Two docks are located on an east-west line 2589 ft apart. From dock A, the bearing coral reef is 60°22'. From dock B, the bearing of the coral reef is 330°22'. Find the distance from dock A to the coral reef.

This problem involves solving a triangle using bearings and the law of sines. Here's how we can approach it:

### Step 1: Interpreting the bearings
- **From dock A**, the bearing to the coral reef is 60°22'. This bearing is measured clockwise from the north, so the angle between the north line at dock A and the line to the coral reef is 60°22'.
- **From dock B**, the bearing to the coral reef is 330°22'. Since bearings are measured clockwise from north, 330°22' can also be interpreted as 29°38' (360° − 330°22') west of north.

### Step 2: Understanding the geometry
We know the two docks are 2589 ft apart. This distance forms the base of a triangle, with one side going from dock A to the coral reef and another from dock B to the coral reef.

To apply the Law of Sines, we need to find the interior angles of the triangle.

### Step 3: Finding the interior angle at the coral reef
Since we are working with bearings, we need to determine the angle at the coral reef formed by the two sight lines from docks A and B. The angle between the bearing at A (60°22') and B (330°22') is:

\[
\text{Angle at the coral reef} = 360° - (60°22' + 29°38') = 360° - 90° = 270°
\]

Thus, the angle at the coral reef is 90°. This means we can use right triangle relationships.

### Step 4: Using the Law of Sines
Now, applying the Law of Sines to solve for the distance from dock A to the coral reef, we can set up the equation:

\[
\frac{d}{\sin(270°)} = \frac{2589 \text{ ft}}{\sin(29°38')}
\]

This will give us the distance \(d\) from dock A to the coral reef.

Let me calculate this.The distance from dock A to the coral reef is approximately 5236.15 feet.

Would you like further details on the steps, or do you have any questions?

Here are 5 related questions to expand on this:
1. How can we calculate distances using non-right triangles?
2. How do bearings differ from standard angle measurements?
3. What is the Law of Sines and when do we apply it?
4. Can we solve this problem using the Law of Cosines instead? If so, how?
5. How do real-world factors like elevation or tides affect such calculations?

**Tip:** In navigation problems involving bearings, always make sure to interpret the angles correctly relative to north!

Two docks are located on an east-west line 2589 ft apart. From dock A, the bearing to the coral reef is 60°22'. From dock B, the bearing of the coral reef is 330°22'. Find the distance from dock A to the coral reef.

Learn how to calculate the distance between dock A and a coral reef using bearings and the Law of Sines. This trigonometry problem involves interpreting bearings, applying right triangle relationships, and using the Law of Sines. Suitable for Grades 10-12.

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