Aprojectile is shot horizontally at 23.4 m/s from the roof of a building 55.0 m tall. what is the time necessary for the projectile to reach the ground below?

The horizontal speed has no effect on the answer. 

It doesn't matter whether you flick a marble horizontally from the roof,
fire a high-power rifle horizontally from the roof, drive a school bus straight
off the roof, or drop a bowling ball from the roof with zero horizontal speed. 
Their vertical speed is completely determined by gravity, (and it happens to
be the same for all of them).

Handy dandy formula for the distance covered by anything that starts out
with zero speed and accelerates to the end:

            Distance = (1/2) (acceleration) x (time)²

If the beginning of the journey is on Earth, then the acceleration is
9.8 m/s² ... the acceleration of gravity on Earth.  We'll assume that
the 55-meter rooftop in the question is part of a building on Earth.

                       55 meters  =  (1/2) (9.8 m/s²) x (time)²           

Divide each side
by  4.9 m/s² :            55 m / 4.9 m/s²  =  (time)²

                                 (time)²  =  (55/4.9)  sec²

Square-root
each side:                time  =  √(55/4.9 sec²)

                                           =      3.35 sec  .

Arise 8 run 1 this means go 8 up and 1 across, which is steepest