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?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Bearing and Direction in Navigation

Vector Addition and Components

Trigonometric Functions and Resultant Magnitude