Rigby has a large target of radius and a laser pointer. He is clumsy, so he points the laser at a uniformly random point on the target. He offers you to play the following game: You point the laser at the target time. Before doing so, you fix a value . If your laser is within distance away from the center, you must pay Rigby . If your laser points outside the circle of radius , Rigby pays you money units. Find the value of that maximizes your expected profit from this game when .