Problem 90 - Precalculus

Question

In calculus, some applications of the derivative require the solution of triangles. Solve each triangle using the Law of Cosines. An airport radar detects two planes approaching. The distance between the planes is 80 miles; the closest plane is 60 miles from the airport and the other plane is 70 miles from the airport. What is the angle (in degrees) formed by the planes and the airport?

Step-by-Step Solution

Answer

The angle formed by the planes and the airport is approximately 75.52 degrees.

Step 1: Understanding the Problem

We are given a triangle formed by two planes and an airport. The side lengths are 60 miles, 70 miles, and 80 miles. We need to find the angle opposite the 80-mile side using the Law of Cosines.

Step 2: Write the Law of Cosines Formula

The Law of Cosines is given by the formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] where \(c\) is the side opposite the angle \(C\). In this problem, \(c = 80\), \(a = 60\), and \(b = 70\).

Step 3: Substitute the Known Values

Substitute the known side lengths into the formula: \[ 80^2 = 60^2 + 70^2 - 2 \times 60 \times 70 \times \cos(C) \] Calculate the squares and product: \[ 6400 = 3600 + 4900 - 2 \times 60 \times 70 \times \cos(C) \]

Step 4: Simplify the Equation

Combine like terms: \[ 6400 = 8500 - 8400 \times \cos(C) \] Rearrange the equation to solve for \(\cos(C)\): \[ \cos(C) = \frac{8500 - 6400}{8400} \]

Step 5: Solve for \(\cos(C)\)

Calculate the value: \[ \cos(C) = \frac{2100}{8400} = \frac{1}{4} \]

Step 6: Calculate Angle \(C\)

Use the inverse cosine to find the angle \(C\): \[ C = \cos^{-1}\left(\frac{1}{4}\right) \] Use a calculator to find that \(C \approx 75.52\) degrees.

Key Concepts

Triangle Solution using the Law of Cosines

When solving triangles, especially in cases where sides are known but not angles, the Law of Cosines becomes a powerful tool. This law allows you to find an unknown angle when all three sides of a triangle are known. In our exercise, imagine the triangle as being made up of two planes approaching an airport. The planes are separated by a distance of 80 miles, while one plane is 60 miles away from, and the other is 70 miles away from the airport.

The Law of Cosines helps us solve this triangle by relating the lengths of the triangle's sides to the cosine of one of its angles. The formula is:

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

In this setup, the angle we are looking for is opposite the longest side, which is 80 miles. Using the formula, we substitute our known values to find the angle opposite this side. Solving for the cosine of the angle first makes it easier to compute, and eventually, you find that the angle is approximately 75.52 degrees.

By using these steps, you break down the task into manageable parts, making it easier to understand and apply the concept to other similar problems.

Derivatives in Calculus

While this problem is primarily about geometry, derivatives in calculus are a fundamental aspect of understanding how quantities change. When calculating derivatives, you are essentially looking at how a function changes at any point. However, there is a connection to our problem, as both involve understanding relationships and changes in measurement.

Though the exercise didn't directly utilize derivatives, a profound understanding of calculus allows you to extend this approach to related problems, like optimizing a solution or calculating rates of change. If you imagine each side of the triangle changing over time, calculus provides methods to find how this affects other properties like angles or area in real-time.

Understanding derivatives and their applications, even in problems that don't directly involve them, can enrich your perspective on solving complex, real-world problems. This background in calculus provides a comprehensive toolkit for tackling dynamic systems whether in physics, engineering, or even advanced geometry.

Angle Calculation

Angle calculation using trigonometry revolves around understanding the relationships within a triangle through sides and angles. Once the cosine of an angle in a triangle is known, finding the angle is straightforward with the use of inverse trigonometric functions.

In our exercise, after determining the cosine value using the Law of Cosines, the next step was finding the actual angle measure. This is done using the inverse cosine function, represented as

• \[C = \cos^{-1}\left(\frac{1}{4}\right)\]

The inverse function allows us to work backwards from the calculated cosine ratio to determine the angle itself.

Calculators often assist in this process by providing a direct input method, but understanding the mathematical reasoning behind it helps reinforce why the calculation leads to the result of approximately 75.52 degrees. With each step understood, finding and verifying angles in various contexts becomes an accessible and insightful process, whether for academics or practical applications.