Answered

The table shows the trajectory of a pitch in baseball, where $x$ represents the horizontal distance the ball travels in feet and $f(x)$ represents the height of the ball in feet.




$x$		$f(x)$

0		2.5

14		4.25

28		5.5

42		6.25

56		6.5

70		6.25






The pitcher releases the ball at the highest point in the curve, and the catcher is located at the $y$-axis. According to the table, how high is the ball when it reaches the catcher?


A0 feet
B2.5 feet
C6.5 feet
D56 feet

Verified Answer :

  • Identify the horizontal distance of the catcher as \(x = 0\).
  • Locate the corresponding height \(f(0)\) in the provided table.
  • The table value at \(x = 0\) is \(2.5\), which gives the height of the ball at that point.
  • The ball is at a height of \(\boxed{2.5}\) feet when it reaches the catcher.

Explanation

  1. Analysis of the Trajectory Data

We are given the trajectory of a baseball pitch through a table of coordinates \((x, f(x))\), where \(x\) represents the horizontal distance in feet and \(f(x)\) represents the height of the ball in feet. The table provided is as follows:

| \(x\) (feet) | \(f(x)\) (feet) |
| :--- | :--- |
| \(0\) | \(2.5\) |
| \(14\) | \(4.25\) |
| \(28\) | \(5.5\) |
| \(42\) | \(6.25\) |
| \(56\) | \(6.5\) |
| \(70\) | \(6.25\) |

Our goal is to determine the height of the ball when it reaches the catcher. The problem states that the catcher is located at the y-axis, which corresponds to the horizontal distance \(x = 0\).

  1. Determining the Height at the Catcher's Location

To find the height of the ball when it reaches the catcher, we look for the value of \(f(x)\) when \(x = 0\). According to the given table, when the horizontal distance \(x\) is \(0\), the corresponding height \(f(0)\) is \(2.5\). Therefore, the ball is at a height of \(2.5\) feet when it reaches the catcher at the y-axis.

  1. Final Answer

Based on our analysis, the height of the ball when it reaches the catcher at \(x = 0\) is \(\boxed{2.5}\) feet.

Examples

Understanding trajectories like the one in this baseball example is fundamental in physics and sports analytics. Coaches use such data to analyze a pitcher's mechanics, release point, and the movement of the ball through the air, while engineers use similar concepts to design projectiles or analyze the motion of objects in sports like basketball or golf. By mapping these paths, one can optimize performance and understand the physical forces at play.

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