Problem 42 - Contemporary Precalculus

Question

Two planes at the same altitude approach an airport. One plane is 16 miles from the control tower and the other is 22 miles from the tower. The angle determined by the planes and the tower, with the tower as vertex, is \(11^{\circ} .\) How far apart are the planes?

Step-by-Step Solution

Answer

Question: Determine the distance between two planes A and B if plane A is 16 miles away from a tower, plane B is 22 miles away from the same tower, and the angle between the two planes at the tower is \(11^{\circ}\). Answer: The two planes are approximately 24.81 miles apart.

Step 1: Identify the information given

First, let's identify the necessary information from the word problem: - Side a: distance from plane A to the tower = 16 miles - Side b: distance from plane B to the tower = 22 miles - Angle C: angle at the tower = \(11^{\circ}\) We are to find the distance between the two planes (side c).

Step 2: Use the law of cosines

Next, we will apply the law of cosines to find the length of side c. The law of cosines states that: \(c^2 = a^2 + b^2 - 2ab \cos{C}\) With the given information, we have: \(c^2 = 16^2 + 22^2 - 2 (16)(22) \cos{11^{\circ}}\)

Step 3: Calculate the value of \(c^2\)

To calculate the value of \(c^2\), we will plug in the values and use a calculator: \(c^2 = 16^2 + 22^2 - 2 (16)(22) \cos{11^{\circ}} \approx 615.75\)

Step 4: Find the distance between the planes

Now we will find the distance between the planes (side c) by taking the square root of \(c^2\): \(c = \sqrt{615.75} \approx 24.81 \text{ miles}\) The two planes are about 24.81 miles apart.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics centered around the study of triangles, particularly right triangles, and the relationships between their angles and sides. The law of cosines is a fundamental theorem within trigonometry that extends the concept of the Pythagorean theorem to non-right triangles. It is expressed in the form \(c^2 = a^2 + b^2 - 2ab \cos{C}\), where \(a\) and \(b\) are the lengths of sides of a triangle, \(c\) is the length of the side opposite angle \(C\), and \(\cos{C}\) is the cosine of angle \(C\).

In everyday applications, trigonometry is used in fields such as engineering, physics, and even geography, helping to solve real-world problems involving distances and angles.

Word Problems

Word problems in mathematics are exercises where information is presented in the form of text, requiring the solver to identify and apply the appropriate mathematical principles to find a solution. To effectively tackle word problems, it's advisable to follow a structured approach:

• Read the problem carefully and identify what is being asked.
• List down known quantities and assign symbols or letters to them.
• Translate the problem's textual information into a mathematical equation using the appropriate formulas or theorems.
• Solve the equation and check if the solution makes sense in the context of the problem.

This strategy helps disentangle complex information and guides towards a logical resolution.

Geometric Applications

The principles of trigonometry are not just theoretical but have many practical applications in various fields, which involve the concept of geometric applications. For instance, the law of cosines can be used to determine the distance between two points in navigation, surveying, or even in the example of the two planes approaching an airport. Understanding the geometric context—such as recognizing when to use the law of cosines in place of the Pythagorean theorem—is crucial when solving real-world geometric problems. It's this interplay between abstract mathematical theory and practical problem-solving that is at the heart of geometric applications in trigonometry.