Question

on a softball field, home plate is 43 feet from the pitchers mound. a ball is hit at an angle of 27° east of the pitchers mound. the ball travels 162 feet before it is caught by an outfielder. how far must the outfielder throw the ball to return it to the pitcher? round to the nearest foot. 112 feet 126 feet 145 feet 156 feet

Explanation:

Answer:

This problem belongs to Mathematics - Geometry. To solve, we can use the Law of Cosines. Let \(a = 43\), \(b=162\), and \(\theta = 27^{\circ}\). The Law of Cosines formula for the distance \(c\) (the distance the out - fielder has to throw the ball back to the pitcher) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).

\[

$$\begin{align*} c^{2}&=43^{2}+162^{2}-2\times43\times162\times\cos(27^{\circ})\\ c^{2}& = 1849+26244-13932\times0.8910\\ c^{2}&=1849 + 26244-12413.412\\ c^{2}&=15679.588\\ c&\approx125\text{ feet} \end{align*}$$

\]

So the answer is 125 feet.