Two docks are located on an​ east-west line 

25892589 

ft apart. From dock​ A, the bearing of a coral reef is 

56 degrees 28 prime56°28′. 

From dock​ B, the bearing of the coral reef is 

326 degrees 28 prime326°28′. 

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.)

This problem involves finding the distance from dock A to a coral reef using bearings, which requires applying the Law of Sines.

### Step 1: Analyze the bearings
- From dock A, the bearing to the coral reef is \( 56^\circ 28' \). Since bearings are measured clockwise from north, the angle between the north-south line at dock A and the coral reef is \( 56^\circ 28' \) (which is equivalent to \( 56.467^\circ \) in decimal form).
  
- From dock B, the bearing is \( 326^\circ 28' \). Since this is measured clockwise from north, the equivalent bearing counterclockwise from the south is \( 360^\circ - 326^\circ 28' = 33^\circ 32' \) or \( 33.533^\circ \).

### Step 2: Calculate the interior angle at the coral reef
The total angle at the coral reef (between the lines from dock A and dock B to the reef) is the difference between the bearings:
\[
180^\circ - (56.467^\circ + 33.533^\circ) = 90^\circ.
\]
This angle is \( 90^\circ \).

### Step 3: Apply the Law of Sines
We know:
- The distance between the two docks (AB) is 25,892,589 feet.
- The angle at the coral reef (angle C) is \( 90^\circ \).

Now we can apply the Law of Sines:
\[
\frac{AB}{\sin C} = \frac{d_A}{\sin B}
\]
where:
- \( AB = 25,892,589 \, \text{ft} \),
- \( C = 90^\circ \),
- \( B = 33.533^\circ \), 
- \( d_A \) is the distance from dock A to the coral reef.

Rearranging for \( d_A \):
\[
d_A = \frac{AB \cdot \sin B}{\sin C} = \frac{25,892,589 \cdot \sin(33.533^\circ)}{\sin(90^\circ)}.
\]
Since \( \sin(90^\circ) = 1 \), the equation simplifies to:
\[
d_A = 25,892,589 \cdot \sin(33.533^\circ).
\]

### Step 4: Compute the result
Now we calculate the distance:

\[
d_A = 25,892,589 \cdot \sin(33.533^\circ).
\]

Let me calculate this.The distance from dock A to the coral reef is approximately 14,303,511 feet (rounded to the nearest integer).

Do you have any questions or would you like more details?

Here are 5 related questions to further explore the topic:
1. How would the calculation change if the distance between the docks was given in miles?
2. Can you use the Law of Cosines to solve this problem instead?
3. What would be the effect on the result if the bearing from dock B was slightly different?
4. How can we convert bearings into true directional angles on a map?
5. How does this approach differ from solving a similar problem using Cartesian coordinates?

**Tip:** Always make sure to convert bearings to angles that match the context of the trigonometric function (clockwise vs. counterclockwise) before applying trigonometric laws.

Two docks are located on an east-west line 25,892,589 ft apart. From dock A, the bearing of a coral reef is 56°28′. From dock B, the bearing of the coral reef is 326°28′. Find the distance from dock A to the coral reef. (Round to the nearest integer as needed.)

Solve the problem of finding the distance between dock A and a coral reef using the Law of Sines. The problem involves calculating angles from bearings and applying trigonometry. Suitable for Grades 10-12, this detailed guide walks through converting bearings and using the Law of Sines.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation