$($b$)$ A machine component is subjected to fluctuating stress that varies from $40\frac{N}{m}{m}^2$

to $100\frac{N}{m}{m}^2.$ The ultimate tensile strength and yield strength of the material of

the component are $600\frac{N}{m}{m}^2$ and $450\frac{N}{m}{m}^2$ respectively. If the endurance

limit stress for the machine component is $270\frac{N}{m}{m}^2,$ find the factor of safety

using $($i$)$ Soderberg line, $($ii$)$ Gerber line and $($iii$)$ Goodman line.