Sure, let's solve this problem step-by-step using the given information and the Law of Sines.
### Given:
- Joe and Bill are 180 feet apart on the beach.
- The angle between the coastline and the line from Joe to the ship is [tex]\( 38^\circ \)[/tex].
- The angle between the coastline and the line from Bill to the ship is [tex]\( 52^\circ \)[/tex].
### Step-by-Step Solution:
1. Determine the third angle of the triangle:
The three points we have form a triangle: Joe, Bill, and the ship. The sum of the angles in any triangle is [tex]\( 180^\circ \)[/tex].
[tex]\[
\text{Angle at the Ship} = 180^\circ - 38^\circ - 52^\circ = 90^\circ
\][/tex]
2. Convert angles to radians:
To apply trigonometric formulas precisely, we often convert degrees to radians. However, in our manual calculation, we don’t need to worry about this step since the angles are crucial as degrees in our context.
3. Apply the Law of Sines:
The Law of Sines states:
[tex]\[
\frac{\sin(\text{Angle at Joe})}{\text{Distance from Bill to the ship}} = \frac{\sin(\text{Angle at the Ship})}{\text{Distance between Joe and Bill}}
\][/tex]
Plugging in the known values:
[tex]\[
\frac{\sin(38^\circ)}{d} = \frac{\sin(90^\circ)}{180 \text{ ft}}
\][/tex]
Here, [tex]\( \sin(90^\circ) = 1 \)[/tex], so the equation simplifies to:
[tex]\[
\frac{\sin(38^\circ)}{d} = \frac{1}{180 \text{ ft}}
\][/tex]
4. Solve for [tex]\( d \)[/tex] (the distance from Bill to the ship):
[tex]\[
d = \frac{180 \text{ ft} \cdot \sin(38^\circ)}{1}
\][/tex]
Using the provided values:
[tex]\[
d = 180 \text{ ft} \cdot \sin(38^\circ)
\][/tex]
5. Calculate the value of [tex]\( d \)[/tex]:
Using a calculator to find [tex]\(\sin(38^\circ)\)[/tex]:
[tex]\[
\sin(38^\circ) \approx 0.61566
\][/tex]
Then,
[tex]\[
d \approx 180 \text{ ft} \cdot 0.61566 \approx 110.82 \text{ ft}
\][/tex]
### Result:
So, the distance from Bill to the ship is approximately [tex]\( 110.82 \)[/tex] feet.
By following these steps, we have found that the ship is around 110.82 feet away from Bill.
