Problem 37 - Trigonometry

Question

Distance to a Ship A ship is anchored off a long straight shoreline that runs north and south. From two observation points 18 miles apart on shore, the bearings of the ship are \(\mathrm{N} 31^{\circ} \mathrm{E}\) and \(\mathrm{S} 53^{\circ} \mathrm{E}\). What is the distance from the ship to each of the observation points?

Step-by-Step Solution

Answer

The ship is approximately 14.43 miles from A and 9.32 miles from B.

Step 1: Sketch the Problem

Draw a diagram with a straight shoreline as a horizontal line. Place two points, A and B, on this line, 18 miles apart. The ship is off the shore, and lines from A and B extend to the ship representing the bearings: - From A, the bearing is N 31° E - From B, the bearing is S 53° E.

Step 2: Calculate Interior Angles

Use the bearing information to determine the angles between the lines and the shore:- Angle at A: Since bearing is N 31° E, the angle between the north line at A and the line to the ship is 31°.- Angle at B: Since bearing is S 53° E, the angle between the line to the shore at B (south line) and the line to the ship is 53°.- The angle between the two bearings at the ship is the supplementary angle: \[ \theta = 180° - (31° + 53°) = 96° \]

Step 3: Apply the Law of Sines

In the triangle formed by the points A, B, and the ship, use the Law of Sines to find the distances from A to the ship (AS) and from B to the ship (BS): \[ \frac{AS}{ ext{sin}(53°)} = \frac{BS}{ ext{sin}(31°)} = \frac{18}{ ext{sin}(96°)} \]

Step 4: Solve for AS and BS

Using the equations from Step 3:- Calculate AS: \[ AS = \frac{18 \times \text{sin}(53°)}{\text{sin}(96°)} \] - Calculate BS: \[ BS = \frac{18 \times \text{sin}(31°)}{\text{sin}(96°)} \] Compute the values using a calculator.

Step 5: Compute the Distances

Perform the calculations:- AS: \[ AS \approx \frac{18 \times 0.7986}{0.9950} \approx 14.430 \]- BS: \[ BS \approx \frac{18 \times 0.5150}{0.9950} \approx 9.318 \]Thus, the distances from the observation points to the ship are approximately 14.43 miles and 9.32 miles.

Key Concepts

Bearing Calculation

To find the position of a ship using bearings, you need to understand the concept of bearings itself. **Bearing** is a way to describe direction. It's measured in degrees from the north in a clockwise direction.

In the given problem, we are dealing with bearings such as N 31° E and S 53° E. Here's what they mean:

• N 31° E indicates that you measure 31 degrees east of due north.
• S 53° E shows that you measure 53 degrees east of due south.

Bearings help in locating the ship with respect to two known observation points on the shore.

An illustration would help, where you picture these bearings as angles drawn from two points on a straight shore toward the ship, allowing the visualization of the triangle that will be used in subsequent calculations.

Interior Angles

Once the bearings are known, the next step is to determine the interior angles of the triangle formed between the ship and the two observation points on the shore. You'll need these angles for distance calculations later using the Law of Sines.

In our problem:

• At point A, we have the bearing N 31° E. This means the angle between the north direction and the line connecting point A to the ship is 31°.
• At point B, the bearing is S 53° E. So, the angle between the south direction and the line from point B to the ship is 53°.
• The ship forms an angle between the lines from A and B, and this angle is the difference from 180°, because angles around a point add up to 360°. Hence, the angle \( \theta = 180° - (31° + 53°) = 96° \).

These angles allow us to apply the Law of Sines, an essential step for figuring out how far the ship is from each point.

Distance Calculation

Now, with the angles calculated, it's time to use the Law of Sines to find the distances from each observation point to the ship. The Law of Sines provides a relationship between the lengths of sides of a triangle and the sines of the opposite angles.

For our scenario, it's expressed as:

• For the distance from A to the ship, \( AS \): \[ AS = \frac{18 \times \sin(53°)}{\sin(96°)} \]
• For the distance from B to the ship, \( BS \): \[ BS = \frac{18 \times \sin(31°)}{\sin(96°)} \]

Computations using these formulas (and a calculator!) yield approximate distances:

• The distance from A to the ship is about 14.43 miles.
• The distance from B to the ship is roughly 9.32 miles.

These results help determine that the ship isn't equidistant from the two observation points, reflecting its distinct positioning along the shoreline.