Problem $1.(20$ points$)$

A geotechnical engineer is evaluating the stability of the slope shown in Figure $1.$This evaluation is considering the potential for a shear failure along the shear surface shown. The soil has $\backslash (\backslash$phi$^\{\backslash$prime$\}=30^\{\backslash$circ$\}\backslash )$ and no cohesive strength.

a$).$ Compute the shear strength at Point A along this surface when the groundwater table is at level C $.$ The unit weight of the soil is $\backslash (120\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}^\{3\}\backslash )$ above the groundwater table and $123\backslash (\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}^\{3\}\backslash )$ below $(\backslash (\backslash$mathbf$\{10\}\backslash )$ points$).$

b$).$ If the shear stress at point A is $2500$ psf $,$ what are the factors of safety against sliding when the water level is at C $?(\backslash (\backslash$mathbf$\{5\}\backslash )$ points$)$

c$).$ Fluctuation of water level affects the factor of safety at point A$.$ Calculate the water level which makes the factor of safety at point A equal to $1.(5$ points$)$

Hint: use effective stress and assume the failure surface at point A is horizontal.

Figure $2.$ Cross$-$section of a potential landslide.