Question $3$: A borehole is drilled through a subsurface formation. The strain components in the formation at a particular point around the borehole are measured in two perpendicular directions using strain gauges. The measured strain components in these directions $($assumed as the $x-$ and $y-$axes$)$ are:

${}_x=450s,{}_y=-300s,{}_{xy}=250s$

a$)$ Using the strain components provided, calculate the principal strains $({}_1$ and ${}_2)$ at this point.

b$)$ Determine the orientation of the principal strain axes relative to the positive $x-$axis.

c$)$ Construct Mohr's circle for strain using the given strain components.

d$)$ From Mohr's circle, determine the maximum shear strain and its orientation.