Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Radio direction finders are set up at two points \(A\) and \(B\), which are 2.50 miles apart on an east-west line. From \(A\), it is found that the bearing of a signal from a radio transmitter is \(\mathrm{N} 36^{\circ} 20^{\prime} \mathrm{E}\). and the bearing of the same signal from \(B\) is \(N 53^{\circ} 40^{\prime} \mathrm{W}\) Find the distance of the transmitter from \(B\).

Step-by-Step Solution

Answer

The distance from point B to the transmitter is approximately 1.48 miles.

Step 1: Interpret Bearings

The bearing from point \( A \) is \( N 36^{\circ} 20' E \), and from point \( B \) it is \( N 53^{\circ} 40' W \). This indicates the direction of the radio signal from each station along the compass.

Step 2: Define Angles in the Triangle

We will place points \( A \) and \( B \) on a coordinate plane where the distance \( AB = 2.50 \) miles. The angle \( \angle BAC \) is equal to \( 36^{\circ} 20' \). The angle \( \angle ABC \) is equal to \( 53^{\circ} 40' \). Use these angles to understand the triangle.

Step 3: Calculate the Remaining Angle

Since the sum of the angles in a triangle must be \( 180^{\circ} \), calculate \( \angle ACB \) using: \[ \angle ACB = 180^{\circ} - (36^{\circ} 20' + 53^{\circ} 40') \]. This results in \( \angle ACB = 90^{\circ} \).

Step 4: Apply the Law of Sines

Use the Law of Sines to find the distance from point \( B \) to the transmitter \( C \): \[ \frac{BC}{\sin \angle BAC} = \frac{AB}{\sin \angle ACB} \] Substitute the known values: \[ \frac{BC}{\sin 36^{\circ} 20'} = \frac{2.50}{\sin 90^{\circ}} \].

Step 5: Solve for BC

Since \( \sin 90^{\circ} = 1 \), the equation simplifies to \[ BC = 2.50 \cdot \sin 36^{\circ} 20' \]. Calculate the sine value and multiply to find \( BC \).

Step 6: Calculate Final Distance

Compute \( \sin 36^{\circ} 20' \) using a calculator (\( \sin 36^{\circ} 20' \approx 0.5927\)), then \( BC = 2.50 \times 0.5927 \approx 1.48175 \).

Key Concepts

Law of Sines

When dealing with non-right triangles, such as the one formed between the radio finders and the signal transmitter, the Law of Sines is a powerful tool. It allows us to find unknown side lengths or angles in a triangle if we know some combinations of them. It is expressed as:

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

This formula shows the relationship between the side lengths and their opposing angles. In simpler terms, the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides in the triangle.
In the provided exercise, we used it to find the distance from point \( B \) to the transmitter \( C \) by substituting the known sides and angles into the equation. This usage of trigonometry proves that as long as two angles and a side are known, you can find other unknown sides or angles in a triangle.

Bearing Calculations

Bearing calculations are essential in navigation and positioning. A bearing is a direction or path along a compass line, expressed in degrees. Bearings are often used to describe the direction between two points. In this exercise, the bearings were expressed with compass directions, like "N 36° 20' E".
Here’s how they work:

• The direction starts from the north line at \(0^{\circ} \), moving clockwise.
• "N 36° 20' E" means 36° 20' towards the east from due north.
• Similarly, "N 53° 40' W" points to 53° 40' towards the west from due north.

This way of describing directions is vital in fields like aviation, marine navigation, and surveying. In this problem, bearing calculations helped to geometrically construct the situation and determine angles crucial for the application of the Law of Sines.

Triangulation

Triangulation is the method of determining the location of a point by forming triangles to it from known points. This technique is widely used in various fields such as navigation, surveying, and even in GPS technology.
In practical situations like in the exercise, triangulation involves:

• Measuring angles from two known positions (radio towers in the problem).
• Knowing the distance between these positions (distance between points \( A\) and \( B \)).
• Using trigonometry or geometric methods to determine unknown distances, like the distance to the transmitter.

This triangulation method allowed us to calculate the distance of the transmitter from point \( B \). The main trick is to use angle measures derived from bearings, in order to determine the exact location.

Angle Measurement

Angle measurement is crucial when dealing with triangles and especially when solving problems involving triangulation or bearings. Angles are typically measured in degrees (°) and minutes ('). Each degree is divided into 60 minutes, which provides more precision in measurement.
In the provided exercise:

• The angles from bearings "N 36° 20' E" and "N 53° 40' W" were crucial for forming the triangle.
• Knowing two angles of the triangle, the third angle was deduced using the rule that the sum of angles in a triangle is always 180°.
• After converting these bearing angles into mathematically usable angles in the triangle, we could solve using the Law of Sines.

Understanding angle measurement helps in applying various trigonometric formulas and methods correctly, which is essential for solving real-world problems.