$3.$ A $\backslash (480-\backslash$mathrm$\{$V$\},60-\backslash$mathrm$\{$Hz$\},\backslash$mathrm$\{$Y$\}\backslash )-$connected, six$-$pole synchronous generator has a synchronc reactance of $\backslash (0\mathrm{.}1\backslash$Omega $\backslash )$ an arnature resistance of $\backslash (0\mathrm{.}015\backslash$Omega $\backslash ),$ At full load, the machine sujepi $1200$ A at $0\mathrm{.}8$ PF lagging. Under full$-$load conditions, the friction and windage lossis: are $40$ kW $,$ and the core losses are $30$ kW $.$ Ignore any field circuit losses.

$($a$)$ What is the speed of rotation of this generator?

$($b$)$ If this load is connected to the generator, find the pxcitation voltage of the generaton

$($c$)$ How much power is the generator supplying $($Pout$)?$ How much power is supplied to the generator by the prime mover $\backslash (\backslash$left$(\backslash$mathrm$\{$P$\}\_\{\backslash$text $\{$in $\}\}\backslash$right$)\backslash )?$ What is this machine's overall efficiency?