Figure $3$ shows a power system operating at steady state consisting of three components in series: an air compressor having an isentropic compressor efficiency of $80\%,$ a heat exchanger, and a turbine having an isentropic turbine efficiency of $90\%.$ Air enters the compressor at $\frac{1}{b}ar,300K$ with a mass flow rate of $5\mathrm{.}8k\frac{g}{s}$ and exits at a pressure of $10$ bar $.$ Air enters the turbine at $\frac{10}{b}ar,1400K$ and exits at a pressure of $1$ bar. Air can be modeled as an ideal gas. Stray heat transfer and kinetic and potential energy effects are negligible. Determine, in kW $,$

$($a$)$ the power required by the compressor,

$($b$)$ the power developed by the turbine, and

$($c$)$ the net power output of the overall power system.