Using Routh’s stability criterion, answer the following parts: (a) Find how many roots with positive real parts are there for Q(s) = s 3 + s 2 + 20s + 78 = 0. (b) Find the range of K such that Q(s) = s 3 + 3Ks2 + (K + 2)s + 4 has all roots in the left half plane. (Here, K is a design parameter.)

To save energy, the air supply to a 2000 ft office has been shut off overnight and the room temperature has dropped to 40°f. in the morning, the thermostat is reset to 70°f and warm air at 120 f begins to flov in at 200ft'/min. the air is well mixed within the room, and an equal mass flow of air at room temperature (changing with time) is withdrawn through a return duct. the air pressure is nearly 1 atm everywhere. ignoring heat transfer with the surroundings and kinetic and potential energy effects, estimate how long it takes for the room temperature to reach 70°f. plot the room temperature as a function of time.