To determine the distance the outfielder throws the ball to the pitcher, we can use the Law of Cosines. This approach is necessary because the hit forms a triangle with the pitcher's mound and the point where the outfielder catches the ball. Here's the step-by-step solution:
1. Identify the given values:
- Distance from the pitcher's mound to home plate (side [tex]\( a \)[/tex]): [tex]\( 60.5 \)[/tex] feet.
- Distance from the pitcher's mound to where the ball is caught (side [tex]\( b \)[/tex]): [tex]\( 214 \)[/tex] feet.
- Angle between these two sides at home plate ([tex]\( \angle C \)[/tex]): [tex]\( 36 \)[/tex] degrees.
2. Convert the angle to radians:
To use the Law of Cosines, angles should be in radians.
[tex]\[
\text{Angle in radians} = 0.6283185307179586 \, \text{radians}
\][/tex]
3. Apply the Law of Cosines:
The Law of Cosines formula is:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two sides, and [tex]\( C \)[/tex] is the included angle. Plugging in the known values:
[tex]\[
c^2 = 60.5^2 + 214^2 - 2 \times 60.5 \times 214 \times \cos(0.6283185307179586)
\][/tex]
4. Calculate the squared distance:
[tex]\[
c^2 = 28507.56394765511 \, \text{square feet}
\][/tex]
5. Take the square root to find the distance [tex]\( c \)[/tex]:
[tex]\[
c = \sqrt{28507.56394765511} \approx 168.84183115464933 \, \text{feet}
\][/tex]
Thus, the distance the outfielder throws the ball to the pitcher is approximately [tex]\( 168.84 \)[/tex] feet.
