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

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
𝑦
=
−
1
42
(
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.
Range: 0≤y≤
0
≤
𝑦
≤

GPT-4o mini · Answer

To interpret the range of the quadratic equation \( y = -\frac{1}{42}(x - 50)^2 + 60 \) in the context of a baseball game, we will analyze the features of the parabola.

Understanding the Equation: The equation is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, \( a = -\frac{1}{42} \), which means the parabola opens downward, indicating that the ball reaches a maximum height.
Finding the Vertex: The vertex of the parabola is at \((50, 60)\). This tells us that the maximum height the ball reaches is \( y = 60 \) feet when it is at a horizontal distance \( x = 50 \) feet from home plate.
Identifying the Range: The equation also expresses a range of heights. Given that the parabola opens downward and has a maximum height of \( 60 \) at \( x = 50 \), the ball will fall back down to the ground as it travels further right (increasing \( x \)). The height \( y \) can reach as low as \( 0 \) feet, which is when the ball lands on the ground.

Therefore, the range \( 0 \leq y \leq 60 \) means that:

0 ≤ y: The height of the ball is never below 0 feet; it cannot travel underground (the ball cannot go lower than the ground level).
y ≤ 60: The maximum height of the ball is 60 feet, the point at which it reaches its apex before descending.

Conclusion: In the context of this baseball game, the range of \( 0 \leq y \leq 60 \) indicates that the ball, once hit, will soar up to a maximum height of 60 feet above the ground before landing on the field at some point where its height reaches 0 feet again.