=1/3+1/2$ (i.e. half-an-hour later), so it follows the shape of $H$ is described by $u(t-1/3)-u(t-(1/3+1/2))$. The magnitude of heat production when turned on lets us fully describe $H$:
$$ H(t) = 60000 (u(t-1/3) - u(t-(1/3+1/2))) $$