Question

A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

Accepted Answer

Draw a diagram to visualize the problem: start at the port (point O), draw a line representing the first leg of the journey at a bearing of 110.0° from north, and mark the endpoint after traveling 8.8 miles (point A). Then, from point A, draw a line due west (270°) for 2.4 miles to point B, the liner's final position. Resolve the first leg of the journey into its north-south and east-west components using trigonometry. Since the bearing is measured clockwise from north, calculate the east (x) and north (y) components as follows:  
$$x_{A} = 8.8 	imes \sin(110.0^\circ)$$  
$$y_{A} = 8.8 	imes \cos(110.0^\circ)$$ Calculate the coordinates of the final position (point B) relative to the port by adjusting the east-west component for the second leg:  
$$x_{B} = x_{A} - 2.4 ($$since the liner travels west, subtract 2.4 miles from the east component)  
$$y_{B} = y_{A} ($$north-south component remains the same as the second leg is due west). Find the straight-line distance from the port to the liner's final position using the Pythagorean theorem:  
$$distance = \sqrt{(x_{B}$$)^{2}$$ + (y_{B}$$)^{2}$$}$$ Determine the bearing from the port to the liner's final position by calculating the angle relative to north:  
$$	heta = \arctan\left(\frac{|x_{B}|}{|y_{B}|}ight)$$  
Adjust the angle based on the quadrant where point B lies to express the bearing as degrees clockwise from north.

A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

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Key Concepts

Bearing and Direction in Navigation

Vector Addition and Components

Trigonometric Functions and Resultant Magnitude