A boat is sailing due east parallel to the shoreline at a...

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

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

Key Concepts

Bearings

Bearings are a way to express direction using a combination of cardinal points and an angular measurement. In navigation, a bearing such as 'S 70° E' indicates that from the south, the direction deviates 70 degrees toward the east. Understanding bearings is crucial for converting navigational information into angles used in mathematical and trigonometric calculations.

Right Triangle Trigonometry

Right triangle trigonometry involves relationships between the angles and sides of a right triangle, often using the tangent, sine, or cosine functions. This concept is essential in problems where distances must be calculated based on observed angles, as one can relate the opposite side and adjacent side with the tangent of the given angle, allowing the determination of unknown distances.

Relative Motion and Displacement

Relative motion refers to the movement of an object in relation to a fixed point or another object, considering both speed and direction. In navigational problems, such as a boat moving along a straight path, the concepts of speed, time, and displacement are used together to determine how far the boat travels over a given interval, which is then integrated into geometric or trigonometric models to solve for other unknowns.

Conversion of Time to Distance

Converting time to distance is a fundamental aspect of motion analysis where the distance traveled is obtained by multiplying speed by the time elapsed. This conversion is critical when a moving object’s change in position over a known time interval is used alongside directional information to solve for unknown distances through methods like trigonometry.

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