Chapter 14.5, Problem 28PPSE

Solution

To find: How far is the pitcher from the third base.

46 ft

Given information: A softball diamond is a square that is 65 ft on a side. Also, the pitcher’s mound is 46 ft from home plate.

Concept used:

Consider a triangle ABC such that the length of side AB is $c$ , the length of side BC is $a$ , and the length of side AC is $b$ .

Then, according to law of cosine $c^2=a^2+b^2-2ab\cos C$ .

Calculation:

Since diamond is a square, the distance from the home base to the pitcher’s mound lies on the diagonal of the square. Also, it will be an angle bisector of the corner.

Below mentioned diagram is the diagrammatic representation of the given question such that $x$ is the distance of pitcher from the third base.

  

Now, apply the Cosine’s rule in the above diagram and solve for $x$ .

  $\begin{array}{c}{x}^2={65}^2+{46}^2-2\left(65\right)\left(46\right)\cos 45°\\ x^2\approx 2112.5\\ x\approx 46\text{ ft}\end{array}$

Hence, the required value is 46 ft.