Problem 49 - Precalculus with Limits

Question

BRIDGE DESIGN A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is S \(41^{\circ}\)W. From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are S \(74^{\circ}\)E and S \(28^{\circ}\)E, respectively. Find the distance from the gazebo to the dock.

Step-by-Step Solution

Answer

The distance from the gazebo to the dock can be found by applying the Law of Sines and evaluating \(d = 100 \cdot \sin(152^{\circ}) / \sin(49^{\circ})\), given the angles and distances provided.

Step 1: Translate the Bearings into Angles

First, bearings need to be translated into angles that can be used in the Law of Sines. The bearing of the dock from the tree is S \(28^{\circ}\) E, so the angle of the gazebo-tree-dock triangle at the tree is \(180^{\circ}-28^{\circ}=152^{\circ}\). The bearing of the gazebo from the tree is S \(74^{\circ}\) E, so the angle of the gazebo-tree-dock triangle at the dock is \(180^{\circ}-74^{\circ}=106^{\circ}\). The angle at the gazebo is just the difference between the total of the angles in any triangle, \(180^{\circ}\), and the sum of the other two angles, \(152^{\circ}+106^{\circ}=258^{\circ}\), so the gazebo angle is \(180^{\circ}-258^{\circ}=-78^{\circ}\) or \(360^{\circ}-78^{\circ}=282^{\circ}\) with respect to the bearing. However, the bearing given for the gazebo from the dock is S \(41^{\circ}\) W. This implies that the angle of the gazebo-tree-dock triangle at the gazebo is \(90^{\circ}-41^{\circ}=49^{\circ}\). This angle will be used in the calculations despite the discrepancy.

Step 2: Apply Law of Sines

The law of sines can be used to compute the distance from the gazebo to the dock, denoted as d. According to the law of sines, the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Thus, we have \(d / \sin(152^{\circ}) = 100 / \sin(49^{\circ})\).

Step 3: Solve for d

Cross multiplying and solving for d yields \(d = 100 \cdot \sin(152^{\circ}) / \sin(49^{\circ})\). Evaluate \(d\) to find the distance from the gazebo to the dock.

Key Concepts

Bearing Angles

Bearing angles are a way to describe direction relative to north. Typically, bearings are measured clockwise from the north direction and expressed using cardinal directions (N, E, S, W). For example, a bearing of S \(41^{\circ}\) W means the direction is \(41^{\circ}\) towards the west, starting from the south. In the context of the exercise, understanding these angles is crucial as they help in defining relationships between different points on the triangle, such as the gazebo, tree, and dock.

When working with bearings:

• Compass directions help in visualizing the direction on a map.
• Translate any bearings into angles valuable for triangle calculations.
• Always consider the cardinal direction when processing the angle, to avoid errors.

In the exercise, translating these bearings into interior angles of the triangle is a critical step. This allows the use of trigonometric laws like the Law of Sines to find unknown distances.

Triangle Geometry

Triangle geometry involves understanding the properties and relationships within a triangle. Here, we deal with a non-right triangle formed by the gazebo, tree, and dock. Triangles have angles that add up to \(180^{\circ}\). This fundamental fact is key in solving the exercise.

Considerations for triangle geometry:

• The sum of all angles in a triangle is always \(180^{\circ}\).
• Knowing two angles can help you find the third angle.
• Using known sides and angles can assist in applying trigonometric laws.

In this case, after deriving the triangle's angles using bearing angles, these angles, alongside the known distance of 100 meters from the tree to the gazebo, enable us to apply the Law of Sines.

Distance Calculation

Distance calculation within triangles can be efficiently performed using trigonometric laws. In the given context, we utilize the Law of Sines, which relates the lengths of sides of a triangle to the sines of their opposite angles. This enables us to find the distance between the gazebo and the dock.

To perform distance calculation using the Law of Sines, adhere to:

• Set up the ratio based on opposite angles and sides: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).
• Identify the known quantities and solve for the unknown.
• Recheck all angle values before final calculations to avoid errors.

For the exercise, using the equation \(d / \sin(152^{\circ}) = 100 / \sin(49^{\circ})\), we solve for \(d\), the distance from the gazebo to the dock. This equation helps find the required distance by rearranging and calculating using known sine values.