Question
You are attending a baseball game. The batter hits the ball high into the air of the infield. You notice the ball, as it is hit, forms a parabolic path from contact with the bat to where it lands in the field. The path the ball traveled can be expressed by the quadratic equation y=−124x2+2x , where x represents the horizontal distance and y represents the height of the ball. Interpret the following key feature of the parabola in terms of the context.(1 point)
Answers
GPT-4o mini
The quadratic equation given for the path of the baseball is \( y = -124x^2 + 2x \). This is a downward-opening parabola, which suggests that the ball reaches a maximum height before falling back to the ground.
### Key Feature: Vertex of the Parabola
In the context of the baseball's flight, the vertex of the parabola represents the **maximum height** that the baseball reaches during its flight. To find the vertex of the parabola represented by the equation \( y = ax^2 + bx + c \), we can use the vertex formula \( x = -\frac{b}{2a} \).
In this case:
- \( a = -124 \)
- \( b = 2 \)
Calculating the x-coordinate of the vertex:
\[
x = -\frac{2}{2 \times -124} = -\frac{2}{-248} = \frac{1}{124}
\]
Now, we can find the corresponding y-coordinate by substituting \( x = \frac{1}{124} \) back into the equation:
\[
y = -124\left(\frac{1}{124}\right)^2 + 2\left(\frac{1}{124}\right)
\]
\[
= -124 \left(\frac{1}{15376}\right) + \frac{2}{124}
\]
\[
= -\frac{124}{15376} + \frac{2}{124}
\]
\[
= -\frac{124}{15376} + \frac{2 \times 124}{15376}
\]
\[
= \frac{248 - 124}{15376} = \frac{124}{15376} = \frac{1}{124}
\]
Thus, the vertex of the parabola, or the maximum height of the ball, occurs at \( \left( \frac{1}{124}, \frac{1}{124} \right) \).
### Interpretation
The vertex indicates that the baseball reaches a maximum height of approximately **0.0008** units (or 0.08 mm) at a horizontal distance of approximately **0.0081** units (or about 0.81 mm) from the point of contact. This height is not substantial, suggesting that the hit was fairly weak or that the units are not in conventional measurements (like feet or meters), which is unusual for a typical baseball hit.
In summary, the vertex of the parabola in this context signifies the peak height that the baseball achieves during its flight before it begins to descend, illustrating the trajectory of the ball in the infield after being struck by the bat.
### Key Feature: Vertex of the Parabola
In the context of the baseball's flight, the vertex of the parabola represents the **maximum height** that the baseball reaches during its flight. To find the vertex of the parabola represented by the equation \( y = ax^2 + bx + c \), we can use the vertex formula \( x = -\frac{b}{2a} \).
In this case:
- \( a = -124 \)
- \( b = 2 \)
Calculating the x-coordinate of the vertex:
\[
x = -\frac{2}{2 \times -124} = -\frac{2}{-248} = \frac{1}{124}
\]
Now, we can find the corresponding y-coordinate by substituting \( x = \frac{1}{124} \) back into the equation:
\[
y = -124\left(\frac{1}{124}\right)^2 + 2\left(\frac{1}{124}\right)
\]
\[
= -124 \left(\frac{1}{15376}\right) + \frac{2}{124}
\]
\[
= -\frac{124}{15376} + \frac{2}{124}
\]
\[
= -\frac{124}{15376} + \frac{2 \times 124}{15376}
\]
\[
= \frac{248 - 124}{15376} = \frac{124}{15376} = \frac{1}{124}
\]
Thus, the vertex of the parabola, or the maximum height of the ball, occurs at \( \left( \frac{1}{124}, \frac{1}{124} \right) \).
### Interpretation
The vertex indicates that the baseball reaches a maximum height of approximately **0.0008** units (or 0.08 mm) at a horizontal distance of approximately **0.0081** units (or about 0.81 mm) from the point of contact. This height is not substantial, suggesting that the hit was fairly weak or that the units are not in conventional measurements (like feet or meters), which is unusual for a typical baseball hit.
In summary, the vertex of the parabola in this context signifies the peak height that the baseball achieves during its flight before it begins to descend, illustrating the trajectory of the ball in the infield after being struck by the bat.