A mountain cimber is sranded on a ledge 30m above the ground. Rescuers on the ground want to shoot a projectie to him with a rope attatched to it. If the projectie is directed upwards at an initial ange of 55-degrees from a horizontal distance of 50m, determine the initial speed the projectile must have in order to land on the ledge

Make a sketch of the situation.

The horizontal displacement component of the projectile is given by:
s_h = v\cos(55) t

The vertical displacement component of the projectile is given by:
s_v = v\sin(55) t - \frac12 gt^2

s_v must be 30 m, s_h must be 50 m. Get an equation of t in terms of v in the first equation, substitute that into the second equation and solve for v.