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 triangle sides and angle

We have a triangle with two sides: \( a = 43 \) feet (distance from home plate to pitcher's mound), \( b = 162 \) feet (distance the ball travels), and the included angle \( C = 27^\circ \). We need to find the third side \( c \) (distance the outfielder must throw) using the Law of Cosines.

Step2: Apply the Law of Cosines

The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \).
Substitute \( a = 43 \), \( b = 162 \), and \( C = 27^\circ \) (convert \( \cos(27^\circ) \approx 0.8910 \)):
\[

$$\begin{align*} c^2&=43^2 + 162^2 - 2\times43\times162\times0.8910\\ &=1849 + 26244 - 2\times43\times162\times0.8910\\ &=28093 - 12334.92\\ &=15758.08 \end{align*}$$

\]

Step3: Find the square root of \( c^2 \)

Take the square root of \( 15758.08 \): \( c=\sqrt{15758.08}\approx125 \) (rounded to the nearest foot).

Answer:

125 feet