$3.$ A $4-$cylinder, single$-$stage, double acting air compressor delivers air at $7$ bar: The pressure and temperature at the end of the suction stroke are $1$ bar and $\backslash (27^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ It delivers $\backslash (30\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ of free air per minute when the compressor is running at $300$ rpm $.$ The pressure and temperature of the ambient air are $1$ bar and $\backslash (17^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The clearance volume is $\backslash (4\backslash \%\backslash )$ of the stroke volume. The stroke$-$to$-$bore ratio is $1\mathrm{.}2.$ The index of compression and expansion is $1\mathrm{.}32.$ Calculate the following:

$($a$)$ Volumetric efficiency of the compressor,

$($b$)$ Indicated power of the compressor;

$($c$)$ Size of motor; if mechanical efficiency is $85\backslash \%,$

$($d$)$ Cylinder dimensions.