Problem 95 - A Graphical Approach to Precalculus with Limits

Question

A ship leaves port on a bearing of \(34.0^{\circ}\) and travels 10.4 miles. The ship then turns due east and travels 4.6 miles. How far is the ship from port, and what is its bearing from port?

Step-by-Step Solution

Answer

The ship is 14.2 miles away, and the bearing is approximately 22 degrees east of north.

Step 1: Breakdown the Problem into Two Segments

The ship travels two segments: the first is 10.4 miles on a bearing of \(34.0^{\circ}\) north of east, and then the ship turns due east to travel 4.6 miles. Visualize these segments as parts of a right-angled triangle where the first segment forms the adjacent side to the angle \(34.0^{\circ}\).

Step 2: Calculate the Eastward and Northward Distances for Segment 1

Using trigonometry, determine the eastward (adjacent) and northward (opposite) distances for the first segment. Use:- Eastward: \( 10.4 \cos(34.0^{\circ}) \)- Northward: \( 10.4 \sin(34.0^{\circ}) \)

Step 3: Add the Eastward Distance from Segment 2

The second segment contributes only to the eastward distance, so add 4.6 miles directly to the eastward distance obtained from the first segment. Hence, the total eastward distance is \( 10.4 \cos(34.0^{\circ}) + 4.6 \).

Step 4: Use Pythagorean Theorem to Find Total Distance from Port

Now, combine the total eastward and northward distances to find the straight-line (hypotenuse) distance from the port using the Pythagorean theorem:\[ \text{Distance} = \sqrt{(\text{Total Eastward})^2 + (\text{Total Northward})^2} \]

Step 5: Calculate the Bearing from the Port

To find the bearing, use trigonometry:\[ \theta = \arctan \left( \frac{\text{Total Northward}}{\text{Total Eastward}} \right) \]Convert \(\theta\) to a compass bearing by adjusting it relative to north.

Key Concepts

Bearing Calculation

In navigation, bearing calculations are essential to determine direction. A bearing is the angle measured in degrees, typically from the north in a clockwise direction. It is used to specify the direction of travel for ships, airplanes, or even land navigation. In our problem, the ship initially departs at a bearing of \(34.0^{\circ}\). This means that the path is angled \(34.0^{\circ}\) from the north towards the east.
Although bearings are given in reference to the north, by mentally adjusting the navigation based on compass directions, bearings provide a universal method to guide journeys. Calculating a bearing involves understanding both where an object starts and how it moves based on angles.

• Bearings are always given as three digits for consistency, such as \(034.0^{\circ}\) for our scenario.
• To compute a new bearing after movement, use trigonometric functions to ascertain exact angles.

After moving north at first, the ship's next move is due east, which requires recalculating its bearing to understand its final position in relation to the starting point.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry and is crucial for determining the distance between two points in a plane when limited information is available. The theorem states that for any right-angled triangle, the square of the hypotenuse (the longest side) is the sum of the squares of the other two sides.
To use this theorem in our scenario:

• Identify the total eastward and total northward distances as two perpendicular sides of a right triangle.
• Apply the formula \[\text{Hypotenuse} = \sqrt{(\text{Eastward})^2 + (\text{Northward})^2}\]The resulting hypotenuse gives you the straight-line distance from the port.

This calculation allows the navigator to determine how far the ship is from the original departure point. This straightforward application of the Pythagorean Theorem helps simplify complex navigation problems by reducing them into comprehensible numbers.

Right-angled Triangle

In practical problems like this one, breaking the scenario into a right-angled triangle helps solve for unknown sides and angles. Recognizing when a right-angled triangle can be formed allows you to employ the Pythagorean Theorem and trigonometric functions effectively.
Key characteristics of right-angled triangles:

• Has one \(90^{\circ}\) angle.
• The side opposite the right angle is the hypotenuse, which is also the longest side.
• The other sides are referred to as the adjacent (next to the angle of interest) and opposite sides.

In the given exercise, the journey of the ship forms such a triangle between its original position, its destination, and its position after turning due east. Leveraging this setup, one can efficiently calculate the ship's straight-line distance from the port, as well as its new bearing.

Trigonometric Functions

Trigonometric functions are essential tools in solving real-world right-angled triangle problems, like the ship's journey. These functions relate the angles of a triangle to the lengths of its sides and are often used to find unknown parts of a triangle.
The primary trigonometric functions are:

• Sine (sin): relates an angle to the ratio of the opposite side to the hypotenuse.
• Cosine (cos): relates an angle to the ratio of the adjacent side to the hypotenuse.
• Tangent (tan): relates an angle to the ratio of the opposite side to the adjacent side.

For example, when calculating how far the ship moves eastward or northward from its bearing, we use cosine and sine:- Eastward distance (adjacent): \(10.4 \cos(34.0^{\circ})\)
- Northward distance (opposite): \(10.4 \sin(34.0^{\circ})\)
Trigonometry also helps in recalculating the ship’s final bearing using the arctan function, which relates the opposite and adjacent sides. This understanding ensures navigational precision over complex paths.