Problem 80 - Precalculus

Question

If a plane takes off bearing \(\mathrm{N} 33^{\circ} \mathrm{W}\) and flies 6 miles and then makes a right \(\left(90^{\circ}\right)\) turn and flies 10 miles further, what bearing will the traffic controller use to locate the plane?

Step-by-Step Solution

Answer

The bearing to locate the plane is 327°.

Step 1: Convert the Initial Bearing

The plane takes off at a bearing of \( \mathrm{N} \, 33^{\circ} \, \mathrm{W} \). This means it is 33 degrees west of due north. To visualize this, consider starting at due north (0°) and rotating 33° towards the west direction.

Step 2: Apply the First Leg of the Journey

From the initial position, the plane flies 6 miles along the bearing \( \mathrm{N} \, 33^{\circ} \, \mathrm{W} \). This does not change the bearing; it simply takes the plane further along that path.

Step 3: Make a Right Turn

A right (90°) turn from the current direction will turn the plane from \( \mathrm{N} \, 33^{\circ} \, \mathrm{W} \) to a new bearing. Since it is a right turn, the plane will now face due south of its initial direction. To calculate this, add 90° to the current direction. Therefore, 33° west of north plus 90° leaves the plane facing 90° - 33° = 57° west, which represents \( \mathrm{W} \, 57^{\circ} \, \mathrm{S} \).

Step 4: Convert to Standard Bearing

In conventional terms, 'W 57° S' can be translated into standard bearing. Bearing is measured clockwise from north. From due north (0°) to due west (270°), you'll cover these steps: 1. From north to west is 270°, and since it's \( \mathrm{W} \, 57^{\circ} \, \mathrm{S} \), the calculation is 270° + 57° = 327° in a clockwise direction.

Key Concepts

Bearings

Bearings are a fantastic tool used in navigation to give precise instructions on direction. They help to describe the location or movement direction of an object, often essential for ships and aircraft. In navigation, bearings are typically measured in degrees, clockwise from the north. There are two main systems:

• Standard Bearings: Measured from 0° at north, moving clockwise up to 360°.
• Compass Bearings: Indicate directions using the points of a compass, such as N 33° W, meaning 33 degrees west of north.

Bearings ensure that everyone involved in navigation, like pilots and traffic controllers, can clearly communicate and understand directions. When a pilot receives the instruction "N 33° W," they know to head 33 degrees towards the west from due north, a simple yet powerful system for safe travel.

Right triangle

In trigonometry, a right triangle is a triangle with one angle measuring exactly 90 degrees. This special characteristic allows us to use specific relationships between the angles and sides to solve various problems.

• The side opposite the right angle, called the hypotenuse, is the longest side of the triangle.
• The other two sides are known as the legs. The relationships between these sides often involve the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
• Triangles and angles can also determine direction changes, like the 90° right turn the plane made in our task.

Understanding right triangles is key in many fields, including navigation, where it helps pilots determine precise paths and needed turns.

Angle measurement

Understanding angle measurement is crucial for making accurate calculations in navigation. Angles are typically measured in degrees, an ancient system that divides a circle into 360 parts. In the context of navigation:

• Bearings use angle measurements from due north going clockwise.
• For instance, an angle of N 33° W means the direction is offset 33° west from true north.
• Making turns requires adding or subtracting degrees. A right turn of 90° changes the plane's current heading, directly impacting how the direction is described with angles.

Mastering angle measurement ensures that directions are followed precisely, which is particularly important for safe travels over long distances.

Navigation

Navigation is the process of accurately ascertaining one's position and planning a route to a desired location. It heavily relies on the concepts of bearings and angle measurements to guide planes, ships, and vehicles safely. In navigation:

• Directions are communicated using bearings, providing a straightforward dialogue between navigators.
• Understanding the changes in direction, like the 90° turn described in plane navigation exercises, is essential to remain accurate in one's path.
• Technological tools such as GPS further aid navigation, but traditional methods remain fundamental in ensuring redundancy and safety.

Good navigation enables pilots and seafarers to choose the most efficient and safe paths, maximizing fuel efficiency and minimizing travel times while ensuring safety at all times.