$($b$)$ A single stage helical gear is transmitting $10$ kW with the following dimensions, $n_p=2400rpm,i=3\mathrm{.}5,$ Material for pinion: $40Cr1S_0=165$MPa and $BHN=250,$ Material for gear: Phosphor Bronze $S_0=80$MPa and $BHN=100,=15,{}_n=20,C_s=1\mathrm{.}25.$ Find out standardized module $($mm$),$ number of teeth and pitch circle diameters, face widths, and centre distance. Considering $F_w=1\mathrm{.}15F_d$ and $E=212$GPa, calculate the required BHN value of the gear tooth. Follow all the necessary design steps. kindly do the answer with proper calculations