Final answer:
To find the vertex of the parabolic path of a projectile, take the midpoint of the horizontal distance between the launch point and the target for the x-coordinate. The y-coordinate will be greater than the highest point provided in the question, but additional data is required to find its exact value.
Step-by-step explanation:
The subject of this question is Mathematics, specifically related to projectile motion and the geometry of parabolas. The task involves finding the vertex of a parabolic path given two points on the path: the launch point (0, 1.5) and the point where the balloon hits the target (6, 15).
The vertex of a parabola in projectile motion is typically the highest point in the trajectory. Since we know the initial and final positions, we can find the vertex by considering that it must be exactly in the middle of the horizontal component (x-axis) between the launch point and the target. This is due to the projectile's symmetric path in absence of air resistance. Thus, with the launch point at x = 0 and the target point at x = 6, the x-coordinate of the vertex would be x = 3.
The y-coordinate of the vertex must be higher than the highest y-value given (since it is the maximum point), so we are looking for a point (3, y) where y > 15. The exact y-coordinate cannot be determined with the given information alone; additional data or assumptions are needed (e.g., the initial velocity, the angle of launch, or the effect of gravity).