Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Aviation An airplane flying at 600 miles per hour has a bearing of \(52^{\circ}\). After flying for 1.5 hours, how far north and how far east has the plane traveled from its point of departure?

Step-by-Step Solution

Answer

The plane would have travelled approximately 543.52 miles east and 705.21 miles north from its point of departure.

Step 1: Calculate the total distance travelled

To calculate the total distance travelled by the plane, multiply the rate by time. Here, the rate is 600 miles per hour, and the time is 1.5 hours. So, the total distance is \(600 \times 1.5 = 900\) miles.

Step 2: Calculate the distance travelled towards east

The cosine of the angle is defined as the ratio of the adjacent side (East) to the hypotenuse (Total distance). In this case, the distance travelled towards East can be found by multiplying the total distance by the cosine of the angle. So, East distance = Total distance \( \times \cos(52^{\circ}) = 900 \times \cos(52^{\circ})\).

Step 3: Calculate the distance travelled towards north

The sine of an angle is defined as the ratio of the opposite side (North) to the hypotenuse (Total Distance). In this case, the distance travelled towards North can be found by multiplying the total distance by the sine of the angle. So, North distance = Total Distance \( \times \sin(52^{\circ}) = 900 \times \sin(52^{\circ})\).

Key Concepts

Bearing in Navigation

Bearings are an essential part of navigation, especially in aviation. A bearing is a direction or path along which something moves or along which it lies. It is usually measured in degrees from the north in a clockwise direction. This is why the compass rose is marked from 0° to 360°. Understanding bearings is crucial for pilots as they determine the course to fly from one location to another.

• An angle like a 52° bearing means the direction of travel is tilted 52 degrees clockwise from north.
• Bearings help to simplify the description of direction and keep the course precise during navigation.

When working with bearings and direction, it’s essential to combine this knowledge with the principles of trigonometry, to accurately calculate distances covered in different directions, which leads us to navigation calculations.

Distance Calculation

Distance calculation is a critical skill in navigation, allowing for effective travel planning. In the context of an airplane flying with a set speed and time, calculating the traveled distance is straightforward.

• First, the basic formula for distance calculation involves multiplying the speed by the time spent traveling. In our example, the plane flying at 600 mph for 1.5 hours results in a total journey of \(600 \times 1.5 = 900\) miles.
• This total distance is then resolved into components that align with the cardinal directions, such as north and east, which involves further calculations using trigonometric functions.

These calculations form the basis for more complex navigational challenges which are often encountered in aviation and marine travels.

Right Triangle Trigonometry

Right triangle trigonometry is a powerful tool that simplifies solving problems involving angles and distances. When you understand the relationships between the angles and sides of a right triangle, you can efficiently solve navigation problems.

• The concepts of sine and cosine in trigonometry are particularly useful here. For a given angle in a right triangle, the sine function calculates the length of the opposite side relative to the hypotenuse, while the cosine function finds the length of the adjacent side relative to the hypotenuse.
• In the problem provided, we calculated the northward and eastward distances using the sine and cosine of the bearing angle (52°). The eastward distance was determined using \(900 \times \cos(52^{\circ})\), while the northward distance was found using \(900 \times \sin(52^{\circ})\).

These versatile trigonometric principles are extremely useful in real-life applications, such as calculating flight paths and ensuring precise navigation.