Two docks are located on an​ east-west line 2581 ft apart. From dock​ A, the bearing of a coral reef is 58 degrees 28 prime. From dock​ B, the bearing of the coral reef is 328 degrees 28 prime. Find the distance from dock A to the coral reef.


To find the distance from dock A to the coral reef, we can use the Law of Sines. The problem involves two bearings from two docks, and these bearings can help us determine the angles in the triangle formed by the two docks and the coral reef.

### Step 1: Determine the angles of the triangle

1. **Angles from the bearings**:
   - The bearing from dock A to the reef is \( 58^\circ 28' \), which is measured clockwise from north.
   - The bearing from dock B to the reef is \( 328^\circ 28' \), which is also measured clockwise from north.

2. **Calculate the angle at the reef**:
   - The bearing from dock B, \( 328^\circ 28' \), can be converted into the equivalent angle by subtracting it from \( 360^\circ \). This gives:
     \[
     360^\circ - 328^\circ 28' = 31^\circ 32'
     \]
   - The angle between the two bearings (i.e., the angle at the coral reef) can be found by subtracting the angle from dock B from the bearing from dock A:
     \[
     58^\circ 28' - 31^\circ 32' = 26^\circ 56'
     \]
     So, the angle \( \angle ARB \) at the coral reef is \( 26^\circ 56' \).

3. **Angle at dock A**:
   - The angle at dock A, \( \angle RAB \), is equal to the difference between \( 90^\circ \) (the east-west line) and the bearing \( 58^\circ 28' \):
     \[
     \angle RAB = 90^\circ - 58^\circ 28' = 31^\circ 32'
     \]

4. **Angle at dock B**:
   - The angle at dock B, \( \angle RBA \), can be calculated by subtracting \( \angle ARB \) and \( \angle RAB \) from \( 180^\circ \):
     \[
     \angle RBA = 180^\circ - 26^\circ 56' - 31^\circ 32' = 121^\circ 32'
     \]

### Step 2: Apply the Law of Sines

Now, using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a = \) distance from dock A to the coral reef \( R \) (what we need to find)
- \( b = \) distance from dock B to the coral reef \( R \)
- \( c = \) distance between the two docks \( = 2581 \, \text{ft} \)
- \( \sin A \), \( \sin B \), and \( \sin C \) are the sines of the angles at the coral reef, dock B, and dock A, respectively.

Using the Law of Sines:
\[
\frac{a}{\sin \angle RBA} = \frac{c}{\sin \angle ARB}
\]
\[
a = \frac{2581 \times \sin 121^\circ 32'}{\sin 26^\circ 56'}
\]

### Step 3: Calculate the Distance

Let's calculate this:
- \( \sin 121^\circ 32' \approx 0.857 \)
- \( \sin 26^\circ 56' \approx 0.453 \)

So,
\[
a = \frac{2581 \times 0.857}{0.453} \approx 4883.13 \, \text{ft}
\]

Thus, the distance from dock A to the coral reef is approximately **4883 feet**.

---

Do you want details or have any questions? Here are some related questions for further exploration:

1. What is the bearing of dock B from the coral reef?
2. How would the distance change if the docks were closer together?
3. What if the bearing from dock A was different—how would that affect the calculation?
4. How can this method be adapted to calculate distances on larger scales, such as nautical miles?
5. How would the result change if the bearings were measured in the opposite direction?

**Tip:** The Law of Sines is particularly useful in solving non-right-angled triangles when you know two angles and one side or two sides and one angle.

Learn how to calculate the distance from dock A to the coral reef using bearings and the Law of Sines. This problem involves trigonometric concepts such as bearings and angles in a triangle. Suitable for high school students studying trigonometry.

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