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.(1 point)range: 0≤y≤
Answers
GPT-4o mini
To interpret the range of the parabola defined by the quadratic equation \( y = -142(x - 50)^2 + 60 \) in the context of a baseball being hit, let's analyze the given equation.
1. **Understanding the Quadratic Equation**:
The equation is in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In this case, \( a = -142 \), \( h = 50 \), and \( k = 60 \).
2. **Vertex of the Parabola**:
The vertex \( (50, 60) \) represents the highest point in the trajectory of the baseball, where \( x = 50 \) feet is the distance from home plate, and \( y = 60 \) feet is the maximum height the ball reaches.
3. **Direction of the Parabola**:
Since \( a \) is negative (\( -142 \)), this means the parabola opens downward, which is characteristic of the trajectory of a ball hit into the air.
4. **Range Interpretation**:
The range indicates the possible heights \( y \) that the baseball reaches as it travels through the air. The given equation shows that the maximum height (the vertex) is \( 60 \) feet. Since the baseball will eventually hit the ground (where \( y = 0 \)), we find that the baseball will be in the air for heights between \( 0 \) and \( 60 \) feet.
Thus, the interpretation of the range \( 0 \leq y \leq 60 \) in this context means:
The baseball travels through the air within a height range of **0 feet** (when it lands on the ground) to a maximum height of **60 feet** (when it reaches its highest point).
1. **Understanding the Quadratic Equation**:
The equation is in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. In this case, \( a = -142 \), \( h = 50 \), and \( k = 60 \).
2. **Vertex of the Parabola**:
The vertex \( (50, 60) \) represents the highest point in the trajectory of the baseball, where \( x = 50 \) feet is the distance from home plate, and \( y = 60 \) feet is the maximum height the ball reaches.
3. **Direction of the Parabola**:
Since \( a \) is negative (\( -142 \)), this means the parabola opens downward, which is characteristic of the trajectory of a ball hit into the air.
4. **Range Interpretation**:
The range indicates the possible heights \( y \) that the baseball reaches as it travels through the air. The given equation shows that the maximum height (the vertex) is \( 60 \) feet. Since the baseball will eventually hit the ground (where \( y = 0 \)), we find that the baseball will be in the air for heights between \( 0 \) and \( 60 \) feet.
Thus, the interpretation of the range \( 0 \leq y \leq 60 \) in this context means:
The baseball travels through the air within a height range of **0 feet** (when it lands on the ground) to a maximum height of **60 feet** (when it reaches its highest point).