The problem provided involves two docks, located on an east-west line 2590 feet apart, and asks to find the distance from dock A to a coral reef based on bearings from both docks. Here's a step-by-step guide on how to solve it using trigonometry, specifically the Law of Sines.

Given:

• Distance between Dock A and Dock B = 2590 ft.
• Bearing from Dock A to the coral reef = 62°24′.
• Bearing from Dock B to the coral reef = 332°24′ (or equivalently, 360° − 332°24′ = 27°36′).

Step-by-step solution:

1. Determine the angles in the triangle:

   • The angle between the two bearings can be found by subtracting the bearings: $\text{Angle at the coral reef} = 62°24′ - 27°36′ = 34°48′.$

2. Apply the Law of Sines: The Law of Sines relates the sides and angles in any triangle: $\frac{A}{\sin(\text{Angle at B})} = \frac{B}{\sin(\text{Angle at A})} = \frac{C}{\sin(\text{Angle at Coral Reef})}.$ In this case:

   • $A$ is the distance from Dock A to the coral reef (which we want to find).
   • $B = 2590$ ft (distance between the two docks).
   • The angle at Dock A is $180° - 62°24′ = 117°36′$.
   • The angle at Dock B is $180° - 27°36′ = 152°24′$.
   • The angle at the coral reef is $34°48′$.

   Using the Law of Sines, we set up the equation: $\frac{2590}{\sin(34°48′)} = \frac{A}{\sin(152°24′)}.$

3. Solve for A: Now we can solve for $A$: $A = \frac{2590 \cdot \sin(152°24′)}{\sin(34°48′)}.$

Let me calculate that for you.The distance from Dock A to the coral reef is approximately 2103 feet, rounded to the nearest integer.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do bearings relate to angles in trigonometry?
2. What is the Law of Sines and when is it used in solving triangles?
3. How do you convert degrees, minutes, and seconds into decimal degrees?
4. How do you determine angles between two bearings in navigation?
5. What is the difference between the Law of Sines and the Law of Cosines?

Tip: When working with bearings, always convert angles to standard position relative to north or east, and use trigonometry to solve for distances or angles.