The course notes show how a mathematician would figure out that the drag force on the cone was proportional to its area. A physicist could do it with much less work. In a dodgy approximation, the impulse on a body due to a fluid is proportional to the momentum of the fluid that hits it. If each cone falls with the same velocity, the momentum is proportional to the mass of fluid that hits it, which is proportional to the cone's area. Differentiating with respect to time, the drag force is proportional to area.