3. (3 points) The temperature in a room is controlled by an on/off heater. The air in the room is well mixed, with no transfer of mass between the room and the environment. Heat is transferred from the room based on the energy difference between the room and the environment, as described by: Rate of heat transfer from the environment =UA[Ta(t)T(t)] where Ta is the temperature outside the room and T is the temperature of the room. In addition, a heater is running and adding energy to the room at a rate of Qh(J/hr). The heater is initially running at a heating rate of Qh,i(J/hr) for an extended period of time, such that the conditions in the room can be considered steady state at T=23C. The following information is available: The outside temperature (Ta) is a constant value of 10C. Properties of air: Cv=0.76J/g/C; density is 1190g/m3 The overall heat transfer coefficient is UA=0.8104J/C/hr The size of the room is 5m5m3m. a. Write the ODE describing the change in room temperature (T, the dependent variable) as a function of the rate of heat addition by the heater, Qh. The ODE should be written in standard form. b. From the initial steady state conditions, determine the initial heating rate (Qhihi (in J/hr) ). c. The room is at a steady state temperature of 23C and the heater switches off. It is set to turn on again when the room reaches 19C. How long will the heater be off