I think this is the wrong interpretation; your answer would be correct if the rope were hovering at a height of 10 km along the full length. My interpretation

I think this is the wrong interpretation; your answer would be correct if the rope were hovering at a height of 10 km along the full length. My interpretation is that the rope is pulled taut to a height of 10 km at a single point, so that it runs in a straight line to the horizon in either direction and lays on the ground the rest of the way; this is also relevant to the question of longest line of sight from a given height.