An archer stands 40.0m from the target. if the arrow is shot horizontally with a velocity of 90.0 m/s, how far above the bull's eye must she aim to compensate for gravity pulling her arrow downward?

Ok, first we need to have in mind that to findthe distance we need to find also the time then we use the kinematic equations in the x-direction 

delta x = Vox*t + 1/2*a*t^2 
since there is no horizontal acceleration, a = 0 m/s/s 
delta x = Vox*t 
40.0 m = 90 m/s * t 
t = 4/9 s 

Now using this time you can calculate the vertical displacement due to gravity. 
delta y = Voy*t + 1/2*g*t^2 
since there is no initial vertical velocity, Voy = 0m/s 
delta y = 1/2*g*t^2 
delta y = 1/2*(-9.81 m/s/s)*(4/9 s)^2 
solve for delta y and you will have the answer