Problem 44 - Algebra and Trigonometry

Question

Airport B is 300 mi from airport A at a bearing \(\mathrm{N} 50^{\circ} \mathrm{E}\) (see the figure). A pilot wishing to fly from \(\mathrm{A}\) to \(\mathrm{B}\) mistakenly flies due east at 200 \(\mathrm{mi} / \mathrm{h}\) for 30 minutes, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? (b) What bearing should he head his plane in order to arrive at airport B?

Step-by-Step Solution

Answer

(a) 247.84 mi, (b) Bearing: N 68° E.

Step 1: Determine Pilot's Initial Eastward Displacement

The pilot flew due east for 30 minutes at 200 mi/h. First, convert 30 minutes to hours: 30 minutes = 0.5 hours. Now, calculate the distance flown: \( 200 \text{ mi/h} \times 0.5 \text{ h} = 100 \text{ mi} \). Thus, the pilot is 100 miles east of Airport A when he realizes his error.

Step 2: Establish Triangle and Apply Law of Cosines

We know the pilot has traveled 100 miles east and needs to reach Airport B which is 300 miles from Airport A at a bearing of \( N 50^{\circ} E \). We can form a triangle where one side is 100 miles, another is 300 miles, and the angle between them is \( 50^{\circ} \). Use the law of cosines to find the direct distance (c) from the pilot to Airport B:\[ c^2 = 100^2 + 300^2 - 2 \times 100 \times 300 \times \cos(50^{\circ}) \].

Step 3: Calculate Pilot's Distance from Destination

Compute the distance from step 2 using the law of cosines:\[ c^2 = 100^2 + 300^2 - 2 \times 100 \times 300 \times \cos(50^{\circ}) \]\[ c^2 = 10000 + 90000 - 60000 \times \cos(50^{\circ}) \]Calculate \( \cos(50^{\circ}) \approx 0.6428 \),\[ c^2 = 10000 + 90000 - 60000 \times 0.6428 \approx 10000 + 90000 - 38568 \approx 61432 \]\[ c \approx \sqrt{61432} \approx 247.84 \text{ mi} \].

Step 4: Calculate Correct Bearing Using Law of Sines

With the distances known, apply the law of sines to find the bearing angle from the due east line the pilot should take:\[ \frac{\sin(\theta)}{300} = \frac{\sin(50^{\circ})}{c} \]\( c \approx 247.84 \), substitute:\[ \frac{\sin(\theta)}{300} = \frac{\sin(50^{\circ})}{247.84} \]Calculate \( \sin(50^{\circ}) \approx 0.7660 \),\[ \sin(\theta) = \frac{300 \times 0.7660}{247.84} \approx 0.927 \]\( \theta \approx \sin^{-1}(0.927) \approx 68 \) degrees from the north direction. Therefore, the pilot should fly \( N 68^{\circ} E \) from his current position.

Key Concepts

Navigation

Navigation is the art and science of determining one's position and directing them to a predetermined destination. In this exercise, the pilot's initial task was to fly from Airport A to Airport B, which required both precise navigation and an understanding of units such as 'miles' and 'hours'. Due to an error, the pilot flew eastward for 100 miles before realizing the mistake. Dealing with longer distances sometimes involves working with units like nautical miles or kilometers. However, in the context of this problem, both distances and speed are provided in statute miles and miles per hour, which are commonly used in terrestrial navigation. Understanding how navigation works, especially when requiring corrective adjustments, often involves geometric perspectives:

• Knowing how to measure angles and distances.
• Understanding direction in terms of compass points and bearings.
• Applying trigonometric laws to redirect accurately.

The pilot's task involved redirecting the flight path to head directly to Airport B after the error. This illustrates the importance of being able to readjust a course when errors are made, ensuring accurate and safe navigation.

Bearings

Bearings are a critical aspect of navigation, providing a way to describe direction using angles. In air navigation, a bearing is typically given as an angle measured clockwise from the north. The problem describes the intended bearing from Airport A to B as \( N 50^{\circ} E \), meaning the direction is 50 degrees clockwise from true north.Bearings are often expressed in one of several formats:

• Three-digit bearing: An angle ranging from 000 to 359 degrees.
• Compass bearing: A direction given as a quadrant bearing, e.g., \( N 50^{\circ} E \).

In correcting his course, the pilot needed to calculate a new bearing using trigonometrical laws. The solution required determining the sine of an angle using distances and angles given, bringing about the correct bearing \( N 68^{\circ} E \) from the pilot's new position.Understanding how to compute bearings ensures pilots can adjust their path using the shortest and safest route, avoiding unnecessary fuel consumption and time loss.

Triangles

Triangles are fundamental in solving navigation problems. The problem encountered by the pilot resulted in the creation of a triangle, defined by:

• The initial incorrect 100-mile flight due east.
• The 300-mile flight line that should have been taken originally.
• The line from the pilot's mistaken position to the desired destination Airport B.

The geometry of the triangle comes into play when applying trigonometric principles like the Law of Cosines and Law of Sines. In this situation, the Law of Cosines helped determine the straight-line distance (\(c\)) from the pilot's erroneous position to Airport B: \[ c^2 = 100^2 + 300^2 - 2 \times 100 \times 300 \times \cos(50^{\circ}) \]After determining the distance, the Law of Sines was used to find the new correct bearing: \[ \frac{\sin(\theta)}{300} = \frac{\sin(50^{\circ})}{c} \]Triangles allow us to model physical problems mathematically. By understanding the properties of triangles and applying the correct laws, one can solve complex navigation challenges with precision.