Two docks are located on an​ east-west line 2583 ft apart. From dock​ A, the bearing of a coral reef is 60 degrees 27 prime. From dock​ B, the bearing of the coral reef is 330 degrees 27 prime. Find the distance from dock A to the coral reef.
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Part 1
The distance from dock A to the coral reef 
  
enter your response here ft.
​(Round to the nearest integer as​ needed.)

We are tasked with finding the distance from dock A to the coral reef, based on the given information. This is a problem of triangulation, and we can use the **Law of Sines** to solve it.

### Step 1: Understand the bearings and the triangle formed
- Dock A and Dock B are 2583 ft apart, forming the base of a triangle.
- The bearings of the coral reef from each dock are given as:
  - From Dock A: **60° 27'**
  - From Dock B: **330° 27'**

### Step 2: Calculate the angle between the two lines of sight
Bearings are measured from the north, so the angle between the two bearings can be calculated as the difference between them:

\[
\text{Angle between the bearings} = 330^\circ 27' - 60^\circ 27' = 270^\circ
\]

So, the angle between the two lines of sight (from Dock A and Dock B) is **270°**.

### Step 3: Apply the Law of Sines
We now have a triangle with:
- The base \(AB = 2583 \, \text{ft}\).
- The angles at Dock A and Dock B: \( \angle A = 60^\circ 27' \) and \( \angle B = 330^\circ 27' \), respectively.
- The angle between the two lines of sight: **270°**.

Using the **Law of Sines**, we can set up the following ratio:

\[
\frac{AB}{\sin(270^\circ)} = \frac{AC}{\sin(60^\circ 27')}
\]

Solving for \(AC\) (the distance from Dock A to the coral reef):

\[
AC = \frac{AB \cdot \sin(60^\circ 27')}{\sin(270^\circ)}
\]

Now we plug in the numbers to compute this value. Let's proceed with the calculation.To continue the calculation, let’s break down the necessary values:

- \( AB = 2583 \, \text{ft} \)
- \( \angle A = 60^\circ 27' \approx 60.45^\circ \)
- \( \angle B = 330^\circ 27' \approx 330.45^\circ \)
- \( \angle \text{between bearings} = 270^\circ \)

We can now use the Law of Sines formula:

\[
AC = \frac{2583 \cdot \sin(60^\circ 27')}{\sin(270^\circ)}
\]

We know that \( \sin(270^\circ) = -1 \), so the equation becomes:

\[
AC = \frac{2583 \cdot \sin(60^\circ 27')}{-1}
\]

### Step 4: Plugging in values for the sine functions
- \( \sin(60^\circ 27') \approx 0.866 \)

Now, substituting this in:

\[
AC = \frac{2583 \cdot 0.866}{-1} \approx 2234.5 \, \text{ft}
\]

So, the distance from Dock A to the coral reef is approximately **2235 feet** (rounded to the nearest integer).

### Final Answer:
The distance from Dock A to the coral reef is approximately **2235 feet**.

---

Would you like further details or clarifications? Here are some related questions for you:

1. How does the Law of Sines work when solving for distances in triangulation problems?
2. What is the significance of using bearings in navigation and geometry?
3. How can you find the distance from a point to a line in geometry?
4. What are the steps for solving a triangulation problem using both the Law of Sines and the Law of Cosines?
5. How do bearings and angles relate to each other in a coordinate plane?

**Tip:** When working with bearings, always ensure the angles are in degrees and minutes, and convert them to decimal degrees when necessary to simplify calculations.

Two docks are located on an east-west line 2583 ft apart. From dock A, the bearing of a coral reef is 60 degrees 27 prime. From dock B, the bearing of the coral reef is 330 degrees 27 prime. Find the distance from dock A to the coral reef.

Learn how to calculate the distance from a dock to a coral reef using bearings and the Law of Sines. This problem involves triangulation, angle calculations, and sine functions to determine an unknown distance in a real-world navigation scenario.

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