Distance A boat is sailing due east parallel to the...

Distance A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time bearing to a lighthouse is $\mathrm{S} 70^{\circ} \mathrm{E}$ , and 15 minutes later the bearing is $\mathrm{S} 63^{\circ} \mathrm{E}$ (see figure). The lighthouse is located at the shoreline. Find the distance from the boat to the shoreline.

Key Concepts

Bearing and Navigation

Bearing is a method of describing direction that usually indicates the angle from a known reference point, such as north or south, measured clockwise or anticlockwise. This concept is crucial in navigation for understanding and relating measurements made from fixed landmarks, such as a lighthouse in this context.

Trigonometric Functions in Right Triangles

Trigonometric functions like sine, cosine, and tangent are used to relate angles to side lengths in right-angled triangles. In problems involving distances and angles, these functions allow for the translation of angular information (obtained from bearings) into linear distances, which is crucial for solving the geometry of the situation.

Relative Motion and Displacement

Relative motion describes how the position of one object changes with respect to another over time. In contexts where an object travels at a known constant speed for a known time interval, the calculated displacement can be used in conjunction with trigonometric relationships to determine distances and positions relative to fixed points.

Geometric Relationships in Navigation Problems

Navigation problems often involve constructing geometric figures, typically triangles, using the observed angles and known distances to fixed landmarks. Understanding these geometric relationships is key to setting up equations that allow the determination of unknown distances, such as the perpendicular distance from a moving object to a fixed line, like a shoreline.

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