$(50)$ Using the soil profile below, design a retaining wall based on Coulomb's equation using the Factor of Safety: $2\mathrm{.}0$ for horizontal sliding and overturning based on the Rankine theory. The unit weight of the backfill is $18\mathrm{.}5k\frac{N}{m^3},$ and the frictional angle of the backfill is $30.$ The excavated depth H is $5$ m $.$

$($a$)$ Calculate Ka & determine the lateral earth force $(k\frac{N}{m})$ per unit length of the wall,

$($b$)$ Draw the lateral earth stress $(k\frac{N}{m^2})$ along the back of the wall,

$($c$)$ Find the location of the resultant, $\frac{?}{\text{b}\text{a}\text{r}}(z),$ which is measured from the bottom of the wall,

$($d$)$ Here is the design of the wall: the bottom width of the wall is $1\mathrm{.}2$ times the top width, The wall height is $10\%$ greater than the slope height, which allows the foot of the wall to be buried below the bottom soil. The unit weight of concrete is $20k\frac{N}{m^3}.$ Determine the top width based on the Factor of Safety for the horizontal sliding or by trial and error $($Using the Excel spreadsheet for the trial and error testing$).$ The frictional angle between the soil and the wall is $21.$ Considering the friction between the backfill and the back of the wall and the bottom of the wall.

$($e$)$ Verify the safety factors for both sliding and toppling.