Question

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Accepted Answer

Draw a diagram to visualize the problem: mark the positions of Mark and Lisa on the same bank, the river flowing north, and the tree on the opposite bank. Label the bearings from Mark and Lisa to the tree as 115.45° and 45.47°, respectively. Understand that bearings are measured clockwise from the north direction. Convert these bearings into angles relative to the river bank (which runs north-south) to help form triangles for calculation. Use the given distance between Mark and Lisa (428.3 m) as one side of a triangle formed by their positions and the tree. The bearings give you the directions from each observer to the tree, allowing you to find the angles inside this triangle. Apply the Law of Sines to the triangle formed by Mark, Lisa, and the tree. The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$, where $$a$$, $$b$$, and $$c$$ are sides opposite angles $$A$$, $$B$$, and $$C$$ respectively. Use this to find the distance from Mark (or Lisa) directly across the river to the tree. Since the river banks are parallel and the tree is on the opposite bank, the distance across the river corresponds to the perpendicular distance from Mark (or Lisa) to the tree along the east-west direction. Use trigonometric relationships (such as sine or cosine of the relevant angle) to calculate this perpendicular distance.

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

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주요 개념

Bearings and Directional Angles

Trigonometric Applications in Triangles

Properties of Parallel Lines and Transversals