Problem 23 - Algebra 2 and Trigonometry

Question

A pole is braced by two wires that extend from the top of the pole to the ground. The lengths of the wires are 16 feet and 18 feet and the measure of the angle between the wires is \(110^{\circ} .\) Find, to the nearest foot, the distance between the points at which the wires are fastened to the ground.

Step-by-Step Solution

Answer

The distance between the points is 29 feet.

Step 1: Identify the Triangle

The problem involves finding the distance between two points on the ground which can be represented as the base of a triangle. The lengths of the wires are the two sides of the triangle, and the angle between them is given as \(110^{\circ} \).

Step 2: Apply the Law of Cosines

To find the distance between the points, apply the Law of Cosines which is \(c^2 = a^2 + b^2 - 2ab\cos(C)\), where \(a\) and \(b\) are the lengths of the wires, \(C\) is the angle between them, and \(c\) is the distance we need to find.

Step 3: Plug in the Values

Substitute \(a = 16\) feet, \(b = 18\) feet, and \(C = 110^{\circ}\) into the Law of Cosines: \[ c^2 = 16^2 + 18^2 - 2 \times 16 \times 18 \times \cos(110^{\circ}). \]

Step 4: Calculate the Cosine Value

Calculate \(- \cos(110^{\circ})\). Since \(\cos(110^{\circ}) \approx -0.342\), this becomes \(+ 0.342\) in the equation: \[ c^2 = 256 + 324 + 246.816. \]

Step 5: Simplify the Expression

Add the terms together: \[ c^2 = 256 + 324 + 246.816 = 826.816 \]

Step 6: Solve for the Distance

Take the square root of both sides to find \(c\): \[ c = \sqrt{826.816} \approx 28.76 \] feet.

Step 7: Round the Results

Round 28.76 feet to the nearest foot, which gives you \(29\) feet.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is fundamental in solving problems involving triangles, especially when the triangle is not a right triangle. The core functions of trigonometry are sine, cosine, and tangent, which are used to derive ratios of the sides of triangles based on their internal angles.
In this particular problem, the cosine function is key. It's especially useful when dealing with the Law of Cosines, which helps in finding unknown lengths or angles in any type of triangle. Understanding how trigonometric functions connect angles with side lengths is crucial for applying mathematical principles to real-world scenarios, such as determining distances in triangle geometry.

Triangle Geometry

Triangles are one of the simplest yet most fundamental shapes in geometry. There are various types of triangles, each defined by its angles and side lengths. In this problem, the focus is on a non-right triangle.
Key features of triangle geometry that we rely on here include:

• Side lengths
• Interior angles
• The relationship between angles and sides

When you understand these features, solving for unknown values in a triangle becomes more intuitive. The problem at hand involves a triangle formed by two wires and the ground, where the lengths of the wires and the angle between them are known, allowing us to apply triangle geometry principles along with trigonometric rules to solve for the unknown base of the triangle.

Applying Cosine Rule

The Law of Cosines, also known as the Cosine Rule, is a crucial tool in triangle geometry, especially in non-right triangles. It can be written mathematically as:\[ c^2 = a^2 + b^2 - 2ab\cos(C) \]Where:

• \(a\) and \(b\) are the lengths of two sides of the triangle
• \(C\) is the angle between them
• \(c\) is the side opposite of angle \(C\)

In our problem, this formula helps find the distance between two points on the ground. By plugging the given values into the cosine rule, we can calculate the base of the triangle, providing a practical application of trigonometric concepts. Remember that after calculating \(c^2\) using the Law of Cosines, taking the square root will yield the distance, which is then rounded to the nearest foot for a clear and concise solution.