A retaining wall with the geometry shown below is $2$ m wide by $4$ m tall, its weight is $123$ kN $($per $m$ into the analysis plane$)$ and its resultant acts $1\mathrm{.}25$ m from the toe. A novel lightweight backfill is used with dry unit weight $10k\frac{N}{m^3}$ and internal friction angle $25.$ The wall sits on granular material and the contact friction angle at the wall's concrete base is $30.$ The design pressure on top of the backfill is $10$ kPa $.$

a$)$ Determine the pressure distributions and their associated resultants and lines of action acting on the wall's back surface. $($Sketch these on the above diagram.$)(5$ pts $)$

b$)$ Determine the wall's Factor of Safety against sliding. $(5$ pts$)$

c$)$ Determine if the wall is safe against overturning. $(5$ pts$)$

d$)$ Determine the pressure distribution acting along the base. $(5$ pts$)$