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 125 feet 145 feet 156 feet

Explanation:

Step1: Identify the law of cosines formula

The law of cosines for a triangle with sides \(a\), \(b\), \(c\) and the included - angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 43\), \(b = 162\), and \(C=27^{\circ}\).

Step2: Calculate the values for the formula

First, find \(\cos(27^{\circ})\approx0.891\). Then, \(a^{2}=43^{2}=1849\), \(b^{2}=162^{2}=26244\), and \(2ab\cos C = 2\times43\times162\times0.891=2\times43\times144.342 = 12393.412\).

Step3: Apply the law of cosines

\(c^{2}=a^{2}+b^{2}-2ab\cos C=1849 + 26244-12393.412=15699.588\).

Step4: Find the value of \(c\)

\(c=\sqrt{15699.588}\approx125.298\approx125\) (rounded to the nearest foot).

Answer:

125 feet