Problem 30 - Algebra 2

Question

A pilot typically flies a route from Bloomington to Rockford, covering a distance of 117 miles. In order to avoid a storm, the pilot first flies from Bloomington to Peoria, a distance of 42 miles, then turns the plane and flies 108 miles on to Rockford. Through what angle did the pilot turn the plane over Peoria?

Step-by-Step Solution

Answer

The pilot turned the plane at an angle of approximately 165.7 degrees.

Step 1: Understand the Triangle

First, visualize the triangle formed by the three cities: Bloomington, Peoria, and Rockford. The sides of this triangle represent the flights between these cities. Bloomington to Peoria is 42 miles, Peoria to Rockford is 108 miles, and Bloomington to Rockford is 117 miles.

Step 2: Apply the Cosine Rule

Since we need to find the angle at Peoria, we'll use the cosine rule to relate the sides of the triangle and the angle. The cosine rule states: \[ c^2 = a^2 + b^2 - 2ab \cos C \]Where \(C\) is the angle at Peoria, \(a\) and \(b\) are the sides adjacent to \(C\), and \(c\) is the side opposite \(C\). In our case, set \(a = 42\), \(b = 108\), and \(c = 117\).

Step 3: Substitute and Solve for Cosine

Substitute the known values into the cosine rule equation:\[ 117^2 = 42^2 + 108^2 - 2 \times 42 \times 108 \times \cos(C) \]Calculate \[ 13689 = 1764 + 11664 - 2 \times 42 \times 108 \times \cos(C) \].

Step 4: Simplify the Equation

Compute the squares and simplify the equation:\[ 13689 = 13428 - 9072 \cos(C) \]Re-arrange to find \( \cos(C) \):\[ 261 = -9072 \cos(C) \].

Step 5: Calculate the Cosine of the Angle

Divide both sides by -9072 to solve for \( \cos(C) \):\[ \cos(C) = \frac{261}{-9072} \]Simplify to determine the cosine value.

Step 6: Find the Angle

Calculate the angle \(C\) using the inverse cosine function:\[ C = \cos^{-1}\left( \frac{261}{-9072} \right) \]Use a calculator to find the angle in degrees. Ensure your calculator is set to degree mode.

Key Concepts

Triangle Geometry

If you can picture a triangle in your mind, you already have a head start on solving many geometrical problems. In the case of the problem with the pilot, triangle geometry helps describe the routes flown between the cities—Bloomington, Peoria, and Rockford. These represent the vertices of the triangle, and the sides are the distances between these cities.

• Bloomington to Peoria: 42 miles
• Peoria to Rockford: 108 miles
• Bloomington to Rockford: 117 miles

Understanding triangle geometry is crucial because it helps break down the flight paths into a relatable shape. Triangles in geometry are determined by their sides and angles. In this exercise, knowing the lengths of all sides, you can find the unknown angle using trigonometric rules. By visualizing and recognizing these elements as a geometric shape, you can apply mathematical rules more easily to solve problems.

Angle Calculation

When dealing with triangles in geometry, calculating angles is often a critical task. In our specific exercise, we needed to calculate the angle at Peoria to understand how the pilot adjusted their flight path. The angle between the flight to Peoria and the continuing path to Rockford reveals how much turn the pilot needs.

To calculate this angle, we use a reliable mathematical tool known as the Cosine Rule. This rule assists with finding unknown angles when you know the lengths of all sides of a triangle.

• First, identify the sides adjacent to the target angle: these are from Bloomington to Peoria (42 miles) and from Peoria to Rockford (108 miles).
• The side opposite to the target angle stretches from Bloomington to Rockford (117 miles).
• Apply the cosine rule formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

By rearranging and solving for \( \cos(C) \), you can find the measure of the angle itself by taking the inverse cosine with your calculator set to degrees. This allows you to convert the cosine value into an actual angle measurement.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. This becomes particularly useful when solving real-world problems such as determining a flight path angle, as in this exercise. By using trigonometric identities and rules, you can calculate unknown quantities in triangles effectively.

In our scenario, the cosine rule is the star of the show. This trigonometric rule derives from the fundamental understanding of the Pythagorean theorem but is extended to work for any triangle, not just right-angled ones. Let's re-cap the cosine rule used here: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] This equation allows you to find the value of cosine for the angle Peoria given all side lengths.

• Upon calculating \( \cos(C) \), the inverse operation, \( \cos^{-1} \), gives the angle measure directly.
• Trigonometric functions such as sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each have their own inverse functions used to derive angles from given ratios.
• Using these tools is essential for translating between side lengths and angle measures in triangle geometry.

The beauty of trigonometry lies in its ability to provide exact results for calculations involving angles and distances, crucial for navigation and many practical engineering tasks.