You are attending a professional baseball game. The batter hits the ball into right field. You notice the ball, as it is hit, forms a parabolic path from the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation y=−142(x−50)2+60 , where x is the distance from home plate, and y is the height in the air as the ball travels in the air. Interpret the following key feature of the parabola in terms of the context.

Question

You are attending a professional baseball game. The batter hits the ball into right field. You notice the ball, as it is hit, forms a parabolic path from the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation  y=−142(x−50)2+60 ,  where x is the distance from home plate, and y is the height in the air as the ball travels in the air. Interpret the following key feature of the parabola in terms of the context.

GPT-4o mini · Answer

The given quadratic equation representing the path of the baseball is \( y = -142(x - 50)^2 + 60 \).

To better understand this equation in the context of the baseball game, let's break down its components:

Vertex: The general form of a parabola expressed as \( y = a(x - h)^2 + k \) allows us to identify the vertex \((h, k)\) of the parabola, which is a key feature. Here, the vertex is at the point \((50, 60)\).
- Interpretation: This vertex point represents the maximum height the baseball reaches and the horizontal distance from home plate where this height occurs. In this case, the ball reaches its highest point of 60 feet at 50 feet from home plate. This indicates that the baseball was hit well, soaring to its peak height about a third of the way to the outfield fence.
Maximum Height: The value \(y = 60\) at \(x = 50\) indicates that the maximum height achieved by the baseball is 60 feet.
- Interpretation: This tells us that at a distance of 50 feet from home plate, the ball is at its highest point, which is significant for both players and spectators, as this height affects the ability of outfielders to catch the ball and the trajectory of the ball as it descends.
Opening Direction: The coefficient of \(-142\) indicates that the parabola opens downwards.
- Interpretation: A downward-opening parabola signifies that after reaching its maximum height (the vertex), the baseball will begin to fall back towards the ground. This is typical for a projectile like a baseball, which will eventually land after reaching its peak height.
Width of the Parabola: The larger the absolute value of the coefficient (in this case \(-142\)), the narrower the parabola.
- Interpretation: This indicates that the ball had a steep trajectory, which may involve higher force upon contact, leading to a greater peak height and sharper descent.

In summary, the vertex \((50, 60)\) of the quadratic equation not only helps us determine the maximum height of the baseball (60 feet) but also indicates the horizontal distance from home plate (50 feet) where this height is reached, providing context for the batter's hit and its significance during the game.