$4.$ A $10-$pitch $\backslash (20^\{\backslash$circ$\}\backslash )$ full$-$depth involute gearset, with $\backslash ($ Q$\_\{$v$\}=10\backslash ),$ and face width of $1\mathrm{.}25$ inches, is being proposed to provide a $\backslash (2$: $1\backslash )$ speed reduction for a conveyor drive unit. The $21-$tooth pinion is to be driven by a $15-$hp$,1725-$rpm electric motor operating steadily at full$-$rated power. A very long life is desired for this gearset, and a reliability of $\backslash (99\backslash \%\backslash )$ is required. Higher precision gearing is to be used. Do the following:

$($a$)$ Using the AGMA refined approach, calculate the tooth bending stress at the tension$-$side root fillet of the driving pinion.

$($b$)$ For tooth bending fatigue, which gear is expected to fail first, the pinion or the gear? If the proposed material for both gears is AISI $4620$ through$-$hardened to BHN $207($Grade $1),$ estimate the existing safety factor at the tension$-$side root fillet based on tooth bending fatigue.