Problem 2 (10 points): (Based on Problem B-7-10) Consider the pneumatic pressure system shown in the figure below. For t > 0, the inlet valve is closed, the outlet valve is fully opened to the atmosphere, and the pressure p2 in the vessel is atmospheric pressure. At t = 0, the inlet valve is fully opened. The inlet pipe is connected to a pressure source that supplies air at a constant (absolute) pressure P1. Assume that the expansion process is isothermal (n = 1) and that the temperature of the entire system stays constant. Inlet Outlet valve valve P2 P P1 P3 Atmospheric pressure (a) Assuming that the capacitance of the vessel is C and the resistance of the two valves is R (i.e., both valves have identical flow characteristics), obtain the corresponding system equations for C and R and obtain the ordinary differential equation for the system as a function of the absolute pressures) P1, P2, and P3. Answer: RCP2 + 2p2 = P1 + P3. (b) Using the Laplace transform and the partial fraction expansion method, determine the pressure p2(t) in the vessel after the inlet valve is fully opened. Note that p2(0) = P3. Answer: p2(t) = (01 +P3)/2 - exp(-2t/RC) (P. + P3)/2. ) Using Python, plot the gauge pressure pz(t). Assume that the air is supplied at a constant gauge pressure P1 = 5.0 MPa. Note that the gauge pressure P3 = 0. Check that the steady-state value of pz is P1/2. Include your Python code. =