A cold air$-$standard Otto cycle has a compression ratio of $12$ and the temperature and pressure at the beginning of the compression process are $520R$ and $14\mathrm{.}2lb\frac{f}{i}n.{?}^2,$ respectively. The heat addition per unit mass of air is $600Bt\frac{u}{l}b.$ Assume constant specific heats evaluted at $520$ R$.$

Determine:

$($a$)$ the maximum temperature, in $R.$

$($b$)$ the maximum pressure, in lbf$/$in$.{?}^2$

$($c$)$ the percent thermal efficiency.

$($d$)$ the mean effective pressure, in Ibf$/$in$.{?}^2$