Problem: $(60$ points$)$

An ideal Diesel cycle with air as the working fluid has a compression ratio of $15$ and a cutoff ratio of $2.$ At the beginning of the compression process, the working fluid is at $14\mathrm{.}7a,85F,$ and $115i{n}^{{?}^{?}}3.$ Utilizing the cold$-$air standard assumptions, determine $($a$)$ the temperature and pressure of air at the end of each process, $($b$)$ the net work output and the thermal efficiency, and $($c$)$ the mean effective pressure.