Sure, I’ll walk you through solving this problem step by step.
### Step 1: Understanding the Problem
You have a baseball field where:
- The pitcher's mound is 60.5 feet away from the home plate.
- A batter hits a ball 226 feet at an angle of 39 degrees to the right of the pitcher's mound.
- The outfielder catches the ball and throws it directly to the pitcher.
You need to find the distance the outfielder throws the ball.
### Step 2: Breaking Down the Problem
Given:
- Distance from pitcher's mound to home plate (a) = 60.5 feet
- Distance the ball is hit (b) = 226 feet
- Angle between these two distances (C) = 39 degrees
We can use the Law of Cosines to find the distance (c) between where the outfielder catches the ball and the pitcher's mound.
### Step 3: Law of Cosines Formula
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
where:
- [tex]\( c \)[/tex] is the distance we need to find,
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the known distances,
- [tex]\( C \)[/tex] is the given angle.
### Step 4: Convert Angle to Radians
Since trigonometric functions generally use radians, we need to convert the angle from degrees to radians. The conversion factor is:
[tex]\[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \][/tex]
Given [tex]\( C = 39^\circ \)[/tex]:
[tex]\[ C_{\text{radians}} \approx 0.6806784082777885 \][/tex]
### Step 5: Applying the Law of Cosines
Now we substitute the known values into the Law of Cosines formula:
[tex]\[ c^2 = (60.5)^2 + (226)^2 - 2 \cdot 60.5 \cdot 226 \cdot \cos(0.6806784082777885) \][/tex]
### Step 6: Calculate the Distance
After performing the calculations:
[tex]\[ c \approx 182.98747645125246 \][/tex]
### Conclusion
The distance that the outfielder throws the ball to the pitcher is approximately 182.99 feet.
