Law of Cosine Word Problem

Question

In aerial navigation, directions are given in degrees clockwise from north. Thus, east is 90°, south is 180°, and so on. A plane leaves South Bend for Buffalo, 400 miles away, intending to fly a straight course in the direction 70°. After flying 200 miles, the pilot realizes that an error has been made and that he has actually been flying in the direction 48°.
(a) At that time, how far is the plane from Buffalo? 
miles

(b)In what direction should the plane now go to reach Buffalo?

Answer 1

Hi Peyton!

We can solve this problem using the Law of Cosines and some trigonometry. Let's work through it step by step!

Given Information:

1. Intended distance from South Bend to Buffalo = 400 miles
2. Intended direction = 70° clockwise from north
3. Actual distance flown = 200 miles
4. Actual direction flown = 48° clockwise from north

The difference between the intended and actual directions is:

70° - 48° = 22°

This creates a triangle with two known sides (400 miles and 200 miles) and an included angle of 22°.

Part (a): How far is the plane from Buffalo?

We use the Law of Cosines:

c² = a² + b² - 2ab cos(C)

Substitute the known values:

c² = 400² + 200² - 2(400)(200)cos(22°)

c² = 160,000 + 40,000 - 160,000(0.92718)

c² = 200,000 - 148,349.42

c² = 51,650.58

Take the square root:

c = √51,650.58

Answer (a): Approximately 227.27 miles

The plane is about 227.27 miles from Buffalo after flying 200 miles in the wrong direction.

Part (b): In what direction should the plane now fly?

We need to find the new compass bearing that takes the plane directly to Buffalo.

First, use the Law of Sines to find the angle at the plane's current location.

sin(B) / 400 = sin(22°) / 227.27

sin(B) = 400 × sin(22°) / 227.27

sin(B) ≈ 0.6593

B ≈ 41.25°

However, the interior angle at the plane is obtuse because the 400-mile side is the longest side of the triangle.

Therefore:

B = 180° - 41.25°

B ≈ 138.75°

The angle between the plane's original direction of travel and the new course is:

180° - 138.75° = 41.25°

Since Buffalo is to the right of the plane's actual flight path, add this angle to the original heading:

New bearing = 48° + 41.25°

Answer (b): Approximately 89.25° clockwise from north

The plane should fly almost directly east to reach Buffalo.

Final Answers

(a) Distance from Buffalo: 227.27 miles

(b) New direction: 89.25° clockwise from north

The key idea is that the Law of Cosines helps us find the remaining distance, while the Law of Sines helps determine the corrected direction.

I hope this helps!