$(25$ pts$.)$ The compression ratio in an ideal Otto cycle is $8\mathrm{.}5.$ At the beginning of the compression stroke the pressure is $14\mathrm{.}7$ psi and the temperature is $70F.$ The heat transfer to the air during the constant$-$volume heat addition is $900$ Btu$/$bm air. Accounting for the variation of specific heats of air with temperature, determine:

$($a$)(15$ pts$.)$ The pressure and temperature at the end of each process of the cycle.

$($b$)(5$ pts$.)$ The thermal efficiency $($solve using both by $=\frac{w_{\text{n}\text{e}\text{t}}}{q_{in}}$ and by using $=1-\frac{1}{r^{k-1}})$ Explain why there is a difference between the two.

$($c$)(5$ pts$.)$ The mean effective pressure.