The problem asks for the distance from Dock A to a coral reef, given the bearings from two docks and their separation.

We have:

• Two docks, A and B, are 2580 feet apart on an east-west line.
• The bearing from Dock A to the coral reef is $65^\circ 23'$ (measured clockwise from north).
• The bearing from Dock B to the coral reef is $335^\circ 23'$.

Step-by-step solution:

1. Understand the Bearings:

   • A bearing of $65^\circ 23'$ from Dock A means the angle from north is about $65.383^\circ$.
   • A bearing of $335^\circ 23'$ from Dock B is equivalent to $360^\circ - 335^\circ 23' = 24.617^\circ$.

2. Draw a Triangle:

   • The two docks (A and B) form the base of the triangle, with the coral reef as the third point.
   • Use the bearings to calculate the angles at each dock and between the lines drawn from A and B to the reef.

3. Solve Using the Law of Sines:

   • We can use trigonometry to solve for the distance from Dock A to the reef by applying the Law of Sines.

Let me calculate the exact distance for you!The distance from Dock A to the coral reef is approximately 2580 feet, which coincidentally is the same as the distance between the docks.

Would you like any further details or explanations about this problem?

Here are 5 related questions for further practice:

1. How would the solution change if the distance between docks A and B was increased?
2. What if the bearings from Dock A and B were swapped? How would that affect the triangle?
3. Can you apply the Law of Cosines to solve this problem instead of the Law of Sines?
4. How do bearings relate to standard angles in trigonometry?
5. How can we check the consistency of the solution using a geometric diagram?

Tip: When dealing with bearings, always visualize the directions on a coordinate plane to better understand the relationships between angles.