Problem $5-1.$ An engine operates on an ideal air$-$standard Diesel cycle which has a compression ratio of $18.$ At the beginning of the isentropic compression process, the pressure is $\backslash (15\backslash$mathrm$\{$lbf$\}/\backslash$mathrm$\{$in$\}^\{2\}\backslash ),$ the temperature is $560$ R and the volume is $\backslash (3\backslash$mathrm$\{$ft$\}^\{3\}\backslash ).$ The maximum temperature in the cycle is $4150$ R $.$ Assuming air as ideal gas with variable specific heats, calculate:

a$)$ the cut$-$off ratio

b$)$ the total net work for the cycle,

b$)$ the thermal efficiency, and

c$)$ the mean effective pressure