Question

A DC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is \(\mathrm{N} 40^{\circ} \mathrm{E} .\) After flying for \(\frac{1}{2}\) mile, the pilot requests permission to turn \(90^{\circ}\) and head toward the southeast. The permission is granted. After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

Short answer

\( \text{S} 50^{\text{∘}} \text{E} \)

Step-by-step solution

- Understand the Initial Bearing

The initial bearing of the aircraft is \(\text{N} 40^{\text{∘}} \text{E} \). This means the plane is heading 40 degrees towards the east from the north direction.

- Calculate the New Direction After the Turn

The aircraft turns \(90^{\text{∘}}\) to head southeast. Southeast is equivalent to \(135^{\text{∘}}\) in standard bearing (clockwise from north). From \( \text{N} 40^{\text{∘}} \text{E} \), turning \(90^{\text{∘}}\) results in \( \text{S} 50^{\text{∘}} \text{E} \).

- Determine the Final Bearing

After adjusting the path in step 2, the aircraft moves towards \( \text{S} 50^{\text{∘}} \text{E} \). Hence, the final bearing used by the control tower to locate the aircraft is \( \text{S} 50^{\text{∘}} \text{E} \).

Key concepts

initial bearing

The initial bearing is the direction in which the aircraft begins its journey. It is crucial to understand this as it sets the reference point for any subsequent directional changes.

In this problem, the aircraft departs with an initial bearing of \(\text{N } 40^{\text{∘}} \text{ E} \). This means the plane is heading 40 degrees towards the east from the north. Visualize a compass: starting at the North (0 degrees), we move 40 degrees clockwise to get the initial bearing direction.

Think of a clock: if 12 o'clock is north, then 1:20 would point you to the direction the aircraft initially follows.

directional turn

In aviation, pilots often need to change directions to follow air traffic control instructions or alter their flight paths.

Here, the aircraft makes a significant change in direction by turning 90 degrees. Understanding the plane's movement after this turn requires knowledge of both the initial bearing and the aviation navigation system.

After the \(90^{\text{∘}}\) turn, the aircraft heads southeast. In standard aviation terms, this is equivalent to \(135^{\text{∘}}\) when measured clockwise from north. Remember, southeast can also be broken down as halfway between south and east, giving us a 45-degree addition to directly south, leading to the final direction.

final bearing

The final bearing is the direction the plane ends up heading after all turns and adjustments.

In this exercise, after the plane turns 90 degrees from its initial bearing, it heads towards the direction known as \( \text{S } 50^{\text{∘}} \text{ E} \). We achieve this angle by first moving 40 degrees from north to east, then adding an additional 90 degrees. This calculation results in \(50^{\text{∘}}\) south from east, which translates to \( \text{S } 50^{\text{∘}} \text{ E} \).

This final bearing is what the control tower will use to locate the aircraft, ensuring safe and accurate tracking of its position.