A $12-$m high rock slope has been excavated at a face angle of $60.$ The rock in which this cut has been made contains persistent bedding planes thatdip at an angle of $35$ into the excavation. The $4\mathrm{.}35-$m deep tension crack is $4$m behind the crest, and is filled with water to a height of $3$m above the sliding surface. The strength parameters of the sliding surface are as follows:

Cohesion $($c$)=25$Kpa

Friction angle$()=37$

The unit weight of the rock is $26$KN$/$m$^3$ and the unit weight of the water is $9\mathrm{.}81$KN$/$m$^3$

Assuming that a plane slope failure is the most likely type of instability, analyze the following stability conditions.

$($a$)$ Calculate the factor of safety of the slope for the conditions given above $(3$ pts$)$

$($b$)$ Determine the factor of safety if the tension crack were completely filled with water due to run$-$off collecting on the crest of the slope $(3$ pts$)$

$($c$)$ Determine the factor of safety if the slope were completely drained

$($d$)$ Determine the factor of safety if the cohesion were to be reduced to zero due to excessive vibrations from nearby blasting operations, assuming that the slope was still completely drained $(2$ pts$)$