A plane leaves South Bend for Buffalo, 400 miles away,...

A plane leaves South Bend for Buffalo, 400 miles away, intending to fly a straight course in the direction $70^{\circ}$ (aerial navigation is explained in Exercise 41 of Section 8.2 ). After flying 180 miles, the pilot realizes that an error has been made and that he has actually been flying in the direction $55^{\circ}$ (a) At that time, how far is the plane from Buffalo? (b) In what direction should the plane now go to reach Buffalo?

A plane leaves South Bend for Buffalo, 400 miles away, intending to fly a straight course in the direction $70^{\circ}$ (aerial navigation is explained in Exercise 41 of Section 8.2 ). After flying 180 miles, the pilot realizes that an error has been made and that he has actually been flying in the direction $55^{\circ}$
(a) At that time, how far is the plane from Buffalo?
(b) In what direction should the plane now go to reach Buffalo?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Navigational Bearings

This concept involves understanding how angles and directions are measured in navigation, typically relative to a fixed point such as true north. It is key to translate between compass directions and standard mathematical angles in order to plot courses or identify deviations from a planned route.

Vector Displacement and Addition

In navigation problems, the overall movement of a vehicle like an aircraft is represented by displacement vectors. By breaking down and then combining these vectors—especially when a course adjustment is made—one can determine the resultant position relative to the intended destination.

Law of Cosines

The Law of Cosines is a fundamental theorem used to relate the lengths of sides in any triangle with the cosine of one of its angles. In navigational contexts, it is used to calculate unknown distances when two sides and the included angle (or alternative combinations) are known, such as when finding the distance from the current position to the destination.

Law of Sines

The Law of Sines connects the ratios of the lengths of sides to the sines of the opposite angles in a triangle. This is particularly useful for determining unknown angles or directions when some sides and angles are known, which is essential for determining a corrected course in navigation.

Transcript

Please give Ace some feedback