Problem 48 - Precalculus

Question

There are two stations along a shoreline and the distance along the beach between the two stations is 200 feet. The angles between the baseline (beach) and the line of sight to the island are \(30^{\circ}\) and \(50^{\circ} .\) Find the shortest distance from the beach to the island. Round to the nearest foot.

Step-by-Step Solution

Answer

The shortest distance from the beach to the island is 102 feet.

Step 1: Setup the Problem

First, visualize the setup with the island forming a triangle with the two stations on the beach. Let's label the two stations as points A and B, and the island as point C. The distance between the stations A and B along the shoreline is 200 feet. The angle at point A (denoted as \( \angle CAB \)) is 30° and at point B (denoted as \( \angle ABC \)) is 50°.

Step 2: Use of the Law of Sines

To find the distance from the beach to the island, we need the length of line segment BC (which represents the shortest distance from the island to the beach, forming a perpendicular from C to line AB). According to the Law of Sines, \[\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle ABC)} = \frac{BC}{\sin(\angle BAC)}.\]Since we need \( BC \), the important part of the equation is:\[\frac{AB}{\sin(\angle ACB)} = \frac{BC}{\sin(\angle BAC)}.\]

Step 3: Calculate Unknown Angle

In any triangle, the sum of angles is 180°. So, calculate \( \angle ACB \) using the expression:\[\angle ACB = 180° - (30° + 50°) = 100°.\]

Step 4: Calculate BC Using Law of Sines

Now that we have all the angles, use the Law of Sines to find BC. Substitute the known values:\[\frac{200}{\sin(100°)} = \frac{BC}{\sin(30°)}\]Solving for \( BC \):\[BC = \frac{200 \times \sin(30°)}{\sin(100°)}.\]Plug in the values:\[BC = \frac{200 \times 0.5}{0.9848} \approx 101.5 \, \text{feet}.\]

Step 5: Round the Result

Round the computed distance BC to the nearest foot:\[\text{Shortest distance} = 102 \, \text{feet}.\]

Key Concepts

Triangle Geometry

Understanding the geometry of triangles is essential in solving problems like finding distances or angles. A triangle consists of three sides and three angles, adding up to a total of 180 degrees. In our problem, the points form a triangle along a shoreline with two stations and an island. This triangle is named as triangle ABC, with stations at the shoreline being points A and B, and the island at point C. Knowing the properties of triangles allows us to relate different sides and angles using various mathematical relationships.

In our specific problem setup:

• Points A and B are 200 feet apart along the beach.
• Angle at point A, denoted as \( \angle CAB \), measures 30 degrees.
• Angle at point B, \( \angle ABC \), is 50 degrees.

These measurements help form a visual understanding of the triangle, setting the stage for using trigonometry to find the shortest distance from the island to the beach.

Trigonometric Functions

Trigonometric functions are key in solving triangle problems, especially those involving angles and distances. The Law of Sines is a trigonometric rule that provides a relationship between the sides and angles of a triangle. This law is helpful when at least one side length and two angles of a triangle are known, which is a common scenario in problems requiring calculations of unknown distances.

The Law of Sines is expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

In this formula, \( a, b, \) and \( c \) represent the lengths of the triangle's sides and \( A, B, \) and \( C \) are the angles opposite those sides. In our problem, applying the Law of Sines becomes handy once we know some of the angles and side AB, allowing us to calculate the shortest distance from the island to the beach.

Angle Calculation

Calculating the angles is a fundamental step in applying the Law of Sines or any trigonometric function to solve a problem. For a triangle, the sum of all interior angles must equal 180 degrees. Given two of the angles in the triangle are known, the third can easily be determined. In our scenario, given:

• \( \angle CAB = 30^{\circ} \)
• \( \angle ABC = 50^{\circ} \)

We calculate the third angle \( \angle ACB \) as follows:

• \( \angle ACB = 180^{\circ} - (30^{\circ} + 50^{\circ}) \)
• \( \angle ACB = 100^{\circ} \)

Armed with this data, we substitute into the Law of Sines to find the length of BC, the shortest distance from the island to the beach. Calculating each angle precisely ensures accurate application of trigonometric principles in solving for unknown elements.