Two docks are located on an​ east-west line 2581 ft apart. From dock​ A, the bearing of a coral reef is 56 degrees 22 prime. From dock​ B, the bearing of the coral reef is 326 degrees 22 prime. Find the distance from dock A to the coral reef.

We can solve this problem using the Law of Sines by considering it as a triangle formed by dock A, dock B, and the coral reef. Here's the detailed approach:

### Step 1: Understand the Bearings

- From dock A, the bearing of the coral reef is \( 56^\circ 22' \), which is equal to \( 56.3667^\circ \) in decimal form.
- From dock B, the bearing of the coral reef is \( 326^\circ 22' \), which is equal to \( 326.3667^\circ \) in decimal form.

Since bearings are measured clockwise from the north, we need to convert these into angles inside the triangle.

- The bearing from dock B is \( 326.3667^\circ \), which means the angle from north is \( 360^\circ - 326.3667^\circ = 33.6333^\circ \).
- Since the docks are on an east-west line, the angle inside the triangle at dock B can be found by adding the angle from north to the bearing from dock A. This is \( 56.3667^\circ + 33.6333^\circ = 90^\circ \).

Thus, the angle at the coral reef, \( \angle R \), is \( 90^\circ \).

### Step 2: Use the Law of Sines

The Law of Sines states that:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a \) is the distance from dock B to the coral reef,
- \( b \) is the distance from dock A to the coral reef,
- \( c = 2581 \) ft is the distance between dock A and dock B,
- \( \angle R = 90^\circ \).

The angle \( \angle A \) at dock A is:
\[
180^\circ - 90^\circ - 56.3667^\circ = 33.6333^\circ.
\]

Now apply the Law of Sines:
\[
\frac{2581}{\sin 90^\circ} = \frac{b}{\sin 33.6333^\circ}
\]
Since \( \sin 90^\circ = 1 \), this simplifies to:
\[
2581 = \frac{b}{\sin 33.6333^\circ}
\]
Solving for \( b \):
\[
b = 2581 \times \sin 33.6333^\circ
\]

Let's calculate that.

\[
b \approx 2581 \times 0.5547 = 1432.47 \text{ ft}.
\]

Thus, the distance from dock A to the coral reef is approximately **1432.47 feet**.

---

Do you want more details or have any questions?

Here are 5 related questions to further your understanding:
1. How does the Law of Sines apply to non-right triangles?
2. How would you solve the problem if the angle at the coral reef wasn't 90 degrees?
3. What are the bearings and how are they typically measured in navigation?
4. Can the Law of Cosines be used to solve this problem? If so, how?
5. How can this method be extended to 3D navigation?

**Tip**: When dealing with bearings, always remember to adjust angles according to the north-south direction to find the interior angles of the triangle.

Two docks are located on an east-west line 2581 ft apart. From dock A, the bearing of a coral reef is 56 degrees 22 prime. From dock B, the bearing of the coral reef is 326 degrees 22 prime. Find the distance from dock A to the coral reef.

Learn how to find the distance between two docks and a coral reef using bearings and the Law of Sines. This problem involves trigonometry, geometry, and bearing conversions. Suitable for students in Grades 10-12.

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