Question

16 baseball a pitcher throws a baseball 60 feet from the pitchers mound to home plate. a batter pops the ball up and it comes down just 24 feet from home plate. what can you determine about how far the ball lands from pitchers mound? explain why the triangle inequality theorem can be used to describe all but the shortest and longest possible distances.

Explanation:

Step1: Apply Triangle Inequality Theorem

Let \(d\) be the distance from the ball to the pitcher's mound. Let \(a = 60\) (distance from pitcher's mound to home - plate) and \(b=24\) (distance from home - plate to the ball). According to the triangle inequality theorem \(|a - b|\lt d\lt a + b\).

Step2: Calculate the range

First, calculate \(a + b\): \(60+24=84\). Then calculate \(|a - b|\): \(|60 - 24|=36\). So \(36\lt d\lt84\).

The Triangle Inequality Theorem can be used because when we consider the three - point system (pitcher's mound, home - plate, and the ball's landing spot), if the three points form a non - degenerate triangle, the theorem gives the range of the third side. The only exceptions are when the three points are collinear. If the ball is on the line segment connecting the pitcher's mound and home - plate (in the direction from home - plate to pitcher's mound), \(d = 60 - 24=36\) (shortest distance). If the ball is on the line segment extended beyond home - plate from the pitcher's mound, \(d=60 + 24 = 84\) (longest distance).

Answer:

The ball lands more than \(36\) feet and less than \(84\) feet from the pitcher's mound.