A $500-V,50Hz,4-$pole, $3-$phase induction motor is drawing $50$ A per phase at $0\mathrm{.}8$ PF lagging. The stator copper losses and rotor copper losses are each $600$ W $.$ The friction and windage losses are $500$ W $,$ the core losses are $1700$ W $,$ and the stray losses are negligible. Find the following quantities:

a$)$ The air$-$gap power $P_{AG}$

b$)$ The power converted $P_{\text{c}\text{o}\text{n}\text{v}}$

c$)$ The output power $P_{\text{o}\text{u}\text{t}}$

d$)$ The efficiency of the motor

e$)$ The synchronous speed $($note $n_s=\frac{120f}{p})$

f$)$ The induced torque $($ note ${}_s=2\frac{n_s}{60},$ and $T_{\text{i}\text{n}\text{d}}=\frac{P_{AG}}{{}_s})$