The problem describes two docks located on an east-west line, with bearings of a coral reef given from each dock. The goal is to find the distance from Dock A to the coral reef.

We can approach this problem using the law of sines in trigonometry. The two docks, Dock A and Dock B, and the coral reef form a triangle. The distance between the two docks, $AB$, is 2586 ft. The bearings of the coral reef from both docks are given, allowing us to find the interior angles of the triangle.

Step 1: Find the angle at the coral reef

The bearings are relative to the north direction:

• From Dock A, the bearing to the coral reef is $63^\circ28'$.
• From Dock B, the bearing to the coral reef is $333^\circ28'$.

Since the bearing from Dock B is measured clockwise from north, the angle between Dock A and Dock B at the coral reef can be calculated as the difference between the bearings: $\theta = 333^\circ28' - 63^\circ28' = 270^\circ.$ Thus, the angle at the coral reef is $\theta = 270^\circ$, which is equivalent to $90^\circ$, as this is the difference in the bearings.

Step 2: Find the interior angles of the triangle

• The angle at Dock A is the bearing measured clockwise from the north (east is 90°), so: $\angle A = 90^\circ - 63^\circ28' = 26^\circ32'.$
• The angle at Dock B is: $\angle B = 333^\circ28' - 270^\circ = 63^\circ28'.$

The three angles in the triangle sum to $180^\circ$, so the angle at the coral reef $C$ can be found as: $\angle C = 180^\circ - \angle A - \angle B.$ Substituting the values: $\angle C = 180^\circ - 26^\circ32' - 63^\circ28' = 90^\circ.$ This means the triangle is a right triangle.

Step 3: Apply the Law of Sines

The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C},$ where $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively.

Since $C$ is $90^\circ$, we know that $c = AB = 2586 \text{ ft}$. To find the distance from Dock A to the coral reef (side $b$), we use: $\frac{b}{\sin B} = \frac{c}{\sin C}.$ Substituting the known values: $\frac{b}{\sin 63^\circ28'} = \frac{2586}{\sin 90^\circ}.$ Since $\sin 90^\circ = 1$, this simplifies to: $b = 2586 \times \sin 63^\circ28'.$

Step 4: Calculate the distance

Now, let's compute $b$, the distance from Dock A to the coral reef.

$b = 2586 \times \sin(63^\circ28').$

I'll calculate that now.The distance from Dock A to the coral reef is approximately $2313.63 \, \text{ft}$.

Would you like more details, or do you have any further questions?

Here are 5 related questions to consider:

1. How would the solution change if the bearing from Dock B were different?
2. How does the Law of Sines help in solving non-right triangles?
3. Can you derive the formula for bearings between two points using trigonometry?
4. What if the distance between the two docks was given in meters instead of feet?
5. How would you find the distance from Dock B to the coral reef?

Tip: In bearing problems, always ensure you interpret angles correctly relative to the north direction when converting them for trigonometric use.