Problem 38 - Precalculus Mathematics for Calculus

Question

Radio Antenna A short-wave radio antenna is supported by two guy wires, \(165 \mathrm{ft}\) and \(180 \mathrm{ft}\) long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?

Step-by-Step Solution

Answer

The anchor points are approximately 224 feet apart.

Step 1: Understand the Problem

We have a tower supported by two wires. One wire is 165 ft long at 67° to the ground, and the other wire is 180 ft long. We need to find the distance between the two anchor points of the wires.

Step 2: Identify Relevant Concepts

We'll use trigonometry to solve this problem. Specifically, the Law of Cosines will help us find the distance between the two anchor points based on the given angles and lengths of the wires.

Step 3: Calculate the Height of the Antenna

The height of the antenna can be calculated using the formula \( h = 165 \times \sin(67^\circ) \). This is because the wire, the height, and the ground form a right triangle.

Step 4: Calculate the Ground Distance from the Antenna to the First Anchor Point

Using the cosine function, calculate this distance: \( d_1 = 165 \times \cos(67^\circ) \). This gives the horizontal distance from the antenna to the anchor point of the first wire.

Step 5: Use the Law of Cosines to Find the Distance Between Anchor Points

Let the distance between anchor points be denoted as \( D \). We know the angle opposite to this distance is 67°, and sides are 165 ft and 180 ft. Use the Law of Cosines: \[ D^2 = 165^2 + 180^2 - 2 \times 165 \times 180 \times \cos(67^\circ) \]Solve to find \( D \).

Step 6: Solve for the Distance Between Anchor Points

Calculate \( D \) from the equation: \( D = \sqrt{165^2 + 180^2 - 2 \times 165 \times 180 \times \cos(67^\circ)} \). This gives the required distance between anchor points.

Key Concepts

Law of Cosines

The Law of Cosines is a vital tool in trigonometry for solving non-right triangles. It's a generalization of the Pythagorean Theorem, applicable to any triangle, not just right triangles. It helps in finding a side of a triangle when two sides and the included angle are known, or finding an angle if all three sides are known. The formula is given by:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Here, \(c\) is the side opposite angle \(C\), while \(a\) and \(b\) are the other two sides of the triangle. This concept is particularly helpful in real-world applications where exact measurement of angles or sides is required, such as in construction or navigation. In our exercise, the law of cosines allowed us to calculate the distance between the anchor points of the wires by considering the angles and lengths of wires forming a triangle with the ground.

Right Triangle

A right triangle is one of the most common geometric shapes in trigonometry, characterized by having one angle equal to 90 degrees. It consists of two legs and a hypotenuse, the longest side opposite the right angle.
Important properties and theorems apply to right triangles that simplify calculations, such as the Pythagorean theorem and basic trigonometric functions.

• **Pythagorean Theorem:** \(a^2 + b^2 = c^2\)
• **Sine, Cosine, and Tangent:** These functions relate the angles and sides of a right triangle.

These aspects make right triangles a fundamental element in solving many real-life problems as they often fit scenarios where measurements are taken vertically and horizontally, such as in our radio antenna exercise. Here, a right triangle helped in calculating the height of the antenna using one of the wires and the ground.

Angle Calculation

Calculating angles is essential when working in trigonometry, especially when dealing with triangles, as knowing angles allows for finding other unknown components using trigonometric functions or laws.
In trigonometry, angles are often measured in degrees or radians and tools like sine, cosine, and tangent functions help relate these angles to the triangle's sides.
To calculate angles in triangles we can use:

• **Inverse Trigonometric Functions:** For finding angles when sides are known.
• **The Law of Sines and Law of Cosines:** To connect sides and angles.

In our exercise, the angle of 67° between the shorter wire and the ground was crucial. It helped in determining the component distances and finally calculating the overall distance between the anchor points. Understanding how angles affect and interact with dimensions in a triangle is essential for accurate measurements in applications ranging from simple computations to sophisticated engineering designs.