Two docks are located on an​ east-west line 2582 ft apart. From dock​ A, the bearing of a coral reef is 64 degrees 25 prime. From dock​ B, the bearing of the coral reef is 334 degrees 25 prime. Find the distance from dock A to the coral reef.
Question content area bottom
Part 1
The distance from dock A to the coral reef 
  
enter your response here ft.
​(Round to the nearest integer as​ needed.)

We can solve this problem by using the **Law of Sines** in trigonometry. Here's a step-by-step breakdown:

### Step 1: Interpreting the Bearings
- The bearing from Dock A to the coral reef is \( 64^\circ 25' \), which is equivalent to \( 64.42^\circ \) (since 25 minutes = 25/60 = 0.42 degrees).
- The bearing from Dock B to the coral reef is \( 334^\circ 25' \), which is equivalent to \( 334.42^\circ \).

### Step 2: Finding the Interior Angle at the Reef
- We are looking at an east-west line, with bearings given from Dock A and Dock B.
- To find the angle between these two bearings at the coral reef, we calculate:
  \[
  \text{Angle at the reef} = 334.42^\circ - 64.42^\circ = 270^\circ
  \]
  This means the angle formed by the reef and the two docks is \( 360^\circ - 270^\circ = 90^\circ \), so it's a right triangle.

### Step 3: Law of Sines
In this case, since the angle is a right angle, we can directly apply trigonometry for a right triangle:
- The distance between the two docks is \( 2582 \, \text{ft} \), which acts as the hypotenuse of a right triangle.
- We can now calculate the distance from Dock A to the coral reef by using the sine of the bearing angle from Dock A.

The formula for the distance \( d_A \) from Dock A to the coral reef is:

\[
d_A = \text{hypotenuse} \times \sin(\text{bearing angle from Dock A})
\]

\[
d_A = 2582 \times \sin(64.42^\circ)
\]

### Step 4: Calculation

Let me calculate that for you.The distance from Dock A to the coral reef is approximately **2,329 feet**.

Would you like further details or have any questions? Here are 5 related questions to explore:

1. How would the problem change if the angle at the reef was not a right angle?
2. How can we apply the Law of Cosines to find the distance in a non-right triangle?
3. What is the bearing of the reef from Dock A if the reef were located north of both docks?
4. How can we compute the distance to the reef from Dock B?
5. What real-world applications are there for solving such triangle-based problems in navigation?

**Tip:** Bearings are always measured clockwise from the north, making them crucial for accurately determining directions on a map or grid.

Two docks are located on an east-west line 2582 ft apart. From dock A, the bearing of a coral reef is 64 degrees 25 prime. From dock B, the bearing of the coral reef is 334 degrees 25 prime. Find the distance from dock A to the coral reef.

Solve a real-world navigation problem by calculating the distance from Dock A to a coral reef using bearings and trigonometry. This detailed guide walks through the Law of Sines and right triangle principles, perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation