A three$-$phase, $\backslash (460\backslash$mathrm$\{$~V$\},60\backslash$mathrm$\{$~Hz$\}\backslash ),4-$pole, $30$ kW induction machine draws $35$ A at a power factor of $0\mathrm{.}9$ lagging when connected to a three$-$phase, $\backslash (460\backslash$mathrm$\{$~V$\},60\backslash$mathrm$\{$~Hz$\}\backslash )$ power supply. The stator core loss is $1000$ W $,$ rotor core loss is $60$ W $,$ stator copper loss is $800$ W $,$ rotor copper loss is $400$ W $,$ and friction and windage loss is $200$ W $.$ Calculate:

$($a$)$ The air gap power.

$($b$)$ The mechanical power developed.

$($c$)$ The output horsepower.

$($d$)$ The load torque if slip is $\backslash (3\backslash \%\backslash ).$

$($e$)$ The efficiency of the motor.