Yes, you have it right, but I think you are over-complicating the calculation. Keep in mind that work is the dot product of force vector and motion vector. The value for theta is the angle of the force applied relative to the direction of movement of the cart - that angle is 30 degrees (the angle of the handle relative to the incline). So to find the work done on the cart by the force simply use F= 20 lb, D = 100 ft, and theta = 30 degrees. There was no need to find vertical and horizontal components of distance. Also be sure to specifythat the answer is in units of foot-pounds.
BTW - the problem statement says the handle is at 30 degrees to the horizontal, but I assume you meant it's at 30 degrees to the direction of motion of the cart.