Problem $4[12$ pts$]$ A simple, ideal Brayton cycle has a pressure ratio of $11$ and a maximum temperature of $1400$ K $.$ The inlet temperature of the cycle is $250$ K and the inlet pressure is $75$ kPa $.$ Assume the compressor and turbine are isentropic and air standard assumptions with variable specific heats.

a$)$ Determine the net work of the cycle

b$)$ Determine the thermal efficiency

c$)$ Determine the back$-$work ratio