Problem 47 - Algebra and Trigonometry

Question

Two fire-lookout stations are 10 miles apart, with station \(\mathrm{B}\) directly east of station A. Both stations spot a fire. The bearing of the fire from station \(A\) is \(N 25^{\circ} \mathrm{E}\) and the bearing of the fire from station \(\mathrm{B}\) is \(\mathrm{N} 56^{\circ} \mathrm{W}\). How far, to the nearest tenth of a mile, is the fire from each lookout station?

Step-by-Step Solution

Answer

The fire is approximately 14.4 miles away from station A and 11.2 miles away from station B.

Step 1: Draw the Diagram

Sketch a triangle with vertices named \(A\), \(B\) and \(F\), which represent the two lookout stations and the fire spot respectively. Label the edge AB as 10 miles. Mark the angles given in the problem, the bearing \(N 25^{\circ} E\) from station \(A\) and bearing \(N 56^{\circ} W\) from station \(B\). Since the angle at \(B\) is 90 degrees (being directly east), then the angle \(ABF\) is \(65^{\circ}\) (90 - 25) and angle \(BAF\) is \(34^{\circ}\) (90 - 56).

Step 2: Apply the Law of Sines

We now apply the Law of Sines to find the distance from station \(A\) to the fire \(AF\). The Law of Sines states that 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. So, \(AF / sin(ABF) = AB / sin(BAF)\). Substituting the known values, we get \(AF = (AB * sin(ABF)) / sin(BAF) = (10 * sin(65)) / sin(34)\).

Step 3: Solve for AF

After calculating, we get the distance of fire from station A, \(AF = 14.4\) approximately.

Step 4: Apply the Law of Cosines

To get the distance from the other station to the fire, \(BF\), we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle with the cosine of one of its angles. As we already know the length of \(AB\), \(AF\) and the angle \(ABF\), we can use the formula: \(BF^2 = AF^2 + AB^2 - 2 * AF * AB * cos(ABF)\).

Step 5: Solve for BF

By substituting the known values, we calculate the value of \(BF = \sqrt{14.4^2 + 10^2 - 2 * 14.4 * 10 * cos(65)} = 11.2\) approximately.

Key Concepts

Understanding Bearings

In navigation and land surveying, a bearing is a way to describe direction. It indicates the angle measured in degrees from the north direction. For instance, if a direction is marked as \(N 25^\circ E\), it means you start facing north and then turn 25 degrees towards the east. Similarly, \(N 56^\circ W\) indicates turning 56 degrees toward the west from north.

Understanding bearings is crucial when reading a compass or planning navigation routes, as it provides a precise direction which can be used practically in finding locations or defining movement paths.

Exploring the Law of Sines

The Law of Sines is a powerful tool to solve triangles, especially non-right triangles. It states that the ratio of a side length to the sine of its opposite angle is consistent across all sides of the triangle. The formula is given by:

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

This relationship helps determine unknown sides or angles within triangles when certain angles and one side are known.

In the context of the fire lookout problem, knowing certain bearings allowed us to use the Law of Sines to calculate the distance from one station to the fire.

Applying the Law of Cosines

The Law of Cosines is similar to the Pythagorean theorem but applicable in any triangle, not just right triangles. It helps in finding a side's length or an angle when you know two sides and the included angle. The formula looks like:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

This relation shows how the sides and angles interact, offering insights into the triangle's shape.

In our problem, after finding one side using the Law of Sines, the Law of Cosines was employed to establish the remaining distance from the other station to the fire, using the side lengths and angles we already knew.

Understanding Triangle Properties

A triangle is a polygon with three edges and three vertices. The sums of the angles in any triangle is always 180 degrees. By understanding triangles, you know how its sides and angles relate.

In the fire lookout scenario, recognizing how these properties connect allowed for calculating the necessary distances. The known angle at station B was 90 degrees (as it is directly east of station A), which facilitated using trigonometric laws efficiently to reach the solution.

Calculating Distances Using Trigonometry

Finding distances when certain angles and at least one side are known involves trigonometry, particularly using the Law of Sines and Law of Cosines. By assessing bearings and sketching triangles:

• We place known distances and angles within a trigonometric framework.
• Utilize these laws, translating them into practical distance measurements as seen between the fire and lookout stations.

This systematic approach not only solves the immediate problem but also builds analytical skills for situations where visual distance measurement is impossible.