Solve each problem. See Examples 1 and 2. Flying Distance The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.

Verified step by step guidance

1. First, understand the bearings given: Bearings are measured clockwise from the north direction. For example, N 64° W means starting from north, rotate 64° towards the west.
2. Calculate the distance from A to B using the plane's speed and time: Distance = Speed × Time, so \(\text{AB} = 350 \times 1.8\) miles.
3. Draw a diagram to visualize points A, B, and C with the given bearings: From A, point C is at N 64° W, and point B is at S 82° W. From B, point C is at N 26° E. This will help in setting up the triangle and angles.
4. Use the bearings to find the angles inside triangle ABC. Convert bearings into standard angles relative to a reference line (like the east or north axis) to find the angle between the paths AB and BC.
5. Apply the Law of Cosines in triangle ABC to find the distance BC: If you know two sides and the included angle, use \(\text{BC}^2 = \text{AB}^2 + \text{AC}^2 - 2 \times \text{AB} \times \text{AC} \times \cos(\theta)\), where \(\theta\) is the angle between sides AB and AC or the appropriate angle found from bearings.

Key Concepts

Understanding Bearings

Bearings describe direction relative to the cardinal points (N, S, E, W) using degrees. They are measured clockwise from north or south towards east or west, such as N 64° W meaning 64 degrees west of north. Correctly interpreting bearings is essential for plotting points and angles in navigation problems.

Using the Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is useful when two sides and the included angle are known or when all three sides are unknown. This law helps find unknown distances when bearings form angles between points.

Distance-Speed-Time Relationship

The formula distance = speed × time calculates the distance traveled when speed and time are known. In this problem, it helps find the distance from A to B, which is necessary to solve for the distance from B to C using trigonometric methods.